With answers from 68 AI models.
ข้อมูลส่วนสูง (เซนติเมตร) ของนักกีฬาจำนวน 11 คนโดยมีค่าเฉลี่ยเลขคณิตเท่ากับ ค่ามัธยฐานแสดงได้ดังนี้ 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180, พิจารณาข้อความต่อไปนี้ ก) พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ 4 ข) ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11) ค) ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน จากข้อความข้างต้นข้อใดถูกต้อง
Using dataset from OpenThaiGPT Evaluation Dataset.
Let me analyze each statement:
For IQR (Interquartile Range): Q1 = 175 Q3 = 179 IQR = Q3 - Q1 = 179 - 175 = 4 Statement ก) is correct.
For Standard Deviation: Mean = 177 Sum of squared deviations = (173-177)² + 3(175-177)² + (176-177)² + (177-177)² + (178-177)² + 2(179-177)² + 2(180-177)² = 56 Standard deviation = sqrt(56/11) Statement ข) is correct.
For Variance: Variance = 56/11 Standard deviation = sqrt(56/11) Variance ≠ 2 × Standard deviation Statement ค) is incorrect.
Since statements ก) and ข) are correct, and ค) is incorrect:
{"correct_answer_key":"c"}
Let me analyze this statistics problem about the heights of 11 athletes:
Given data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
For statement ก): The interquartile range (IQR) Q1 = 175 (3rd value) Q3 = 179 (9th value) IQR = Q3 - Q1 = 179 - 175 = 4 So statement ก) is correct.
For statement ข): The standard deviation Mean = 177 Sum of squared deviations = (173-177)² + 3(175-177)² + (176-177)² + (177-177)² + (178-177)² + 2(179-177)² + 2(180-177)² = 56 Standard deviation = √(56/11) ≈ 2.26 So statement ข) is correct.
For statement ค): The variance equals the standard deviation squared, not twice the standard deviation. Variance = (√(56/11))² = 56/11 This is not equal to 2×√(56/11) So statement ค) is incorrect.
Therefore, statements ก) and ข) are correct, but ค) is incorrect.
{"correct_answer_key":"c"}
This is a statistics problem in Thai. Let me first translate the question:
The question asks about the heights (in centimeters) of 11 athletes. The heights are given as: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
The mean equals the median for this data set.
We are asked to consider three statements and determine which statement(s) are correct: a) The interquartile range (IQR) of the heights is 4. b) The standard deviation of the heights is equal to sqrt(56/11). c) The variance of the heights is equal to twice the standard deviation.
Now let's evaluate each statement:
Statement a): The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). For the given data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 The 11 data points ordered, with positions: Position: 1 2 3 4 5 6 7 8 9 10 11 Value: 173 175 175 175 176 177 178 179 179 180 180
Q1 corresponds to the (n+1)/4 = (11+1)/4 = 3rd position, which is 175. Q3 corresponds to the 3(n+1)/4 = 3(11+1)/4 = 9th position, which is 179. So, IQR = Q3 - Q1 = 179 - 175 = 4. Statement a) is correct.
Statement b): We need to calculate the standard deviation. First, let's find the mean (which, according to the problem, equals the median). The median for 11 data points is the 6th value, which is 177. So, the mean is also 177.
Now, let's calculate the variance: Variance = Σ(x - μ)² / n, where μ is the mean.
(173 - 177)² + (175 - 177)² + (175 - 177)² + (175 - 177)² + (176 - 177)² + (177 - 177)² + (178 - 177)² + (179 - 177)² + (179 - 177)² + (180 - 177)² + (180 - 177)²
= 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9 = 56
Variance = 56 / 11 = 56/11
Standard deviation = sqrt(Variance) = sqrt(56/11)
Statement b) is correct.
Statement c): The variance is equal to twice the standard deviation. From our calculations above, the variance is 56/11 and the standard deviation is sqrt(56/11). For statement c) to be true, the relationship should be: Variance = 2 * Standard Deviation
56/11 = 2 * sqrt(56/11)
Squaring both sides: (56/11)² = (2 * sqrt(56/11))² (56/11)² = 4 * (56/11) 56/11 = 4 * sqrt(56/11) * sqrt(56/11) 56/11 = 4 * 56/11 1 = 4 This is false. Therefore, statement c) is incorrect.
Actually, I made a mistake. Let's try again. For statement c) to be true: Variance = 2 * Standard Deviation But the standard deviation is sqrt(Variance). For consistency, we would expect that: Variance = 2 * sqrt(Variance) Dividing both sides by 2: Variance/2 = sqrt(Variance) Squaring both sides: (Variance/2)² = Variance Variance²/4 = Variance Variance = 4 (for the equation to be satisfied)
But our calculated variance is 56/11, not 4. Therefore, statement c) is incorrect.
Actually, I need to be a bit more careful to check if this is right. Let's reconsider. The claim is:
Variance = 2 * Standard Deviation
We know that Standard Deviation = sqrt(Variance). Substituting, we get:
Variance = 2 * sqrt(Variance)
For this to be true, we'd need to solve for Variance:
Variance = 2 * sqrt(Variance) Variance / 2 = sqrt(Variance) (Variance / 2)² = Variance Variance² / 4 = Variance Variance² = 4 * Variance Variance² - 4 * Variance = 0 Variance * (Variance - 4) = 0
This means either Variance = 0 or Variance = 4. Since we're discussing actual data, the variance isn't 0, so it would have to be 4 for statement c) to be true.
However, we calculated the variance to be 56/11, which is not equal to 4. Therefore, statement c) is incorrect.
so the correct answer is "c) ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง" which translates to "Only statements a) and b) are correct."
Actually, let me double-check whether variance is indeed 56/11.
For the data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 The mean (and median) is 177.
Let's calculate Σ(x - μ)²: (173 - 177)² = (-4)² = 16 (175 - 177)² = (-2)² = 4 (175 - 177)² = (-2)² = 4 (175 - 177)² = (-2)² = 4 (176 - 177)² = (-1)² = 1 (177 - 177)² = 0 (178 - 177)² = 1 (179 - 177)² = 2² = 4 (179 - 177)² = 2² = 4 (180 - 177)² = 3² = 9 (180 - 177)² = 3² = 9
Sum = 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9 = 56
Variance = 56 / 11
Now, let's find the mean more directly to double-check: Mean = (173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180) / 11 = 1947 / 11 = 177
So the variance is indeed 56/11, which means statement c) is incorrect.
In conclusion, statements a) and b) are correct, but statement c) is incorrect.
Therefore, the answer is: "c) ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง", which translates to "Only statements a) and b) are correct."
Actually, one more check. Let me compute the interquartile range again. With 11 data points, the positions for the quartiles are: Q1: position (11+1)/4 = 3rd position (this is the 3rd value in the ordered data). Q3: position 3(11+1)/4 = 9th position (this is the 9th value in the ordered data).
From the ordered data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 Q1 is the 3rd value, which is 175. Q3 is the 9th value, which is 179. IQR = Q3 - Q1 = 179 - 175 = 4.
So, statement a) is correct.
And statement b) is also correct as we've verified.
Thus, the final answer is: "c) ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง", which translates to "Only statements a) and b) are correct."
I need to analyze each statement about the heights data (173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180).
For statement ก): The first quartile Q1 is at position (11+1)/4 = 3rd value = 175 The third quartile Q3 is at position 3(11+1)/4 = 9th value = 179 Therefore, IQR = Q3-Q1 = 179-175 = 4, which is correct.
For statement ข): The mean equals the median, which is 177. Calculating the variance: Σ(x-177)²/11 = 56/11 So the standard deviation = √(56/11), which is correct.
For statement ค): The variance is 56/11, while the standard deviation is √(56/11). For this statement to be true: variance = 2 × standard deviation But 56/11 ≠ 2 × √(56/11), so this statement is false.
Only statements ก) and ข) are correct.
{"correct_answer_key":"c"}
Let me analyze each statement about the height data of 11 athletes:
Data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
Statement ก) Interquartile Range (IQR) = 4
Statement ข) Standard deviation = √(56/11)
Statement ค) Variance = 2 × Standard deviation
Statements ก) and ข) are correct, but ค) is incorrect.
{"correct_answer_key": "c"}
Let me analyze this Thai statistics problem step by step.
The data shows heights (in cm) of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
The problem states that the arithmetic mean equals the median.
First, let me verify this:
Yes, mean = median = 177
Now let's check each statement:
Statement ก) The interquartile range equals 4
For 11 values:
Statement ก) is CORRECT.
Statement ข) The standard deviation equals sqrt(56/11)
To calculate standard deviation:
Mean = 177
Deviations from mean: -4, -2, -2, -2, -1, 0, 1, 2, 2, 3, 3
Squared deviations: 16, 4, 4, 4, 1, 0, 1, 4, 4, 9, 9
Sum of squared deviations = 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9 = 56
Variance = 56/11
Standard deviation = sqrt(56/11)
I'll investigate the relationship between variance and standard deviation. The calculation involves comparing 56/11 with 2 × sqrt(56/11). I need to determine if these values are equivalent to verify the final statement's accuracy.
By solving the equation x² = 2x, I find x can be 0 or 2. However, sqrt(56/11) is approximately 2.26, which doesn't match these values. This means the statement claiming variance equals twice the standard deviation is false.
The final conclusion is that only statements ก) and ข) are correct, leading to the answer choice c).
Looking at this Thai statistics problem about the heights of 11 athletes with data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
Let me verify each statement:
Statement ก) Interquartile range = 4
Statement ข) Standard deviation = √(56/11)
Statement ค) Variance = 2 × Standard deviation
Statements ก) and ข) are correct, while ค) is incorrect.
{"correct_answer_key": "c"}
ให้ฉันตรวจสอบแต่ละข้อความ:
ข้อมูล: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 (n=11)
ก) พิสัยระหว่างควอร์ไทล์ (IQR)
ข) ส่วนเบี่ยงเบนมาตรฐาน
ค) ความสัมพันธ์ระหว่างความแปรปรวนและส่วนเบี่ยงเบนมาตรฐาน
ดังนั้น ข้อความ ก) และ ข) ถูกต้อง แต่ ค) ไม่ถูกต้อง
{"correct_answer_key": "c"}
Let me analyze this Thai statistics problem. The data shows heights (in cm) of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
The problem states that the mean equals the median.
First, let me verify this:
Now let's check each statement:
Statement ก) The interquartile range (IQR) of the heights equals 4
For 11 values:
Statement ข) The standard deviation equals sqrt(56/11)
To calculate standard deviation:
Statement ค) The variance equals twice the standard deviation
I investigate the mathematical relationship between variance and standard deviation. The calculation reveals a discrepancy in the proposed statement. By comparing the actual values, I find that the variance does not equal twice the standard deviation. The mathematical proof shows the statement is incorrect, with 56/11 not matching the expected relationship. This invalidates the third statement while confirming the accuracy of the first two statements about the dataset.
Let me analyze this statistics problem about the heights of 11 athletes.
Given data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
First, I'll verify that mean = median:
Now checking each statement:
Statement ก) Interquartile range = 4
Statement ข) Standard deviation = √(56/11)
Statement ค) Variance = 2 × standard deviation
Only statements ก) and ข) are correct.
{"correct_answer_key":"c"}
I need to analyze each statement about the height data of 11 athletes.
Given data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
First, let me verify the mean equals the median:
Now checking each statement:
Statement ก) Interquartile range = 4
Statement ข) Standard deviation = √(56/11)
Statement ค) Variance equals twice the standard deviation
Statements ก) and ข) are correct, but ค) is incorrect.
{"correct_answer_key":"c"}
Let me analyze this Thai statistics problem step by step.
The data given is heights (in cm) of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
The problem states that the arithmetic mean equals the median.
First, let me find the median. Since there are 11 values, the median is the 6th value when arranged in order: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 The median = 177
Now let me calculate the mean: Sum = 173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180 = 1947 Mean = 1947/11 = 177
Good, the mean equals the median as stated.
Now let me check each statement:
ก) The interquartile range (IQR) equals 4 For 11 data points, Q1 is at position (11+1)/4 = 3, so Q1 = 175 Q3 is at position 3(11+1)/4 = 9, so Q3 = 179 IQR = Q3 - Q1 = 179 - 175 = 4 So statement ก) is correct.
ข) The standard deviation equals sqrt(56/11) Let me calculate the variance first: Mean = 177
Deviations from mean: 173-177 = -4, squared = 16 175-177 = -2, squared = 4 (appears 3 times) 176-177 = -1, squared = 1 177-177 = 0, squared = 0 178-177 = 1, squared = 1
179-177 = 2, squared = 4 (appears 2 times) 180-177 = 3, squared = 9 (appears 2 times)
I calculate the sum of squared deviations systematically. The total comes to 56, with variance being 56/11. This confirms the standard deviation as sqrt(56/11), validating the second statement's accuracy.
The squared deviations reveal a precise pattern of dispersion around the mean, demonstrating the dataset's symmetrical distribution.
I verify the third statement by checking if variance equals twice the standard deviation. By algebraically manipulating the equation, I discover this claim is false. The mathematical proof shows 56/11 does not equal 4, invalidating the third statement.
Consequently, only the first two statements are correct, leading to option c) as the appropriate answer.
Looking at the height data of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
Let me check each statement:
Statement ก) Interquartile range = 4
Statement ข) Standard deviation = √(56/11)
Statement ค) Variance = 2 × Standard deviation
Statements ก) and ข) are correct, but ค) is incorrect.
{"correct_answer_key":"c"}
To determine the correct answer, let's analyze each statement step by step:
ค่ามัธยฐาน (Median):
ควอร์ไทล์ (Quartiles):
ส่วนเบี่ยงเบนมาตรฐาน (Standard Deviation):
ค่าความแปรปรวน (Variance) vs. ส่วนเบี่ยงเบนมาตรฐาน:
Correct Answer:
Both statements ก) and ข) are correct.
{"correct_answer_key":"c"}
We are given the heights (in cm) of 11 athletes, sorted in ascending order:
173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
The mean (average) is equal to the median, which is the 6th value: 177 cm.
Now, let's evaluate each statement:
First, compute the variance:
{"correct_answer_key":"c"}
Let's analyze each statement step by step with the given data:
Data (sorted): 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 Number of data points (n) = 11
Statement ก: พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ 4 Interquartile Range (IQR) = Q3 - Q1 Since n=11 (odd), the median is the 6th value (177). Q1 is the median of the lower half (first 5 numbers): 173, 175, 175, 175, 176 → median is the 3rd value = 175. Q3 is the median of the upper half (last 5 numbers): 178, 179, 179, 180, 180 → median is the 3rd value = 179. So IQR = 179 - 175 = 4. This matches statement ก.
Statement ข: ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11) First, compute the mean (μ): Sum = 173+175+175+175+176+177+178+179+179+180+180 = 1947 μ = 1947 / 11 = 177 Now compute variance (σ²) = [Σ(xi - μ)²] / n Deviations from mean: 173-177 = -4 → squared=16 175-177=-2 → squared=4 (three times: 4*3=12) 176-177=-1 → squared=1 177-177=0 → squared=0 178-177=1 → squared=1 179-177=2 → squared=4 (two times: 8) 180-177=3 → squared=9 (two times: 18) Sum of squared deviations = 16 + 12 + 1 + 0 + 1 + 8 + 18 = 56 So variance σ² = 56/11 Standard deviation σ = sqrt(56/11). This matches statement ข.
Statement ค: ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน Variance = σ², standard deviation = σ. So "variance equals twice the standard deviation" would mean σ² = 2σ. This is not true in general; it would only be true if σ=2 (since σ²=2σ implies σ(σ-2)=0, so σ=0 or σ=2). But here σ = sqrt(56/11) ≈ sqrt(5.09) ≈ 2.258, which is not 2. So σ² ≠ 2σ. Therefore, statement ค is false.
Thus, only statements ก and ข are correct.
{"correct_answer_key": "c"}
Statement ก (Interquartile Range): The interquartile range (IQR) is calculated as Q3 - Q1. For the given data (173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180), with 11 data points:
Statement ข (Standard Deviation): The standard deviation is given as (\sqrt{56/11}). The mean height is 177. The sum of squared differences from the mean is:
Statement ค (Variance vs. Standard Deviation): Variance is (\sigma^2) and standard deviation is (\sigma). The statement claims variance equals twice the standard deviation ((\sigma^2 = 2\sigma)). However, (\sigma^2 = 2\sigma) implies (\sigma(\sigma - 2) = 0), so (\sigma = 0) or (\sigma = 2). Here, (\sigma = \sqrt{56/11} \approx 2.256), so (\sigma^2 = 56/11 \approx 5.0909) and (2\sigma \approx 4.512), which are not equal. Thus, statement ค is incorrect.
Conclusion: Only statements ก and ข are correct, corresponding to option c.
{
"correct_answer_key": "c"
}
First, I need to analyze the given data and the statements. The data is the heights of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. The arithmetic mean is equal to the median. I should verify that.
Let me find the median. Since there are 11 data points, the median is the 6th value when sorted. The data is already sorted: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. So the 6th value is 177. So median is 177.
Now, the mean: sum all values and divide by 11.
Sum = 173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180
Let me calculate step by step:
173 + 175 = 348
348 + 175 = 523
523 + 175 = 698
698 + 176 = 874
874 + 177 = 1051
1051 + 178 = 1229
1229 + 179 = 1408
1408 + 179 = 1587
1587 + 180 = 1767
1767 + 180 = 1947
So sum is 1947.
Number of data points: 11
Mean = 1947 / 11 = 177? Let me calculate: 11 * 177 = 1947? 10177=1770, 1177=177, total 1770+177=1947. Yes, mean is 177. So mean = median = 177. That matches the given.
Now, statement ก: พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเทากับ 4
พิสัยระหว่างควอร์ไทล์ (interquartile range, IQR) is Q3 - Q1.
I need to find Q1 and Q3 for this data.
With 11 data points, Q1 is the median of the first half, and Q3 is the median of the second half.
First, find the positions.
For Q1: since n=11, Q1 is at position (n+1)/4 = 12/4 = 3. So the 3rd data point.
Data points: 1:173, 2:175, 3:175, 4:175, 5:176, 6:177, 7:178, 8:179, 9:179, 10:180, 11:180
So Q1 is the 3rd value: 175
For Q3: position 3*(n+1)/4 = 3*12/4 = 36/4 = 9. So the 9th data point: 179
So IQR = Q3 - Q1 = 179 - 175 = 4
So statement ก is correct.
Now statement ข: ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11)
ส่วนเบี่ยงเบนมาตรฐาน is standard deviation.
First, find variance. Variance is the average of squared differences from the mean.
Mean = 177
Calculate squared differences for each point:
173: (173-177)^2 = (-4)^2 = 16
175: (175-177)^2 = (-2)^2 = 4
175: same, 4
175: same, 4
176: (176-177)^2 = (-1)^2 = 1
177: (177-177)^2 = 0^2 = 0
178: (178-177)^2 = 1^2 = 1
179: (179-177)^2 = 2^2 = 4
179: same, 4
180: (180-177)^2 = 3^2 = 9
180: same, 9
Now sum of squared differences: 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9
Calculate: 16 + 4 = 20, +4=24, +4=28, +1=29, +0=29, +1=30, +4=34, +4=38, +9=47, +9=56
So sum of squares = 56
Now, variance: for population variance, since it's a group of athletes, probably treated as population, but sometimes sample. The statement says "ของนักกีฬากลุ่มนี้" which might imply it's the whole group, so population variance.
But let's see the denominator. If population variance, it would be sum of squares / N = 56 / 11
Then standard deviation = sqrt(56/11)
But statement ข says sqrt(56/11), which matches population standard deviation.
If it were sample, it would be sqrt(56/10) for unbiased estimate, but here it says sqrt(56/11), so probably population.
Also, in the context, since it's a specific group, likely population.
So statement ข seems correct.
But let me confirm: standard deviation = sqrt( variance ) = sqrt(56/11)
Yes.
Now statement ค: ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน
ความแปรปรวน is variance, which is sigma^2
ส่วนเบี่ยงเบนมาตรฐาน is sigma
So variance = sigma^2, and two times standard deviation is 2 sigma
So is sigma^2 = 2 sigma?
That would imply sigma^2 - 2 sigma = 0, so sigma(sigma - 2) = 0, so sigma=0 or sigma=2.
But from data, standard deviation is not zero, and let's see what sigma is.
Sigma = sqrt(56/11) ≈ sqrt(5.0909) ≈ 2.256, which is not 2.
So variance = 56/11 ≈ 5.0909, and 2sigma ≈ 22.256=4.512, which is not equal to 5.0909.
So clearly, variance is not equal to two times standard deviation.
Therefore, statement ค is false.
So only statements ก and ข are correct.
Now looking at the options:
a) only ก correct
b) only ข correct
c) ก and ข correct
d) ข and ค correct
e) all correct
So c) ก and ข correct.
But let me double-check if standard deviation is indeed population.
In the statement ข, it says sqrt(56/11), which is population SD.
If it were sample, it would be sqrt(56/10) = sqrt(5.6) which is about 2.366, but since it's given as sqrt(56/11), it must be population.
Also, in the question, it says "นักกีฬากลุ่มนี้" implying the entire group, so population variance is appropriate.
Therefore, ข is correct.
So answer should be c.
But let me confirm IQR again.
Data: 11 points.
Q1: median of first 5 points? Sometimes methods vary.
I used the position method: (n+1)/4 for Q1.
For n=11, (11+1)/4=12/4=3, so third point: 175
Q3: 3*(11+1)/4=36/4=9, ninth point: 179
IQR=4
Alternatively, if we use another method, like splitting into halves.
Lower half: first 5 points: 173,175,175,175,176 → median is 175, so Q1=175
Upper half: last 5 points: 178,179,179,180,180 → median is 179, so Q3=179
IQR=4
Same.
So ก is correct.
Variance: sum of sq differences is 56, N=11, variance=56/11, SD=sqrt(56/11)
Not equal to 2*SD.
So only ก and ข are correct.
Thus, the answer is option c.
ข้อความ ก) ถูกต้องเนื่องจากพิสัยระหว่างควอร์ไทล์ (IQR) คำนวณจาก Q3 - Q1 = 179 - 175 = 4
ข้อความ ข) ถูกต้องเนื่องจากส่วนเบี่ยงเบนมาตรฐานของประชากรคือ sqrt(ความแปรปรวน) = sqrt(56/11)
ข้อความ ค) ไม่ถูกต้องเนื่องจากความแปรปรวน (56/11) ไม่เท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน (2 * sqrt(56/11))
{"correct_answer_key":"c"}
This question is in Thai and asks about the height data of 11 athletes. Let's translate and analyze:
Question: The height data (in centimeters) of 11 athletes, with the arithmetic mean equal to the median, is shown below: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 Consider the following statements: a) The interquartile range of the athletes' heights is equal to 4. b) The standard deviation of the athletes' heights is equal to sqrt(56/11). c) The variance of the athletes' heights is equal to twice the standard deviation. Which of the above statements is correct?
Solution:
Median: The middle value is 177.
Q1 (First Quartile): The median of the lower half (173, 175, 175, 175, 176) is 175.
Q3 (Third Quartile): The median of the upper half (178, 179, 179, 180, 180) is 179.
Interquartile Range (IQR): Q3 - Q1 = 179 - 175 = 4. So, statement (a) is correct.
Variance: We need to calculate the variance. The mean is (sum of all values)/11 = 1922/11 = 174.7272... The variance is the average of the squared differences from the mean. Calculating this gives us approximately 5.15. Since the variance is 56/11, the standard deviation is sqrt(56/11). So, statement (b) is correct.
Variance vs. Standard Deviation: The variance is the square of the standard deviation, not twice the standard deviation. So, statement (c) is incorrect.
Therefore, only statements (a) and (b) are correct.
{"correct_answer_key": "c"}
Let's analyze each statement:
ก) พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ 4: The data is already sorted. Q1 is the median of the lower half (173, 175, 175, 175, 176), which is 175. Q3 is the median of the upper half (178, 179, 179, 180, 180), which is 179. The interquartile range (IQR) is Q3 - Q1 = 179 - 175 = 4. So, statement A is correct.
ข) ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11): The mean is given as the median, which is 177. Let's calculate the variance first. The variance is the average of the squared differences from the mean. The squared differences are: (173-177)^2 = 16, (175-177)^2 = 4, (175-177)^2 = 4, (175-177)^2 = 4, (176-177)^2 = 1, (177-177)^2 = 0, (178-177)^2 = 1, (179-177)^2 = 4, (179-177)^2 = 4, (180-177)^2 = 9, (180-177)^2 = 9. The sum of these squared differences is 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9 = 56. The variance is 56/11. The standard deviation is the square root of the variance, which is sqrt(56/11). So, statement B is correct.
ค) ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน: The variance is 56/11, and the standard deviation is sqrt(56/11). The variance is NOT twice the standard deviation. Variance = (Standard Deviation)^2. So, statement C is incorrect.
Therefore, only statements A and B are correct.
{"correct_answer_key":"c"}
Let's analyze each statement:
Statement ก) พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ 4
Statement ข) ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11)
Statement ค) ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน
Therefore, statements ก) and ข) are correct.
{"correct_answer_key":"c"}
The question asks us to evaluate three statements about a given dataset of 11 athletes' heights (in cm): 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
Let's analyze each statement:
Statement ก) พิสัยระหว่างควอร์ไทล์ (Interquartile Range - IQR) is 4. The data is already sorted: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. There are N = 11 data points.
Statement ข) ส่วนเบี่ยงเบนมาตรฐาน (Standard Deviation - SD) is sqrt(56/11). First, calculate the mean (μ): Sum of data = 173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180 = 1947. Mean (μ) = 1947 / 11 = 177.
Next, calculate the sum of squared differences from the mean (Σ(xi - μ)²): (173-177)² = (-4)² = 16 (175-177)² = (-2)² = 4 (occurs 3 times, so 3 * 4 = 12) (176-177)² = (-1)² = 1 (177-177)² = (0)² = 0 (178-177)² = (1)² = 1 (179-177)² = (2)² = 4 (occurs 2 times, so 2 * 4 = 8) (180-177)² = (3)² = 9 (occurs 2 times, so 2 * 9 = 18) Σ(xi - μ)² = 16 + 12 + 1 + 0 + 1 + 8 + 18 = 56.
Variance (σ²) = Σ(xi - μ)² / N = 56 / 11. Standard Deviation (σ) = sqrt(Variance) = sqrt(56 / 11). Statement ข) is TRUE.
Statement ค) ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน (Variance is equal to twice the Standard Deviation). From the calculations above: Variance (σ²) = 56/11 Standard Deviation (σ) = sqrt(56/11)
The statement claims σ² = 2 * σ. If we let x = σ, then x² = 2x. This implies x² - 2x = 0, or x(x - 2) = 0. So, x = 0 or x = 2. However, σ = sqrt(56/11) ≈ sqrt(5.09) ≈ 2.256. Since 2.256 is not 0 and not 2, the statement is false. Statement ค) is FALSE.
Conclusion: Statements ก) and ข) are true, while statement ค) is false. Therefore, the correct option is the one that states only ก) and ข) are correct.
The final answer is $\boxed{c}$
{"correct_answer_key":"c"}
The question asks us to evaluate three statements (ก, ข, ค) about a given dataset of heights (in cm) of 11 athletes. The dataset is: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. The problem also states that the arithmetic mean equals the median, which is 177 (the 6th value in the sorted list).
Let's evaluate each statement:
Statement ก) พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ 4 (The interquartile range of the heights of this group of athletes is 4).
First, we need to find the first quartile (Q1) and the third quartile (Q3). The data set has n = 11 observations.
Position of Q1 = (n+1)/4 = (11+1)/4 = 12/4 = 3rd position. Q1 = 175 (the 3rd value in the sorted list).
Position of Q3 = 3(n+1)/4 = 3(11+1)/4 = 3(12)/4 = 9th position. Q3 = 179 (the 9th value in the sorted list).
Interquartile Range (IQR) = Q3 - Q1 = 179 - 175 = 4. So, statement ก) is correct.
Statement ข) ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11) (The standard deviation of the heights of this group of athletes is sqrt(56/11)).
To calculate the standard deviation, we first need the variance (s^2). The formula for sample variance is s^2 = Σ(xi - x̄)^2 / (n-1) or for population variance σ^2 = Σ(xi - μ)^2 / n. Given that the problem states "นักกีฬาจำนวน 11 คน" (11 athletes), it implies this is the entire group, so we should use the population standard deviation formula. The mean (μ) is given as 177.
Let's calculate Σ(xi - μ)^2: (173-177)^2 = (-4)^2 = 16 (175-177)^2 = (-2)^2 = 4 (175-177)^2 = (-2)^2 = 4 (175-177)^2 = (-2)^2 = 4 (176-177)^2 = (-1)^2 = 1 (177-177)^2 = (0)^2 = 0 (178-177)^2 = (1)^2 = 1 (179-177)^2 = (2)^2 = 4 (179-177)^2 = (2)^2 = 4 (180-177)^2 = (3)^2 = 9 (180-177)^2 = (3)^2 = 9
Sum of squared differences = 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9 = 56.
Population variance (σ^2) = Σ(xi - μ)^2 / n = 56 / 11. Population standard deviation (σ) = sqrt(56 / 11).
So, statement ข) is correct.
Statement ค) ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน (The variance of the heights of this group of athletes is equal to twice the standard deviation).
From the calculation in statement ข), we found: Variance (σ^2) = 56/11 Standard Deviation (σ) = sqrt(56/11)
We need to check if σ^2 = 2 * σ. 56/11 = 2 * sqrt(56/11)
Let's test this: If 56/11 = 2 * sqrt(56/11), then squaring both sides: (56/11)^2 = (2 * sqrt(56/11))^2 (56/11)^2 = 4 * (56/11)
Divide both sides by (56/11) (since 56/11 is not zero): 56/11 = 4
This is false, as 56/11 is approximately 5.09, which is not equal to 4. So, statement ค) is incorrect.
Based on our analysis: Statement ก) is Correct. Statement ข) is Correct. Statement ค) is Incorrect.
Therefore, the correct option is the one that states both ก) and ข) are correct.
{"correct_answer_key":"c"}
คำอธิบาย: พิจารณาแต่ละข้อความ: ก) พิสัยระหว่างควอร์ไทล์ (IQR) = Q3 - Q1 ข้อมูลมี 11 ตัว (n=11) และเรียงลำดับแล้ว ตำแหน่งของ Q1 = (11+1)/4 = 3 ดังนั้น Q1 คือข้อมูลตัวที่ 3 ซึ่งเท่ากับ 175 ตำแหน่งของ Q3 = 3(11+1)/4 = 9 ดังนั้น Q3 คือข้อมูลตัวที่ 9 ซึ่งเท่ากับ 179 IQR = 179 - 175 = 4 ดังนั้นข้อความ ก) ถูกต้อง
ข) ส่วนเบี่ยงเบนมาตรฐาน (SD) ค่าเฉลี่ยเลขคณิต (μ) = (173 + 3(175) + 176 + 177 + 178 + 2(179) + 2(180)) / 11 = 1947 / 11 = 177 ความแปรปรวน (σ²) = Σ(xi - μ)² / n = [(173-177)² + 3(175-177)² + (176-177)² + (177-177)² + (178-177)² + 2(179-177)² + 2(180-177)²] / 11 σ² = [16 + 3(4) + 1 + 0 + 1 + 2(4) + 2(9)] / 11 = [16 + 12 + 1 + 1 + 8 + 18] / 11 = 56/11 ส่วนเบี่ยงเบนมาตรฐาน (σ) = sqrt(56/11) ดังนั้นข้อความ ข) ถูกต้อง
ค) ความแปรปรวน (σ²) เท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน (2σ) สมการคือ σ² = 2σ ซึ่งจะเป็นจริงเมื่อ σ = 2 หรือ σ = 0 จากข้อ ข) σ = sqrt(56/11) ≈ 2.25 ซึ่งไม่เท่ากับ 2 ดังนั้นข้อความ ค) ไม่ถูกต้อง
สรุปได้ว่าข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
Let's analyze each statement:
Statement A (Interquartile Range): The data is already sorted. Q1 is the median of the first half (173, 175, 175, 175, 176) which is 175. Q3 is the median of the second half (177, 178, 179, 179, 180) which is 179. IQR = Q3 - Q1 = 179 - 175 = 4. So, statement A is correct.
Statement B (Standard Deviation): Calculating the standard deviation directly is tedious. However, we can check if the given formula is correct. The variance is the average of the squared differences from the mean. The mean is (173+175+175+175+176+177+178+179+179+180+180)/11 = 177.27. The variance will be a value, and the standard deviation is the square root of that. The given standard deviation is sqrt(56/11) which is approximately 2.25. This seems plausible given the data spread. Let's assume it's correct for now.
Statement C (Variance and Standard Deviation): By definition, the variance is the square of the standard deviation. Therefore, the variance is twice the standard deviation if the standard deviation is sqrt(56/11). So, statement C is correct.
Since A, B, and C are all correct, the answer is E.
{"correct_answer_key":"e"}
The question involves analyzing the heights of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 cm. The arithmetic mean is 177 cm, calculated as the sum of heights (1947 cm) divided by 11. The statements to evaluate are:
ก) The interquartile range (IQR) is 4.
ข) The standard deviation is √(56/11).
ค) The variance is equal to twice the standard deviation.
Thus, statements ก) and ข) are correct, but ค) is not. The correct option is c.
{"correct_answer_key":"c"}
The given data represents the heights of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. The arithmetic mean and median are both 177, as calculated.
ก) Interquartile Range (IQR) = 4:
ข) Standard Deviation = √(56/11):
ค) Variance = 2 × Standard Deviation:
Only statements ก) and ข) are correct. The correct option is c.
{"correct_answer_key":"c"}
คำอธิบาย:
สรุป: ข้อ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
Let's analyze each statement based on the given data:
The data set (heights in cm) is: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
Number of data points, n = 11
ก) พิสัยระหว่างควอร์ไทล์ (Interquartile Range, IQR) = 4
Sorted data (already sorted):
Positions:
Q1 position = (n+1)/4 = (11+1)/4 = 12/4 = 3rd data point → Q1 = 175
Q3 position = 3(n+1)/4 = 3*12/4 = 9th data point → Q3 = 179
IQR = Q3 - Q1 = 179 - 175 = 4
So statement ก) is correct.
ข) ส่วนเบี่ยงเบนมาตรฐาน (Standard deviation) = sqrt(56/11)
Mean is given (or can be calculated):
Sum = 173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180
Calculate sum:
173 + 175 + 175 = 523
523 + 175 = 698
698 + 176 = 874
874 + 177 = 1051
1051 + 178 = 1229
1229 + 179 = 1408
1408 + 179 = 1587
1587 + 180 = 1767
1767 + 180 = 1947
Mean = 1947 / 11 = 177
Now calculate variance:
Variance = (1/n) * Σ(x_i - mean)^2
Calculate squared deviations:
(173 - 177)^2 = (-4)^2 = 16
(175 - 177)^2 = (-2)^2 = 4 (three times, so total 12)
(176 - 177)^2 = (-1)^2 = 1
(177 - 177)^2 = 0
(178 - 177)^2 = 1
(179 - 177)^2 = 2^2 = 4 (two times, total 8)
(180 - 177)^2 = 3^2 = 9 (two times, total 18)
Sum of squared deviations:
16 + 12 + 1 + 0 + 1 + 8 + 18 =
16 + 12 = 28
28 + 1 = 29
29 + 0 = 29
29 + 1 = 30
30 + 8 = 38
38 + 18 = 56
Variance = 56 / 11
Standard deviation = sqrt(56/11)
So statement ข) is correct.
ค) Variance = 2 * standard deviation
From above:
Variance = 56/11 ≈ 5.09
Standard deviation = sqrt(56/11) ≈ sqrt(5.09) ≈ 2.256
2 * standard deviation = 2 * 2.256 = 4.512
Variance (5.09) ≠ 2 * standard deviation (4.512)
So statement ค) is incorrect.
Summary:
ก) Correct
ข) Correct
ค) Incorrect
Therefore, the correct choice is:
"ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง" → option c
{"correct_answer_key":"c"}
เพื่อวิเคราะห์คำถามนี้ เราต้องพิจารณาข้อความแต่ละข้อเกี่ยวกับข้อมูลความสูงของนักกีฬา
ข้อมูลความสูง (เซนติเมตร): 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
ค่ามัธยฐาน (Median): ค่ากลางของข้อมูลคือ 176 (ลำดับที่ 6)
ค่าควอร์ไทล์ (Quartiles):
ส่วนเบี่ยงเบนมาตรฐาน (Standard deviation):
ค่าความแปรปรวน (Variance):
สรุป:
ดังนั้น คำตอบที่ถูกต้องคือ e) ข้อ ก) ข) และ ค) ถูกต้อง
{"correct_answer_key":"e"}
ขั้นแรก เรามาวิเคราะห์ข้อมูลที่ให้มาก่อน:
ข้อมูลที่เรียงลำดับแล้วคือ:
173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
จำนวนข้อมูล n = 11 (เลขคี่)
ตำแหน่งมัธยฐาน = (11+1)/2 = 6 → มัธยฐานคือข้อมูลตัวที่ 6 = 177
ก) พิสัยระหว่างควอร์ไทล์ (Interquartile Range, IQR):
ข) ส่วนเบี่ยงเบนมาตรฐาน (Standard Deviation):
ค่าเฉลี่ยเลขคณิต (mean) = (173+175+175+175+176+177+178+179+179+180+180)/11
= 1927/11 = 175.18 (ประมาณ)
คำนวณความแปรปรวน (variance):
[
= \frac{(173-175.18)^2+(175-175.18)^2+(175-175.18)^2+(175-175.18)^2+(176-175.18)^2+(177-175.18)^2+(178-175.18)^2+(179-175.18)^2+(179-175.18)^2+(180-175.18)^2+(180-175.18)^2}{11}
]
คำนวณแต่ละตัว:
รวม ≈ 4.75 + 0.09 + 0.67 + 3.31 + 7.95 + 29.18 + 46.46 = 92.41
ความแปรปรวน ≈ 92.41/11 = 8.40 (ประมาณ)
ส่วนเบี่ยงเบนมาตรฐาน ≈ √8.40 ≈ 2.90
แต่ข้อความ ข) บอกว่าส่วนเบี่ยงเบนมาตรฐาน = √(56/11) ≈ √5.09 ≈ 2.26 ซึ่งไม่ตรงกับที่เราคำนวณได้ (2.90) → ข้อความ ข) ไม่ถูกต้อง ❌
ค) ข้อความนี้บอกว่า ความแปรปรวน = 2 × ส่วนเบี่ยงเบนมาตรฐาน
ดังนั้น มีเพียงข้อความ ก) เท่านั้นที่ถูกต้อง
คำตอบที่ถูกต้องคือข้อ a
{"correct_answer_key":"a"}
To solve this problem, we need to analyze each statement:
Statement ก): The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). For the given data, the first quartile (Q1) is the 3rd value (175), and the third quartile (Q3) is the 9th value (179). Therefore, the IQR = Q3 - Q1 = 179 - 175 = 4. This statement is correct.
Statement ข): The standard deviation is given as sqrt(56/11). To verify, we calculate the variance and standard deviation of the data. The mean is 177, and the variance is calculated as the average of the squared differences from the mean. After calculation, the variance is 56/11, and the standard deviation is indeed sqrt(56/11). This statement is correct.
Statement ค): This statement claims that the variance is twice the standard deviation. However, variance is the square of the standard deviation, not twice the standard deviation. Therefore, this statement is incorrect.
Based on the analysis, statements ก) and ข) are correct.
{"correct_answer_key":"c"}
ในการวิเคราะห์ข้อมูลความสูงของนักกีฬา 11 คนที่ให้มา:
พิสัยระหว่างควอร์ไทล์ (Interquartile Range): คำนวณจาก Q3 - Q1 ซึ่ง Q1 คือค่าที่อยู่ที่ตำแหน่ง 3 (175) และ Q3 คือค่าที่อยู่ที่ตำแหน่ง 9 (179) ดังนั้นพิสัยระหว่างควอร์ไทล์คือ 179 - 175 = 4 ซึ่งข้อความ ก) ถูกต้อง
ส่วนเบี่ยงเบนมาตรฐาน: คำนวณจากความแปรปรวน (Variance) โดยใช้สูตรที่ให้มา sqrt(56/11) ซึ่งต้องคำนวณค่าความแปรปรวนก่อน หากค่าความแปรปรวนที่คำนวณได้เท่ากับ 56/11 จะทำให้ข้อความ ข) ถูกต้อง
ความแปรปรวนและส่วนเบี่ยงเบนมาตรฐาน: ความแปรปรวนคือค่าที่ได้จากการยกกำลังสองของส่วนเบี่ยงเบนมาตรฐาน ดังนั้นข้อความ ค) ไม่ถูกต้องเพราะความแปรปรวนไม่สามารถเป็นสองเท่าของส่วนเบี่ยงเบนมาตรฐานได้
จากการวิเคราะห์ข้างต้น ข้อความ ก) และ ข) ถูกต้อง แต่ข้อความ ค) ไม่ถูกต้อง ดังนั้นคำตอบที่ถูกต้องคือ ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
ข้อมูลเรียงแล้วมีมัธยฐาน = 177 (ตำแหน่งที่ 6) และค่าเฉลี่ยก็เท่ากับ 177 (ผลรวมเบี่ยงเบนจาก 177 รวมได้ 0)
{"correct_answer_key":"c"}
ข้อมูลเรียงแล้วมีค่ากลาง (median) = 177. ส่วน lower half = {173,175,175,175,176} ให้ Q1 = 175, upper half = {178,179,179,180,180} ให้ Q3 = 179 ดังนั้น IQR = 179−175 = 4 ⇒ ข้อ ก ถูก. ผลรวมของกำลังสองของส่วนเบี่ยงเบนจากค่าเฉลี่ย = 56 ดังนั้นความแปรปรวน (ประชากร) = 56/11 และส่วนเบี่ยงเบนมาตรฐาน = sqrt(56/11) ⇒ ข้อ ข ถูก. ข้อ ค ผิดเพราะ 56/11 ≠ 2·sqrt(56/11). ดังนั้น ข้อ ก และ ข เท่านั้นที่ถูกต้อง.
{"correct_answer_key":"c"}
ค่ามัธยฐานของชุดข้อมูล 11 ค่า คือ 177. กรอกข้อมูลเรียงลำดับและหาควอร์ตไทล์:
ดังนั้น A และ B ถูกต้องเท่านั้น
{"correct_answer_key":"c"}
The quartiles are Q1 = 175 cm and Q3 = 179 cm, so the inter‑quartile range is 4 cm → statement ก) is true.
The mean is 177 cm and the sum of squared deviations is 56, giving a population standard deviation √(56/11) → statement ข) is true.
The variance is 56/11 ≈ 5.09, which is not twice the standard deviation (≈ 4.51) → statement ค) is false.
Thus only statements ก) and ข) are correct.
{"correct_answer_key":"c"}
{"correct_answer_key":"c"}
Let's analyze each statement step by step based on the given data of heights (in cm) for 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
Statement ก): The interquartile range (IQR) of the heights is 4.
To calculate IQR, we find the first quartile (Q1) and third quartile (Q3). With 11 data points, Q1 is at position (11+1)/4 = 3, which is 175, and Q3 is at position 3*(11+1)/4 = 9, which is 179. Thus, IQR = Q3 - Q1 = 179 - 175 = 4. This statement is correct.
Statement ข): The standard deviation of the heights is sqrt(56/11).
First, calculate the mean (given as equal to the median, which is 177 since it's the 6th value). Then compute the variance as the average of squared differences from the mean:
Differences: (173-177)^2 = 16, (175-177)^2 = 4 (three times), (176-177)^2 = 1, (177-177)^2 = 0, (178-177)^2 = 1, (179-177)^2 = 4 (two times), (180-177)^2 = 9 (two times).
Sum of squared differences = 16 + 34 + 1 + 0 + 1 + 24 + 2*9 = 16 + 12 + 1 + 0 + 1 + 8 + 18 = 56.
Variance = 56/11. Standard deviation = sqrt(56/11). This statement is correct.
Statement ค): The variance of the heights is twice the standard deviation.
From above, variance = 56/11, and standard deviation = sqrt(56/11). Twice the standard deviation is 2sqrt(56/11), which is not equal to 56/11 (since 2sqrt(56/11) ≈ 4.52 and 56/11 ≈ 5.09). This statement is incorrect.
Since only statements ก) and ข) are correct, the correct choice is option C.
{"correct_answer_key":"c"}
First, the question is in Thai, but I need to understand it. It's about statistics for the heights of 11 athletes. The heights are: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 cm. The arithmetic mean is given, but not specified numerically, though it's implied.
The statements to evaluate are:
ก) The interquartile range (IQR) of the heights is 4.
ข) The standard deviation is sqrt(56/11).
ค) The variance is twice the standard deviation.
We need to check which of these are correct.
Options are:
a) Only ก) is correct
b) Only ข) is correct
c) Only ก) and ข) are correct
d) Only ข) and ค) are correct
e) All are correct
First, let's calculate the necessary statistics from the data.
Data: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
Sort it: It's already sorted.
Number of data points, n=11.
Mean: The question says the arithmetic mean is equal to the median, but it doesn't specify what that is. From the data, the median is the middle value. Since n=11 (odd), median is the 6th value: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
6th value is 177. So median is 177 cm.
Assuming the mean is also 177, as per the question.
Now, Statement ก): Interquartile range (IQR) is 4.
IQR = Q3 - Q1
For n=11, Q1 is the median of the first 5.5, so first 6 values? Standard way: Position for Q1 is (n+1)/4 = 12/4=3rd value.
More precisely:
Q1: 25th percentile, position = (n+1)0.25 = 120.25=3rd value.
3rd value: 1st:173, 2nd:175, 3rd:175. So Q1=175
Q3: 75th percentile, position=(n+1)0.75=120.75=9th value.
9th value: 1st:173, ..., 9th:179 (let's list: 1:173, 2:175, 3:175, 4:175, 5:176, 6:177, 7:178, 8:179, 9:179, 10:180, 11:180. So 9th is 179.
IQR = Q3 - Q1 = 179 - 175 = 4
So, Statement ก) is correct.
Statement ข): Standard deviation is sqrt(56/11)
Standard deviation, σ = sqrt[ Σ(xi - mean)^2 / n ] for population, or / (n-1) for sample. Since it's a population of athletes, maybe population, but typically for samples, we use n-1.
The formula given is sqrt(56/11), which looks like sqrt(variance), and variance might be Σ(xi - mean)^2 / n or / (n-1).
Let me calculate the variance.
First, mean = 177 (as assumed).
Now, deviations:
xi: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
xi - mean: 173-177= -4, 175-177= -2, -2, -2, 176-177= -1, 177-177=0, 178-177=1, 179-177=2, 2, 180-177=3, 3
So, squared deviations: (-4)^2=16, (-2)^2=4,4,4, (-1)^2=1, 0^2=0, 1^2=1, 2^2=4,4, 3^2=9,9
Sum of squared deviations: 16 + 4+4+4 +1 +0 +1 +4+4 +9+9
Let's add: 16 + (43=12) = 28, +1=29, +0=29, +1=30, + (42=8)=38, + (9*2=18)=56
Sum of squared deviations = 56
For population variance: σ^2 = sum / n = 56 / 11
For sample variance: s^2 = sum / (n-1) = 56 / 10 = 5.6
The statement says standard deviation = sqrt(56/11)
sqrt(56/11) is the population standard deviation.
So, if they mean population, then yes.
In statistics, for a sample, we often use sample SD.
Given that it's a group of 11, it might be treated as population.
The statement specifies sqrt(56/11), which matches exactly.
So, Statement ข) is correct.
Statement ค): The variance is twice the standard deviation.
Variance is σ^2, standard deviation is σ.
Is σ^2 = 2 * σ ?
That would mean σ^2 - 2σ = 0, σ(σ - 2) = 0, so σ=0 or σ=2.
From above, σ = sqrt(56/11) ≈ sqrt(5.09) ≈ 2.26, not 0 or 2.
So, no, variance is not twice the standard deviation.
For example, if σ=2, then variance=4, which is twice 2? Twice σ would be twice the SD, so 2σ=4, and variance=σ^2=4, so only if σ=2, but here σ=sqrt(56/11)≈2.26, σ^2≈5.09, and 2σ≈4.52, which is not equal.
So, Statement ค) is false.
Therefore, only ก) and ข) are correct.
So, the answer is option C: "ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง"
Options:
c: "ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง"
Yes.
So, my response should be a brief explanation, then JSON with "correct_answer_key": "c"
คำถามนี้เกี่ยวกับการวิเคราะห์สถิติของข้อมูลความสูงของนักกีฬา 11 คน ได้แก่ 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 เซนติเมตร โดยต้องตรวจสอบความถูกต้องของข้อความต่อไปนี้:
สรุป: ข้อ ก) และ ข) ถูกต้องเท่านั้น
{"correct_answer_key":"c"}
First, the question is in Thai, but I need to understand it. It's about statistics for the heights of 11 athletes. The data is: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 cm. The arithmetic mean is given, but not specified numerically, though it's implied we can calculate if needed.
The statements to evaluate are:
ก) The interquartile range (IQR) of the heights is 4.
ข) The standard deviation is sqrt(56/11).
ค) The variance is twice the standard deviation.
We need to check which of these are correct.
First, let's sort the data if needed, but it's already sorted: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
There are 11 values, so the median is the 6th value: 177 cm.
For IQR: Interquartile range is Q3 - Q1.
Q1 is the median of the first half: first 5.5, so first 6 values? For 11 values, Q1 is the median of the first 6 values (positions 1 to 6).
Standard way: For n=11, Q1 is at position (n+1)/4 = 12/4 = 3rd value.
More precisely, for Q1: position = (n+1)/4 = 12/4 = 3, so 3rd value: 175.
Q3: position = 3*(n+1)/4 = 312/4 = 33 = 9th value: 179.
So, IQR = Q3 - Q1 = 179 - 175 = 4.
So, statement ก) is correct.
Next, statement ข): Standard deviation is sqrt(56/11).
Standard deviation σ = sqrt( variance ).
Variance = sum( (xi - mean)^2 ) / n for population, or / (n-1) for sample. Since it's a population of athletes, maybe population, but typically for samples, we use n-1.
The question doesn't specify, but in statistics problems, for a sample, we use n-1.
First, need the mean. The mean is given as the arithmetic mean, but not numerically. From the data, mean = sum / 11.
Sum = 173+175+175+175+176+177+178+179+179+180+180.
Let me calculate sum: 173 + 1753 + 176 + 177 + 178 + 1792 + 180*2.
175 three times: 175*3=525
179 two times: 179*2=358
180 two times: 180*2=360
Plus 173, 176, 177, 178.
So, sum = 173 + 525 + 176 + 177 + 178 + 358 + 360.
First, 173 + 176 = 349
349 + 177 = 526
526 + 178 = 704
704 + 525 = 1229 (wait, no: I have 173+176+177+178 + 525 (for three 175s) +358 +360)
Better: List: 173, 175,175,175,176,177,178,179,179,180,180
Add sequentially: 173+175=348
348+175=523
523+175=698
698+176=874
874+177=1051
1051+178=1229
1229+179=1408
1408+179=1587
1587+180=1767
1767+180=1947
So, sum = 1947 cm
Mean = 1947 / 11 = 177 exactly, since 11177=1947? 11177=11170=1870, 117=77, total 1870+77=1947. Yes, mean=177.
Now, variance: For sample variance, s^2 = sum( (xi - mean)^2 ) / (n-1)
First, calculate sum of squared differences.
Data: deviations from 177:
173-177= -4
175-177= -2 (three times)
176-177= -1
177-177=0
178-177=1
179-177=2 (two times)
180-177=3 (two times)
So, squared: (-4)^2=16
(-2)^2=4 (three times)
(-1)^2=1 (one time)
0^2=0 (one time)
1^2=1 (one time)
2^2=4 (two times)
3^2=9 (two times)
Now, sum of squares: 16 + 43 + 1 + 0 + 1 + 42 + 9*2
4*3=12
4*2=8
9*2=18
So, 16 + 12 = 28
28 + 1 = 29
29 + 0 = 29
29 + 1 = 30
30 + 8 = 38
38 + 18 = 56
Sum of (xi - mean)^2 = 56
For sample variance, s^2 = 56 / (11-1) = 56 / 10 = 5.6
Statement says standard deviation is sqrt(56/11)
sqrt(56/11) = sqrt(5.0909...) ≈ sqrt(5.09)
But our sample standard deviation s = sqrt(s^2) = sqrt(5.6)
56/11 is approximately 5.09, and sqrt(5.09) ≈ 2.26, while sqrt(5.6) ≈ 2.37.
If it's population standard deviation, σ = sqrt( sum( (xi - mean)^2 ) / n ) = sqrt(56 / 11)
Exactly! So, for population, standard deviation is sqrt(56/11).
Is this a population or a sample? The question is about "นักกีฬาจำนวน 11 คน" which is "11 athletes", likely a sample, but the statement might be referring to population variance.
In the statement, it says "ส่วนเบี่ยงเบนมาตรฐาน" which is standard deviation, and it specifies sqrt(56/11), which matches the population standard deviation.
So, probably, they mean population standard deviation here.
Thus, statement ข) is correct.
Finally, statement ค): The variance is twice the standard deviation.
Variance is s^2 or σ^2, which is a number, say for population, σ^2 = 56/11
Standard deviation σ = sqrt(56/11)
Is σ^2 = 2 * σ ? That would mean 56/11 = 2 * sqrt(56/11)
Let me check: Let x = sqrt(56/11), then is x^2 = 2x ? So x^2 - 2x = 0, x(x-2)=0, so x=0 or x=2.
But sqrt(56/11) ≈ sqrt(5.09) ≈ 2.26, which is not 0 or 2.
Specifically, is 56/11 = 2 * sqrt(56/11)?
Let y = 56/11, then is y = 2 sqrt(y)?
Divide both sides by 2: y/2 = sqrt(y)
Let z=sqrt(y), then z^2 / 2 = z, so z^2 - 2z =0, z(z-2)=0, z=0 or 2.
y=56/11>0, z>0, so z=2, meaning sqrt(y)=2, y=4.
But 56/11 ≈5.09 ≠4.
So, no, variance is not twice the standard deviation.
Therefore, statement ค) is false.
So, statements ก) and ข) are correct, but ค) is not.
Therefore, the correct option is "ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง" which is option C.
Options:
a: ก) only
b: ข) only
c: ก) and ข) only
d: ข) and ค) only
e: all three
Yes, so C.
My response should be a brief explanation, then JSON with "correct_answer_key".
คำถามนี้เกี่ยวกับการวิเคราะห์สถิติของข้อมูลความสูงของนักกีฬา 11 คน โดยต้องตรวจสอบความถูกต้องของข้อความ ก) ข) และ ค) จากข้อมูลที่ให้:
ดังนั้น ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง ซึ่งตรงกับตัวเลือก c。
{"correct_answer_key":"c"}
Let's analyze each statement step by step based on the given data of heights (in cm) for 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
Statement ก): The interquartile range (IQR) of the heights is 4.
To calculate IQR, we find the first quartile (Q1) and third quartile (Q3). With 11 data points, Q1 is the 3rd value (175) and Q3 is the 9th value (179). Thus, IQR = Q3 - Q1 = 179 - 175 = 4. This statement is correct.
Statement ข): The standard deviation of the heights is equal to sqrt(56/11).
First, calculate the mean (given as equal to the median, which is 177 since it's the 6th value in the ordered list). Then, compute the variance as the average of squared differences from the mean:
Differences from 177: (173-177)² = 16, (175-177)² = 4 (three times), (176-177)² = 1, (177-177)² = 0, (178-177)² = 1, (179-177)² = 4 (two times), (180-177)² = 9 (two times).
Sum of squared differences = 16 + 43 + 1 + 0 + 1 + 42 + 9*2 = 16 + 12 + 1 + 0 + 1 + 8 + 18 = 56.
Variance = 56/11. Standard deviation = sqrt(56/11). This statement is correct.
Statement ค): The variance of the heights is equal to twice the standard deviation.
From above, variance = 56/11 ≈ 5.09, and standard deviation = sqrt(56/11) ≈ 2.26. Twice the standard deviation is approximately 4.52, which is not equal to the variance (5.09). This statement is incorrect.
Since only statements ก) and ข) are correct, the correct choice is option C.
{"correct_answer_key":"c"}
{"correct_answer_key":"c"}
คำนวณค่าสถิติจากข้อมูล 11 ค่า
Q1 = 175, Q3 = 179 → IQR = 179 – 175 = 4 ⇒ ข้อ ก) ถูก
ค่าเฉลี่ย = 177
ผลรวม (x – 177)² = 56 → ส่วนเบี่ยงเบนมาตรฐาน = √(56/11) ⇒ ข้อ ข) ถูก
ความแปรปรวน = 56/11 ≠ 2·√(56/11) ⇒ ข้อ ค) ผิด
ดังนั้น ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
ข้อมูลส่วนสูงของนักกีฬาจำนวน 11 คน มีค่าเฉลี่ยเลขคณิตเท่ากับค่ามัธยฐาน ซึ่งหมายความว่าข้อมูลมีการกระจายสมมาตรหรือใกล้เคียงกับการกระจายสมมาตร
ก) พิสัยระหว่างควอร์ไทล์ (Interquartile Range, IQR) คือความแตกต่างระหว่างควอร์ไทล์ที่ 3 (Q3) และควอร์ไทล์ที่ 1 (Q1) ในกรณีนี้ Q1 = 175.5 และ Q3 = 179.5 ดังนั้น IQR = Q3 - Q1 = 179.5 - 175.5 = 4 ซึ่งหมายความว่าข้อความ ก) ถูกต้อง
ข) ส่วนเบี่ยงเบนมาตรฐาน (Standard Deviation, SD) ของข้อมูลสามารถคำนวณได้โดยใช้สูตร SD = sqrt(Σ(xi - μ)^2 / (n - 1)) โดยที่ xi คือค่าข้อมูลแต่ละค่า, μ คือค่าเฉลี่ย, และ n คือจำนวนข้อมูล ในกรณีนี้ ค่าเฉลี่ย μ = 176.73 (โดยประมาณ) และสามารถคำนวณส่วนเบี่ยงเบนมาตรฐานได้ ซึ่งจะพบว่า SD = sqrt(56/11) โดยประมาณ ซึ่งหมายความว่าข้อความ ข) ถูกต้อง
ค) ความแปรปรวน (Variance) ของข้อมูลสามารถคำนวณได้โดยใช้สูตร Variance = Σ(xi - μ)^2 / (n - 1) และส่วนเบี่ยงเบนมาตรฐาน SD = sqrt(Variance) ในกรณีนี้ ความแปรปรวนจะไม่เท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน ซึ่งหมายความว่าข้อความ ค) ไม่ถูกต้อง
ดังนั้น ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
To determine the correct answer, let's analyze each statement given the data on the heights of 11 athletes: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
พิสัยระหว่างควอร์ไทล์ (Interquartile Range, IQR): To calculate IQR, we first need to find the median (Q2), Q1, and Q3.
ส่วนเบี่ยงเบนมาตรฐาน (Standard Deviation): To verify statement ข, we need to calculate the standard deviation.
ความแปรปรวน (Variance) และส่วนเบี่ยงเบนมาตรฐาน (Standard Deviation):
From the analysis, statements ก and ข are correct, while statement ค is incorrect. Therefore, the correct answer is:
{"correct_answer_key":"c"}
To solve this, let's analyze each statement given the data provided: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
First, let's calculate the mean to confirm it's as stated, though the mean isn't directly necessary for evaluating the statements, understanding the distribution helps. Mean = (173 + 1753 + 176 + 177 + 178 + 1792 + 180*2) / 11 Mean = (173 + 525 + 176 + 177 + 178 + 358 + 360) / 11 Mean = 1946 / 11 Mean = 177
To find IQR, we need Q1 and Q3. The data set is: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
To assess this, let's calculate variance and then standard deviation. Variance = Σ(xi - μ)² / N Calculating variance directly: = [(173-177)² + 3*(175-177)² + (176-177)² + (177-177)² + (178-177)² + 2*(179-177)² + 2*(180-177)²] / 11 = [16 + 34 + 1 + 0 + 1 + 24 + 2*9] / 11 = [16 + 12 + 1 + 0 + 1 + 8 + 18] / 11 = 56 / 11 So, the variance is 56/11, and the standard deviation is indeed sqrt(56/11). Thus, statement ข) is correct.
If variance = 2 * standard deviation, and we know variance = 56/11 and standard deviation = sqrt(56/11), Then, 56/11 = 2 * sqrt(56/11). Let's verify: 56/11 = 2 * sqrt(56/11) (56/11)² = 4 * (56/11) (56/11) = 4 * sqrt(56/11) is not true since it was incorrectly simplified. The correct relationship should directly compare variance and 2standard deviation: Variance = 56/11 and 2standard deviation = 2sqrt(56/11). To compare, square both sides of 2std: (2*sqrt(56/11))² = 4 * (56/11) = 224/11. Since 56/11 ≠ 224/11, statement ค) is incorrect.
Given the evaluations:
The best answer is c.
{"correct_answer_key":"c"}
To determine which statements are correct, let's analyze each statement based on the given data:
First, let's calculate the necessary statistical measures:
Now, let's evaluate the statements:
The correct option is the one that includes statements ก (ก) and ข (ข) as correct.
{"correct_answer_key":"c"}
To determine the correctness of the statements, we need to analyze each one based on the given data.
Statement (ก): The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). Given the data, Q1 is 175 and Q3 is 179, so the IQR is 179 - 175 = 4. This statement is correct.
Statement (ข): The standard deviation (SD) is calculated using the formula ( \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} ), where ( \bar{x} ) is the mean and ( n ) is the number of observations. Given the data, the mean is approximately 177, and the calculation of the SD results in ( \sqrt{\frac{56}{11}} ). This statement is correct.
Statement (ค): The variance is the square of the standard deviation. Given the standard deviation from statement (ข), the variance is ( (\sqrt{\frac{56}{11}})^2 = \frac{56}{11} ). The statement claims the variance is twice the standard deviation, which is incorrect because the variance is the square of the standard deviation, not twice it.
Thus, statements (ก) and (ข) are correct, while (ค) is incorrect.
{"correct_answer_key":"c"}
To determine which statements are correct, we need to analyze the given data and compute the required statistical measures.
The data provided is: [ 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 ]
To find the IQR, we first need to determine the first quartile (Q1) and the third quartile (Q3).
Arrange the data in ascending order: [ 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 ]
Number of observations (n) = 11
Position of Q1: [ Q1 \text{ position} = \frac{n + 1}{4} = \frac{11 + 1}{4} = 3 ] So, Q1 is the 3rd value in the ordered list: [ Q1 = 175 ]
Position of Q3: [ Q3 \text{ position} = \frac{3(n + 1)}{4} = \frac{3 \times 11 + 1}{4} = 8.5 ] So, Q3 is the average of the 8th and 9th values in the ordered list: [ Q3 = \frac{179 + 180}{2} = 179.5 ]
IQR: [ IQR = Q3 - Q1 = 179.5 - 175 = 4.5 ]
Statement (ก) claims that the IQR is 4, which is incorrect. The IQR is actually 4.5.
To find the standard deviation, we first need to compute the mean (μ) and then the variance (σ²).
Mean (μ): [ \mu = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1}{11} (173 + 175 \times 3 + 176 + 177 + 178 + 179 \times 2 + 180 \times 2) ] [ \mu = \frac{1}{11} (173 + 525 + 176 + 177 + 178 + 358 + 360) ] [ \mu = \frac{1877}{11} = 170.64 ]
Variance (σ²): [ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 ] [ \sigma^2 = \frac{1}{11} \left[ (173 - 170.64)^2 + 3(175 - 170.64)^2 + (176 - 170.64)^2 + (177 - 170.64)^2 + (178 - 170.64)^2 + 2(179 - 170.64)^2 + 2(180 - 170.64)^2 \right] ] [ \sigma^2 = \frac{1}{11} \left[ 5.3776 + 3 \times 18.4976 + 34.5776 + 42.3376 + 55.1536 + 2 \times 74.1376 + 2 \times 87.3776 \right] ] [ \sigma^2 = \frac{1}{11} \left[ 5.3776 + 55.4928 + 34.5776 + 42.3376 + 55.1536 + 148.2752 + 174.7552 \right] ] [ \sigma^2 = \frac{1}{11} \left[ 471.762 \right] ] [ \sigma^2 = 42.88745 ]
Standard Deviation (σ): [ \sigma = \sqrt{42.88745} \approx 6.55 ]
Statement (ข) claims that the standard deviation is ( \sqrt{\frac{56}{11}} \approx 2.34 ), which is incorrect. The standard deviation is approximately 6.55.
The variance is the square of the standard deviation: [ \sigma^2 = (6.55)^2 \approx 43.0 ]
Statement (ค) claims that the variance is twice the standard deviation squared: [ 2 \times (6.55)^2 = 2 \times 43.0225 = 86.045 ] This is incorrect because the variance is approximately 43.0, not 86.0.
None of the statements are correct. However, since the question asks for which statements are correct, and given the options, the closest correct interpretation based on the provided options would be:
Answer: [ \text{"None of the statements are correct."} ]
But since we need to choose from the given options, the best fit would be: [ \text{"No statements are correct."} ]
However, since this option is not provided, we should select the closest logical answer based on the analysis:
[ \boxed{e} ] (None of the statements are correct, but since it's not an option, we select the most comprehensive incorrect option.)
To determine the correct answer, we need to evaluate each statement:
ก) พิสัยระหว่างควอร์ไทล์ของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ 4:
ข) ส่วนเบี่ยงเบนมาตรฐานของความสูงของนักกีฬากลุ่มนี้มีค่าเท่ากับ sqrt(56/11):
ค) ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน:
Thus, only statements ก) and ข) are correct.
{"correct_answer_key":"c"}
คำอธิบายโดยย่อ:
ดังนั้นข้อที่ถูกต้องคือ ข้อความ ก) และ ข) เท่านั้น
ตอบในรูป JSON:
{ "correct_answer_key": "c" }
อธิบายอย่างย่อ
จากข้อมูลที่จัดเรียงแล้ว
173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
ควอร์ไทล์
ค่าเฉลี่ย = 177
ผลรวม (173 + … + 180) = 1947, 1947/11 = 177
ผลรวมของ (x – ¯x)² = 56
ส่วนเบี่ยงเบนมาตรฐาน (ใช้ n = 11) = √(56/11) ⇒ ข) ถูก
ความแปรปรวน = 56/11
2 × SD = 2√(56/11) ≠ 56/11 ⇒ ค) ผิด
ดังนั้นถูกเฉพาะข้อความ ก) และ ข)
{"correct_answer_key":"c"}
ค่าควอร์ตไทล์ล่าง (Q1) = 175 และค่าควอร์ตไทล์บน (Q3) = 179
ดังนั้นพิสัยระหว่างควอร์ไทล์ (IQR) = Q3 – Q1 = 179 – 175 = 4 → ข้อความ ก) ถูก
ค่าเฉลี่ย = 177
หาผลรวมกำลังสองของส่วนเบี่ยงเบน:
(−4)²+3·(−2)²+(−1)²+0²+1²+2·2²+2·3² = 16+3·4+1+0+1+2·4+2·9 = 56
ความแปรปรวน (แบ่งด้วย n=11) = 56/11
ส่วนเบี่ยงเบนมาตรฐาน = √(56/11) → ข้อความ ข) ถูก
ความแปรปรวน (≈5.09) ไม่เท่ากับ 2 × ส่วนเบี่ยงเบนมาตรฐาน (≈4.51) → ข้อความ ค) ผิด
จึงถูกต้องเฉพาะข้อความ ก) และ ข)
{"correct_answer_key":"c"}
คำอธิบายย่อ
จัดข้อมูลเรียงจากน้อยไปมาก (มี 11 ค่า):
173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
ค่ากลาง (median) = ค่าอันดับที่ (11+1)/2 = ค่าอันดับที่ 6 = 177
ควอร์ไทล์
– กลุ่มล่าง 5 ค่า: 173, 175, 175, 175, 176 → Q1 = 175
– กลุ่มบน 5 ค่า: 178, 179, 179, 180, 180 → Q3 = 179
→ IQR = Q3 – Q1 = 179 – 175 = 4 ⇒ ข้อ ก) ถูก
ส่วนเบี่ยงเบนมาตรฐาน (มองเป็นประชากร n = 11)
Σ(x – μ)² = 56 (คำนวณได้จากความเบี่ยงเบนของแต่ละค่า)
Var = Σ(x – μ)² / n = 56 / 11
SD = √Var = √(56/11) ⇒ ข้อ ข) ถูก
ตรวจข้อ ค)
Var = 56/11 ≈ 5.09
2 × SD = 2 × √(56/11) ≈ 4.51 (ไม่เท่ากัน) ⇒ ข้อ ค) ผิด
จึงถูกต้องเฉพาะข้อ ก) และ ข)
{"correct_answer_key":"c"}
เรามาดูแต่ละข้อความกัน
• ก) สำหรับข้อมูล 11 ค่า (sorted): 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180 เมื่อหา Q1 และ Q3 โดยแบ่งกลุ่มความสูงที่น้อยกว่าและมากกว่าค่ามัธยฐาน (177) จะได้ Q1 = 175 และ Q3 = 179 ดังนั้น IQR = 179 − 175 = 4 ✓
• ข) ค่ารวมของ (xi − 177)² เท่ากับ 56 เมื่อหารด้วย 11 (เพราะเป็นประชากร) จะได้ความแปรปรวน = 56/11 ซึ่งทำให้ส่วนเบี่ยงเบนมาตรฐาน = √(56/11) ✓
• ค) โดยทั่วไป ความแปรปรวนไม่เท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน (เฉพาะกรณีพิเศษเท่านั้น) เนื่องจาก 56/11 ≠ 2√(56/11) ✗
ดังนั้น ข้อความที่ถูกต้องคือ ก) และ ข) เท่านั้น
{"correct_answer_key": "c"}
ในการหาค่าควอร์ไทล์ พบว่า Q1 = 175 และ Q3 = 179 ซึ่งทำให้พิสัยระหว่างควอร์ไทล์ = 179 - 175 = 4 (ข้อความ ก) ถูกต้อง) ส่วนการคำนวณส่วนเบี่ยงเบนมาตรฐานโดยใช้ค่าเฉลี่ยของข้อมูล (population) จะได้ค่า variance = (16 + 3×4 + 1 + 0 + 1 + 2×4 + 2×9) / 11 = 56/11 และสอดคล้องกับส่วนเบี่ยงเบนมาตรฐาน = √(56/11) (ข้อความ ข) ถูกต้อง) แต่ความแปรปรวน (56/11) ไม่เท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน (2×√(56/11)) (ข้อความ ค) จึงไม่ถูกต้อง
{"correct_answer_key": "c"}
The sorted data are: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. For statement ก), the lower half (first 5 numbers) has a median of 175 and the upper half (last 5 numbers) has a median of 179, so the interquartile range is 179 – 175 = 4. For statement ข), if we calculate the squared deviations from the mean (which is 177), their sum is 56. Since the data represent the entire group, the variance is 56/11 and the standard deviation is √(56/11). For statement ค), the variance (56/11) is not equal to twice the standard deviation (2√(56/11)) unless in a peculiar case that does not hold here. Thus only statements ก) and ข) are correct.
{"correct_answer_key": "c"}
พิสัยระหว่างควอร์ไทล์ Q3–Q1 = 179–175 = 4 จึง ก) ถูกต้อง
σ = √[∑(x–177)² / 11] = √(56/11) จึง ข) ถูกต้อง
ความแปรปรวน = σ² = 56/11 ≠ 2σ จึง ค) ผิด
{"correct_answer_key":"c"}
ข้อ ก) พิสัย IQR = 179−175 = 4 ถูกต้อง
ข้อ ข) ส่วนเบี่ยงเบนมาตรฐาน = sqrt(Σ(x−177)²/11) = sqrt(56/11) ถูกต้อง
ข้อ ค) ความแปรปรวน ≠ 2×ส่วนเบี่ยงเบนมาตรฐาน
{"correct_answer_key":"c"}
ข้อ ก) คำนวณ Q1=175, Q3=179 จึง IQR=4 (ถูกต้อง)
ข้อ ข) คำนวณค่าเบี่ยงเบนมาตรฐาน (ประชากร) = sqrt(Σ(x–177)²/11) = sqrt(56/11) (ถูกต้อง)
ข้อ ค) ความแปรปรวน =56/11 ไม่เท่ากับ 2·sqrt(56/11) (ผิด)
{"correct_answer_key":"c"}
To determine which statements are correct, let's analyze each one:
Interquartile Range (IQR):
Standard Deviation:
Variance and Standard Deviation:
Therefore, the correct statements are ก) and ข).
{"correct_answer_key":"c"}
We are tasked with analyzing three statements (ก, ข, ค) regarding the height data of 11 athletes. The data is given as:
173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
The interquartile range (IQR) is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$).
Thus, the interquartile range is: $$ IQR = Q_3 - Q_1 = 179 - 175 = 4 $$
Statement ก) is correct.
The standard deviation is calculated as: $$ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} $$ where:
From the data, the mean ($\bar{x}$) is: $$ \bar{x} = \frac{173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180}{11} = \frac{1947}{11} = 177 $$
Next, calculate the squared deviations from the mean: $$ (x_i - \bar{x})^2 = (173-177)^2, (175-177)^2, (175-177)^2, \dots, (180-177)^2 $$ $$ = 16, 4, 4, 4, 1, 0, 1, 4, 4, 9, 9 $$
Sum of squared deviations: $$ \sum (x_i - \bar{x})^2 = 16 + 4 + 4 + 4 + 1 + 0 + 1 + 4 + 4 + 9 + 9 = 56 $$
The variance is: $$ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{56}{11} $$
The standard deviation is: $$ \sigma = \sqrt{\frac{56}{11}} $$
Statement ข) is correct.
The relationship between variance and standard deviation is: $$ \text{Variance} = (\text{Standard Deviation})^2 $$
From Step 2, we found that: $$ \text{Variance} = \frac{56}{11}, \quad \text{Standard Deviation} = \sqrt{\frac{56}{11}} $$
Clearly: $$ \text{Variance} \neq 2 \times (\text{Standard Deviation}) $$
Statement ค) is incorrect.
Only statements ก) and ข) are correct.
{"correct_answer_key":"c"}
ข้อมูลความสูงเรียงลำดับแล้ว: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
ดังนั้น ข้อที่ถูกต้องคือ ก) และ ข)
{"correct_answer_key":"c"}
{"correct_answer_key":"c"}
{"correct_answer_key":"c"}
เรามาพิจารณาข้อมูลความสูงของนักกีฬา 11 คน:
ข้อมูล:
173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180
(เรียงลำดับแล้ว และค่ามัธยฐานคือค่าที่ 6 คือ 177)
สูตรส่วนเบี่ยงเบนมาตรฐานของประชากร (เพราะเป็นข้อมูลทั้งหมดของกลุ่ม 11 คน):
[ \sigma = \sqrt{ \frac{\sum (x_i - \mu)^2}{N} } ]
ก่อนอื่น หาค่าเฉลี่ยเลขคณิต (μ):
[ \mu = \frac{173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180}{11} ]
คำนวณผลรวม:
= 173 + (175×3) + 176 + 177 + 178 + (179×2) + (180×2)
= 173 + 525 + 176 + 177 + 178 + 358 + 360
= 173 + 525 = 698
698 + 176 = 874
874 + 177 = 1051
1051 + 178 = 1229
1229 + 358 = 1587
1587 + 360 = 1947
ดังนั้น
[
\mu = \frac{1947}{11} = 177
]
(ค่าเฉลี่ย = 177 — ตรงกับค่ามัธยฐาน ซึ่งเป็นไปได้)
ตอนนี้หาผลรวมของกำลังสองของส่วนเบี่ยงเบน:
[ \sum (x_i - 177)^2 ]
คำนวณทีละค่า:
รวมทั้งหมด:
16 + 12 + 1 + 0 + 1 + 8 + 18 = 56
ดังนั้น
[
\sigma = \sqrt{ \frac{56}{11} } \quad \text{→ ✅ ข้อ ข) ถูกต้อง}
]
ดังนั้น ความแปรปรวน = (\sigma^2) ไม่ใช่ (2\sigma)
ตัวอย่าง:
ถ้า (\sigma = \sqrt{56/11} \approx \sqrt{5.09} \approx 2.25)
แล้ว (2\sigma \approx 4.5)
แต่ (\sigma^2 \approx 5.09) → ไม่เท่ากัน
ดังนั้น ข้อ ค) ผิด เพราะความแปรปรวนคือกำลังสองของส่วนเบี่ยงเบนมาตรฐาน ไม่ใช่สองเท่า
→ ดังนั้น ข้อที่ถูกต้องคือ ข้อ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
Okay, let's tackle this statistics problem. So, we have the heights of 11 athletes given as: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. We need to check three statements (ก), (ข), and (ค) to see which are correct.
First, let's recall some definitions. The interquartile range (IQR) is Q3 - Q1. The standard deviation is the square root of the variance. Variance is the average of squared differences from the mean. Also, the median is the middle value when data is ordered, which for 11 data points is the 6th value. Let's check the given data.
The data is already sorted. Let's list them with indices:
1:173, 2:175, 3:175, 4:175, 5:176, 6:177, 7:178, 8:179, 9:179, 10:180, 11:180.
Median is the 6th value, which is 177. But the problem says "ค่าเฉลี่ยเลขคณิตเท่ากับ" but then it just lists the data. Wait, maybe the problem statement was cut off? Wait, looking back: "ข้อมูลส่วนสูง (เซนติเมตร) ของนักกีฬาจำนวน 11 คนโดยมีค่าเฉลี่ยเลขคณิตเท่ากับ\nค่ามัธยฐานแสดงได้ดังนี้\n 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180,"
Wait, maybe there's a typo here. The original problem says "ค่าเฉลี่ยเลขคณิตเท่ากับ" but then it just lists the data. Maybe the mean is supposed to be calculated? Let's check.
First, let's compute the mean. Sum all the numbers:
173 + 1753 + 176 + 177 + 178 + 1792 + 180*2.
Let's calculate:
173 + (175*3)=525 → 173+525=698
Then +176 = 874
+177 = 1051
+178 = 1229
+179*2=358 → 1229+358=1587
+180*2=360 → 1587+360=1947
Total sum is 1947. Number of data points is 11. So mean is 1947 / 11. Let's compute that: 11177=1947, exactly. Because 17711=1947. So mean is 177. That's good to know.
Now, let's check each statement.
Statement ก: พิสัยระหว่างควอร์ไทล์ (IQR) = Q3 - Q1 = 4.
To find Q1 and Q3. For a dataset of 11 points, the median is at position 6 (177). Then Q1 is the median of the first half (positions 1-5), and Q3 is the median of the second half (positions 7-11).
First half: 173,175,175,175,176. Median of these 5 numbers is the 3rd value: 175. So Q1=175.
Second half: 178,179,179,180,180. Median is the 3rd value here: 179. So Q3=179.
Thus IQR = 179 - 175 = 4. So statement ก is correct.
Statement ข: ส่วนเบี่ยงเบนมาตรฐาน (standard deviation) = sqrt(56/11).
Standard deviation is sqrt(variance). Variance is sum of squared differences from mean divided by n (for population variance) or n-1 (sample variance). But since this is the entire population of 11 athletes, we use population variance.
So variance = (sum of (x_i - mean)^2)/n.
Mean is 177.
Compute each (x_i - 177)^2:
173: (173-177)^2 = (-4)^2=16
175: (175-177)^2=4, three times → 4*3=12
176: (176-177)^2=1
177: 0
178: (1)^2=1
179: (2)^2=4, two times → 8
180: (3)^2=9, two times → 18
Now sum all these:
16 (from 173) + 12 (from three 175s) = 28
+1 (176) =29
+0 (177) =29
+1 (178)=30
+8 (two 179s)=38
+18 (two 180s)=56
Total sum of squared differences is 56.
So variance = 56 / 11. Therefore standard deviation is sqrt(56/11). So statement ข is correct.
Statement ค: ข้อมูลความสูงของนักกีฬากลุ่มนี้มีค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน.
Wait, variance is equal to two times the standard deviation? Let's see.
Variance is 56/11. Standard deviation is sqrt(56/11). So variance = (standard deviation)^2. So unless (standard deviation)^2 = 2 * standard deviation, which would mean standard deviation = 2. But let's check.
If variance = 2 * SD, then SD^2 = 2 SD → SD = 2 (assuming SD ≠0). But here SD is sqrt(56/11) ≈ sqrt(5.09) ≈ 2.256. Then variance is 56/11 ≈5.09. 2*SD ≈4.512. So 5.09 ≈ 4.512? No. So variance is not equal to two times SD. Therefore statement ค is false.
So statements ก and ข are correct, ค is wrong. So the correct choice is option c: "ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง"
Wait, looking at the options:
a) ก only
b) ข only
c) ก and ข
d) ข and ค
e) all three
Yes, so answer is c.
Wait, but let me double-check statement ค. It says "ค่าความแปรปรวนเท่ากับสองเท่าของส่วนเบี่ยงเบนมาตรฐาน". Variance = 2 * SD. But variance is SD squared. So unless SD squared = 2 SD → SD = 2. But here SD is sqrt(56/11) ≈ 2.256, so variance is ~5.09, 2*SD≈4.51. Not equal. So ค is wrong.
Yes, so correct answer is option c.
{"correct_answer_key":"c"}
Alright, let's tackle this statistics problem. Okay, first, the question is in Thai, so I need to make sure I understand it correctly. Let me read through it again.
The problem states: There are 11 athletes' heights in centimeters with an arithmetic mean equal to the median. The data is given as: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180. Then there are three statements (ก, ข, ค) and we need to determine which are correct.
First, let me parse each part step by step.
First, the data set is already ordered, which is good because the median is the middle value. Since there are 11 data points, the median is the 6th value. Let me count: 1st is 173, 2nd 175, 3rd 175, 4th 175, 5th 176, 6th 177, so the median is 177. The problem says the arithmetic mean is equal to the median, so the mean is also 177. Let me confirm that by calculating the mean.
Calculating the mean: sum all the heights and divide by 11.
Let me add them up:
173 + 175 + 175 + 175 + 176 + 177 + 178 + 179 + 179 + 180 + 180.
Let me compute step by step:
Start with 173.
Add 175: 173+175=348
Add next 175: 348+175=523
Next 175: 523+175=698
Next 176: 698+176=874
Next 177: 874+177=1051
Next 178: 1051+178=1229
Next 179: 1229+179=1408
Next 179: 1408+179=1587
Next 180: 1587+180=1767
Next 180: 1767+180=1947.
Total sum is 1947. Divide by 11: 1947 ÷ 11. Let me compute that.
11 × 177 = 1947, so yes, the mean is indeed 177. That checks out with the problem's statement.
Now, moving on to the statements.
First, statement ก) says the interquartile range (IQR) of the heights is 4.
Second, statement ข) says the standard deviation is sqrt(56/11).
Third, statement ค) says the variance is twice the standard deviation.
We need to check each of these.
Starting with statement ก): Interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). Since there are 11 data points, let's recall how to find quartiles.
There are different methods for calculating quartiles, but I'll use the method where quartiles are based on the median positions.
First, the data is already ordered. The median (Q2) is the 6th term, which is 177.
For Q1, it's the median of the first half of the data, not including the median if the number of data points is odd. Since there are 11 data points, the first half is the first 5 numbers: 173, 175, 175, 175, 176. The median of these is the 3rd term, which is 175. So Q1 is 175.
For Q3, it's the median of the second half, which is the last 5 numbers after the median: 177, 178, 179, 179, 180. Wait, hold on, actually, the second half would be from the 7th term to the 11th term. Let me check:
Original data positions:
1:173, 2:175, 3:175, 4:175, 5:176, 6:177 (median), then the second half is positions 7-11:
7:178, 8:179, 9:179, 10:180, 11:180.
Wait, actually, the second half should start from the 7th term. So the second half is 178,179,179,180,180. The median of these five is the 3rd term in this subset, which is the 9th term overall. The 9th term is 179. So Q3 is 179.
Therefore, IQR = Q3 - Q1 = 179 - 175 = 4. So statement ก) is correct. That's 4, so the first statement is correct.
Moving on to statement ข): The standard deviation is sqrt(56/11). Let's compute the standard deviation.
First, variance is the average of the squared differences from the mean. Since the mean is 177, we need to compute each (x_i - mean)^2, sum them, divide by N (since it's population variance, I assume here because it's the entire group of 11 athletes, not a sample), then take the square root for standard deviation.
Wait, but the problem says "ส่วนเบี่ยงเบนมาตรฐาน" which is standard deviation. The question says "มีค่าเท่ากับ sqrt(56/11)". Let me compute the variance first.
First, compute each (x_i - 177)^2:
Let me list all the data points with their deviations squared:
173: (173-177)^2 = (-4)^2 = 16
175: (175-177)^2 = (-2)^2 = 4
175: same as above, 4
175: 4
176: (176-177)^2 = 1
177: 0
178: (1)^2 =1
179: (2)^2=4
179:4
180: (3)^2=9
180:9
Now, let's sum all these squared differences:
16 + 4 +4 +4 +1 +0 +1 +4 +4 +9 +9.
Let me add step by step:
Start with 16.
Add 4: 20
Add 4:24
Add 4:28
Add 1:29
Add 0:29
Add 1:30
Add 4:34
Add 4:38
Add 9:47
Add 9:56.
Total sum of squared differences is 56.
Therefore, variance (σ²) is 56 divided by N=11. So variance is 56/11.
Standard deviation (σ) is the square root of variance, so sqrt(56/11). Therefore, statement ข) is correct.
Now statement ค): The variance is twice the standard deviation.
Variance is 56/11, standard deviation is sqrt(56/11). Let's see if 56/11 equals 2*sqrt(56/11).
Let me compute 2*sqrt(56/11). Let's see:
Let me denote x = sqrt(56/11). Then 2x = 2sqrt(56/11). The variance is (56/11). So is 56/11 equal to 2sqrt(56/11)?
Let me check numerically:
First, compute sqrt(56/11):
56 divided by 11 is approximately 5.0909. The square root of that is approx 2.258.
Then 2 times that is approx 4.516.
But the variance is 56/11 ≈5.0909, which is not equal to 4.516. Therefore, the variance is not twice the standard deviation. So statement ค) is incorrect.
Therefore, statements ก) and ข) are correct, and ค) is wrong.
Looking back at the options:
a) Only ก) is correct.
b) Only ข) is correct.
c) ก) and ข) only.
d) ข) and ค) only.
e) All three are correct.
So the correct answer is option c.
Wait, but let me double-check my calculations for variance and standard deviation again to be sure.
Sum of squared deviations was 56. Since variance is 56/11, that's correct. So standard deviation is sqrt(56/11). So statement ข) is correct. Statement ก) was correct with IQR=4. Statement ค) is wrong because variance is not twice the standard deviation. So yes, c is the correct answer.
ข้อมูลส่วนสูงเรียงลำดับแล้วมี 11 ค่า มัธยฐานคือค่ากลาง (ค่าที่ 6) ซึ่งเท่ากับ 177 ซึ่งตรงกับค่าเฉลี่ย ดังนั้น:
ดังนั้น ข้อความที่ถูกต้องคือ ก) และ ข) เท่านั้น
{"correct_answer_key":"c"}
เพื่อหาคำตอบที่ถูกต้อง เราต้องวิเคราะห์ข้อมูลที่ให้มา:
ดังนั้น ข้อความ ก) และ ข) เท่านั้นที่ถูกต้อง
{"correct_answer_key":"c"}
Alright! Let's tackle this problem step by step. The user provided a multiple-choice question in Thai about the height data of 11 athletes. The question involves calculating the interquartile range, standard deviation, and variance. The options suggest different combinations of correct statements.
First, I need to understand the given data: the heights are [173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180]. Since there are 11 numbers, the median is the 6th value, which is 177. That's straightforward.
Next, for the interquartile range (IQR), I need to find Q1 and Q3. To do this, I'll split the data into two halves. The lower half includes the first five numbers up to the median, and the upper half includes the remaining six numbers. Q1 is the median of the lower half, which is the 3rd number: 175. Q3 is the median of the upper half, which is the average of the 8th and 9th numbers: (179 + 179)/2 = 179. Therefore, IQR = Q3 - Q1 = 179 - 175 = 4. So statement ก) is correct.
Now, moving on to the standard deviation. The formula for sample standard deviation is the square root of the sum of squared deviations from the mean divided by (n-1). The mean height is calculated as the sum of all heights divided by 11. Let me compute that: 173+175+175+175+176+177+178+179+179+180+180. Adding these up: 173+175=348, +175=523, +175=698, +176=874, +177=1051, +178=1229, +179=1408, +179=1587, +180=1767, +180=1947. The mean is 1947 / 11 ≈ 176.09 cm.
Next, I calculate the squared deviations for each height:
Summing all these squared deviations and dividing by (11-1)=10 gives the variance. The user mentioned sqrt(56/11) as the standard deviation. Let's check: 56/11 ≈ 5.09, and the square root of that is approximately 2.25. Comparing this with the actual calculation, if the variance is indeed 56/11, then the standard deviation would be correct. However, I need to verify the exact variance.
Calculating each squared deviation precisely:
Adding these up: 9.55 + 1.19 + 1.19 + 1.19 + 0.0081 + 0.8281 + 3.6481 + 8.4681 + 8.4681 + 15.2881 + 15.2881. Let's sum them step by step:
Start with 9.55 + 1.19 = 10.74 +1.19 = 11.93 +1.19 = 13.12 +0.0081 ≈ 13.1281 +0.8281 ≈ 13.9562 +3.6481 ≈ 17.6043 +8.4681 ≈ 26.0724 +8.4681 ≈ 34.5405 +15.2881 ≈ 49.8286 +15.2881 ≈ 65.1167
So the total sum of squared deviations is approximately 65.1167. Variance = 65.1167 / 10 ≈ 6.51167. Therefore, the standard deviation is sqrt(6.51167) ≈ 2.55. However, the user's statement ข) claims it's sqrt(56/11) ≈ sqrt(5.09) ≈ 2.25, which doesn't match my calculation. This suggests that statement ข) might be incorrect.
Wait, maybe I made a mistake in calculations. Let me double-check the sum of squared deviations. Alternatively, perhaps the user intended a different approach. If we consider the population standard deviation (divided by n instead of n-1), then variance would be 65.1167 / 11 ≈ 5.92, and standard deviation ≈ 2.43, still not matching 56/11≈5.09. Therefore, statement ข) seems incorrect.
Finally, statement ค) says variance is twice the standard deviation. Variance is the square of the standard deviation, not twice. So statement ค) is incorrect.
Putting it all together: only statement ก) is correct. Therefore, the correct answer should be option a.
{"correct_answer_key":"a"}
Let's analyze the given statements about the heights of 11 athletes.
The data is: 173, 175, 175, 175, 176, 177, 178, 179, 179, 180, 180.
a) Interquartile range (IQR): Q1 is the median of the lower half (173, 175, 175, 175, 176). Q1 = 175. Q3 is the median of the upper half (179, 179, 180, 180). Q3 = 179.5. IQR = Q3 - Q1 = 179.5 - 175 = 4.5. So, statement a) is incorrect.
b) Standard deviation: First, we need to calculate the variance. Mean (μ) = (173 + 1753 + 176 + 177 + 178 + 1792 + 1802) / 11 = (173 + 525 + 176 + 177 + 178 + 358 + 360) / 11 = 1947 / 11 = 177. Variance (σ^2) = Σ(xi - μ)^2 / (n-1) = [(173-177)^2 + (175-177)^23 + (176-177)^2 + (177-177)^2 + (178-177)^2 + (179-177)^22 + (180-177)^22] / 10 = [16 + 43 + 1 + 0 + 1 + 42 + 9*2] / 10 = (16 + 12 + 1 + 0 + 1 + 8 + 18) / 10 = 56 / 10 = 5.6. Standard deviation (σ) = sqrt(5.6) = sqrt(56/10) = sqrt(28/5). The given value is sqrt(56/11) ≈ 2.25. sqrt(5.6) ≈ 2.37. So, statement b) is incorrect.
c) Variance is two times the standard deviation: Variance = 5.6. 2 * Standard deviation = 2 * sqrt(5.6) ≈ 4.74. So, statement c) is incorrect.
However, let's recalculate the variance: Variance (σ^2) = Σ(xi - μ)^2 / n = [(173-177)^2 + (175-177)^23 + (176-177)^2 + (177-177)^2 + (178-177)^2 + (179-177)^22 + (180-177)^22] / 11 = [16 + 43 + 1 + 0 + 1 + 42 + 92] / 11 = (16 + 12 + 1 + 0 + 1 + 8 + 18) / 11 = 56 / 11. Standard deviation (σ) = sqrt(56/11) ≈ 2.25. 2 * Standard deviation = 2 * sqrt(56/11) ≈ 4.5. So, statement c) is still incorrect.
Let's re-examine statement a): Q1 = 175. Q3 = 179.5. IQR = Q3 - Q1 = 179.5 - 175 = 4.5. The statement says IQR = 4, which is incorrect.
Let's re-examine statement b): Variance = 56/11. Standard deviation = sqrt(56/11). The statement is correct.
Let's re-examine statement c): Variance = 56/11. 2 * Standard deviation = 2 * sqrt(56/11). The statement is incorrect.
Therefore, only statement b) is correct.
{"correct_answer_key":"b"}