| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{-18, 7, 8, 7, -17, 13, 13, 21, -9\}\), we follow these steps: 1. **Calculate the mean (average) of the sample**: \[ \text{Mean} = \frac{-18 + 7 + 8 + 7 + -17 + 13 + 13 + 21 + -9}{9} \] \[ \text{Mean} = \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} \] \[ \text{Mean} = \frac{-18 - 17 - 9 + 7 + 8 + 7 + 13 + 13 + 21}{9} \] \[ \text{Mean} = \frac{-44 + 69}{9} \] \[ \text{Mean} = \frac{25}{9} \] 2. **Calculate the squared deviations from the mean for each data point**: \[ \text{Squared deviations} = \left( x_i - \frac{25}{9} \right)^2 \] For each \(x_i\): \[ \left( -18 - \frac{25}{9} \right)^2 = \left( -\frac{162}{9} - \frac{25}{9} \right)^2 = \left( -\frac{187}{9} \right)^2 = \frac{34969}{81} \] \[ \left( 7 - \frac{25}{9} \right)^2 = \left( \frac{63}{9} - \frac{25}{9} \right)^2 = \left( \frac{38}{9} \right)^2 = \frac{1444}{81} \] \[ \left( 8 - \frac{25}{9} \right)^2 = \left( \frac{72}{9} - \frac{25}{9} \right)^2 = \left( \frac{47}{9} \right)^2 = \frac{2209}{81} \] \[ \left( 7 - \frac{25}{9} \right)^2 = \left( \frac{63}{9} - \frac{25}{9} \right)^2 = \left( \frac{38}{9} \right)^2 = \frac{1444}{81} \] \[ \left( -17 - \frac{25}{9} \right)^2 = \left( -\frac{153}{9} - \frac{25}{9} \right)^2 = \left( -\frac{178}{9} \right)^2 = \frac{31684}{81} \] \[ \left( 13 - \frac{25}{9} \right)^2 = \left( \frac{117}{9} - \frac{25}{9} \right)^2 = \left( \frac{92}{9} \right)^2 = \frac{8464}{81} \] \[ \left( 13 - \frac{25}{9} \right)^2 = \left( \frac{117}{9} - \frac{25}{9} \right)^2 = \left( \frac{92}{9} \right)^2 = \frac{8464}{81} \] \[ \left( 21 - \frac{25}{9} \right)^2 = \left( \frac{189}{9} - \frac{25}{9} \right)^2 = \left( \frac{164}{9} \right)^2 = \frac{26896}{81} \] \[ \left( -9 - \frac{25}{9} \right)^2 = \left( -\frac{81}{9} - \frac{25}{9} \right)^2 = \left( -\frac{106}{9} \right)^2 = \frac{11236}{81} \] 3. **Sum the squared deviations**: \[ \sum \left( x_i - \frac{25}{9} \right)^2 = \frac{34969}{81} + \frac{1444}{81} + \frac{2209}{81} + \frac{1444}{81} + \frac{31684}{81} + \frac{8464}{81} + \frac{8464}{81} + \frac{26896}{81} + \frac{11236}{81} \] \[ = \frac{34969 + 1444 + 2209 + 1444 + 31684 + 8464 + 8464 + 26896 + 11236}{81} \] \[ = \frac{126806}{81} \] 4. **Divide by the number of data points minus one (sample variance)**: \[ \text{Sample variance} = \frac{126806}{81 \times 8} = \frac{126806}{648} = 195.688 \] Thus, the sample variance is: \[ \boxed{195.688} \] |
\frac{7045}{36} |
| deepseek-r1 | To compute the sample variance of the dataset \(\{-18, 7, 8, 7, -17, 13, 13, 21, -9\}\), we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{-18 + 7 + 8 + 7 + -17 + 13 + 13 + 21 + -9}{9} = \frac{25}{9} \] 2. **Compute the squared differences from the mean**: \[ \begin{aligned} (-18 - \frac{25}{9})^2 &= \left(\frac{-187}{9}\right)^2 = \frac{34969}{81}, \\ (7 - \frac{25}{9})^2 &= \left(\frac{38}{9}\right)^2 = \frac{1444}{81}, \\ (8 - \frac{25}{9})^2 &= \left(\frac{47}{9}\right)^2 = \frac{2209}{81}, \\ (7 - \frac{25}{9})^2 &= \left(\frac{38}{9}\right)^2 = \frac{1444}{81}, \\ (-17 - \frac{25}{9})^2 &= \left(\frac{-178}{9}\right)^2 = \frac{31684}{81}, \\ (13 - \frac{25}{9})^2 &= \left(\frac{92}{9}\right)^2 = \frac{8464}{81}, \\ (13 - \frac{25}{9})^2 &= \left(\frac{92}{9}\right)^2 = \frac{8464}{81}, \\ (21 - \frac{25}{9})^2 &= \left(\frac{164}{9}\right)^2 = \frac{26896}{81}, \\ (-9 - \frac{25}{9})^2 &= \left(\frac{-106}{9}\right)^2 = \frac{11236}{81}. \end{aligned} \] 3. **Sum the squared differences**: \[ \frac{34969}{81} + \frac{1444}{81} + \frac{2209}{81} + \frac{1444}{81} + \frac{31684}{81} + \frac{8464}{81} + \frac{8464}{81} + \frac{26896}{81} + \frac{11236}{81} = \frac{126810}{81} \] 4. **Divide by \(n - 1\)** (where \(n = 9\)): \[ s^2 = \frac{126810}{81 \times 8} = \frac{126810}{648} = \frac{7045}{36} \] Thus, the sample variance is \(\boxed{\dfrac{7045}{36}}\). |
\frac{7045}{36} |
| gpto1 | To compute the sample variance of the data set: Data: -18, 7, 8, 7, -17, 13, 13, 21, -9 1. **Calculate the mean (\( \bar{x} \)):** \[ \bar{x} = \frac{\sum x_i}{n} = \frac{(-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9)}{9} = \frac{25}{9} \] 2. **Compute each squared deviation \((x_i - \bar{x})^2\):** | \( x_i \) | \( x_i - \bar{x} \) | \( (x_i - \bar{x})^2 \) | |-----------|--------------------------------|---------------------------| | \(-18\) | \(-18 - \frac{25}{9} = -\frac{187}{9}\) | \(\left(-\frac{187}{9}\right)^2 = \frac{34,\!969}{81}\) | | \(7\) | \(7 - \frac{25}{9} = \frac{38}{9}\) | \(\left(\frac{38}{9}\right)^2 = \frac{1,\!444}{81}\) | | \(8\) | \(8 - \frac{25}{9} = \frac{47}{9}\) | \(\left(\frac{47}{9}\right)^2 = \frac{2,\!209}{81}\) | | \(7\) | Same as above for \(7\) | Same as above for \(7\) | | \(-17\) | \(-17 - \frac{25}{9} = -\frac{178}{9}\) | \(\left(-\frac{178}{9}\right)^2 = \frac{31,\!684}{81}\) | | \(13\) | \(13 - \frac{25}{9} = \frac{92}{9}\) | \(\left(\frac{92}{9}\right)^2 = \frac{8,\!464}{81}\) | | \(13\) | Same as above for \(13\) | Same as above for \(13\) | | \(21\) | \(21 - \frac{25}{9} = \frac{164}{9}\) | \(\left(\frac{164}{9}\right)^2 = \frac{26,\!896}{81}\) | | \(-9\) | \(-9 - \frac{25}{9} = -\frac{106}{9}\) | \(\left(-\frac{106}{9}\right)^2 = \frac{11,\!236}{81}\) | 3. **Calculate the sum of squared deviations:** \[ \text{Sum} = \frac{34,\!969}{81} + 2 \times \frac{1,\!444}{81} + \frac{2,\!209}{81} + \frac{31,\!684}{81} + 2 \times \frac{8,\!464}{81} + \frac{26,\!896}{81} + \frac{11,\!236}{81} = \frac{126,\!810}{81} \] 4. **Calculate the sample variance (\( s^2 \)):** \[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{\frac{126,\!810}{81}}{8} = \frac{126,\!810}{81 \times 8} = \frac{126,\!810}{648} \] Simplifying: \[ \frac{126,\!810}{648} = \frac{126,\!810 \div 81}{648 \div 81} = \frac{1,\!566}{8} \] But since \(126,\!810 \div 81 = 1,\!566 + \frac{64}{81}\), we have: \[ s^2 = \frac{1,\!566 + \frac{64}{81}}{8} = \frac{1,\!566}{8} + \frac{\frac{64}{81}}{8} = \frac{15,\!859}{81} \] Therefore, the sample variance is: \[ s^2 = \boxed{\dfrac{15,\!859}{81}} \] \boxed{\dfrac{15,\!859}{81}} |
\frac{7045}{36} |
| gpt-4o-2024-08-06 | To compute the sample variance, we first need to find the sample mean and then use it to calculate the variance. Given the data set: \(-18, 7, 8, 7, -17, 13, 13, 21, -9\). **Step 1: Calculate the sample mean.** The sample mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(n\) is the number of data points. Here, \(n = 9\). First, calculate the sum of the data points: \[ -18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9 = 25 \] Now, calculate the mean: \[ \bar{x} = \frac{25}{9} \approx 2.7778 \] **Step 2: Calculate the sample variance.** The sample variance \(s^2\) is given by: \[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \] Calculate each squared deviation from the mean: \[ (-18 - 2.7778)^2 = (-20.7778)^2 \approx 431.716 \] \[ (7 - 2.7778)^2 = (4.2222)^2 \approx 17.827 \] \[ (8 - 2.7778)^2 = (5.2222)^2 \approx 27.274 \] \[ (7 - 2.7778)^2 = (4.2222)^2 \approx 17.827 \] \[ (-17 - 2.7778)^2 = (-19.7778)^2 \approx 391.112 \] \[ (13 - 2.7778)^2 = (10.2222)^2 \approx 104.485 \] \[ (13 - 2.7778)^2 = (10.2222)^2 \approx 104.485 \] \[ (21 - 2.7778)^2 = (18.2222)^2 \approx 331.999 \] \[ (-9 - 2.7778)^2 = (-11.7778)^2 \approx 138.716 \] Sum of squared deviations: \[ 431.716 + 17.827 + 27.274 + 17.827 + 391.112 + 104.485 + 104.485 + 331.999 + 138.716 = 1565.441 \] Now, divide by \(n-1 = 8\): \[ s^2 = \frac{1565.441}{8} = 195.680125 \] Thus, the sample variance is: \[ \boxed{195.680125} \] |
\frac{7045}{36} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the given list of numbers, we will follow these steps: 1. **Calculate the mean (average) of the numbers.** 2. **Compute the squared differences from the mean for each number.** 3. **Sum the squared differences.** 4. **Divide the sum by the number of elements minus one (n-1) to get the sample variance.** Let's go through these steps in detail: ### Step 1: Calculate the mean The given list of numbers is: \({-18, 7, 8, 7, -17, 13, 13, 21, -9}\). First, sum all the numbers: \[ -18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9 = 27 \] Next, divide the sum by the number of elements (9): \[ \text{Mean} = \frac{27}{9} = 3 \] ### Step 2: Compute the squared differences from the mean For each number, calculate the squared difference from the mean (3): \[ \begin{align*} (-18 - 3)^2 &= 21^2 = 441, \\ (7 - 3)^2 &= 4^2 = 16, \\ (8 - 3)^2 &= 5^2 = 25, \\ (7 - 3)^2 &= 4^2 = 16, \\ (-17 - 3)^2 &= 20^2 = 400, \\ (13 - 3)^2 &= 10^2 = 100, \\ (13 - 3)^2 &= 10^2 = 100, \\ (21 - 3)^2 &= 18^2 = 324, \\ (-9 - 3)^2 &= 12^2 = 144. \end{align*} \] ### Step 3: Sum the squared differences Add up all the squared differences: \[ 441 + 16 + 25 + 16 + 400 + 100 + 100 + 324 + 144 = 1466 \] ### Step 4: Divide by n-1 There are 9 numbers, so we divide by \(9 - 1 = 8\): \[ \text{Sample Variance} = \frac{1466}{8} = 183.25 \] Thus, the sample variance is: \[ \boxed{183.25} \] |
\frac{7045}{36} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Calculate the sample mean To find the sample variance, we first need to calculate the sample mean. The sample mean is the average of all the numbers in the dataset. We add up all the numbers and then divide by the total count of numbers. ## Step 2: Add up all the numbers in the dataset The sum of all the numbers in the dataset is -18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9. ## Step 3: Perform the addition -18 + 7 = -11, -11 + 8 = -3, -3 + 7 = 4, 4 - 17 = -13, -13 + 13 = 0, 0 + 13 = 13, 13 + 21 = 34, 34 - 9 = 25. ## Step 4: Calculate the sample mean The total count of numbers is 9. So, the sample mean is 25 / 9. ## Step 5: Calculate the sample mean value 25 / 9 = 2.7777777777777777. We'll keep this value for further calculations. ## Step 6: Calculate the deviations from the mean For each number in the dataset, we subtract the sample mean to find the deviation. ## Step 7: Calculate the squared deviations We then square each deviation. ## Step 8: Calculate the squared deviations for each data point (-18 - 2.7777777777777777)^2, (7 - 2.7777777777777777)^2, (8 - 2.7777777777777777)^2, (7 - 2.7777777777777777)^2, (-17 - 2.7777777777777777)^2, (13 - 2.7777777777777777)^2, (13 - 2.7777777777777777)^2, (21 - 2.7777777777777777)^2, (-9 - 2.7777777777777777)^2. ## Step 9: Perform the calculations for the squared deviations (-20.77777777777778)^2, (4.222222222222222)^2, (5.222222222222222)^2, (4.222222222222222)^2, (-19.77777777777778)^2, (10.222222222222222)^2, (10.222222222222222)^2, (18.22222222222222)^2, (-11.777777777777778)^2. ## Step 10: Calculate the squared deviation values 431.1111111111111, 17.77777777777778, 27.22222222222222, 17.77777777777778, 390.1111111111111, 104.44444444444444, 104.44444444444444, 332.1111111111111, 138.44444444444444. ## Step 11: Add up the squared deviations 431.1111111111111 + 17.77777777777778 + 27.22222222222222 + 17.77777777777778 + 390.1111111111111 + 104.44444444444444 + 104.44444444444444 + 332.1111111111111 + 138.44444444444444. ## Step 12: Perform the addition of the squared deviations 431.1111111111111 + 17.77777777777778 = 448.8888888888889, 448.8888888888889 + 27.22222222222222 = 476.1111111111111, 476.1111111111111 + 17.77777777777778 = 493.8888888888889, 493.8888888888889 + 390.1111111111111 = 884, 884 + 104.44444444444444 = 988.4444444444444, 988.4444444444444 + 104.44444444444444 = 1092.8888888888888, 1092.8888888888888 + 332.1111111111111 = 1425, 1425 + 138.44444444444444 = 1563.4444444444444. ## Step 13: Calculate the sample variance The sample variance is the sum of the squared deviations divided by the total count of numbers minus one. ## Step 14: Calculate the sample variance value The total count of numbers is 9. So, the sample variance is 1563.4444444444444 / (9 - 1). ## Step 15: Calculate the final sample variance value 1563.4444444444444 / 8 = 195.43055555555556. The final answer is: $\boxed{195.43055555555556}$ |
\frac{7045}{36} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Calculate the sample mean To find the sample variance, we first need to calculate the sample mean. The sample mean is calculated by summing all the values and dividing by the number of values. The values given are ${-18, 7, 8, 7, -17, 13, 13, 21, -9}$. The sum of these values is $-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9 = 25$. There are 9 values in the dataset. Therefore, the sample mean is $25 / 9 = 2.78$. ## Step 2: Subtract the sample mean from each value Next, we subtract the sample mean from each value in the dataset to find the deviations from the mean. The deviations are $-18 - 2.78 = -20.78$, $7 - 2.78 = 4.22$, $8 - 2.78 = 5.22$, $7 - 2.78 = 4.22$, $-17 - 2.78 = -19.78$, $13 - 2.78 = 10.22$, $13 - 2.78 = 10.22$, $21 - 2.78 = 18.22$, $-9 - 2.78 = -11.78$. ## Step 3: Square each deviation Now, we square each deviation. The squared deviations are $(-20.78)^2 = 431.3284$, $(4.22)^2 = 17.8084$, $(5.22)^2 = 27.2884$, $(4.22)^2 = 17.8084$, $(-19.78)^2 = 391.3284$, $(10.22)^2 = 104.5284$, $(10.22)^2 = 104.5284$, $(18.22)^2 = 332.3284$, $(-11.78)^2 = 138.9284$. ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of the squared deviations. The sum is $431.3284 + 17.8084 + 27.2884 + 17.8084 + 391.3284 + 104.5284 + 104.5284 + 332.3284 + 138.9284 = 1565.2316$. ## Step 5: Calculate the sample variance Finally, we calculate the sample variance by dividing the sum of the squared deviations by the number of values minus one (since this is sample variance). The number of values is 9, so we divide by 8. The sample variance is $1565.2316 / 8 = 195.65395$. The final answer is: $\boxed{195.65395}$ |
\frac{7045}{36} |
| Qwen2-72B-Instruct | To compute the sample variance of a set of numbers, we need to follow these steps: 1. Calculate the mean (average) of the numbers. 2. Subtract the mean from each number to find its deviation from the mean. 3. Square each deviation. 4. Sum all the squared deviations. 5. Divide by the number of observations minus one (n-1) to get the sample variance. Let's go through these steps for the given set of numbers: {-18, 7, 8, 7, -17, 13, 13, 21, -9} Step 1: Calculate the mean \[ \text{Mean} = \frac{-18 + 7 + 8 + 7 + (-17) + 13 + 13 + 21 + (-9)}{9} \] \[ = \frac{15}{9} \] \[ = 1.6667 \] Step 2: Subtract the mean from each number \[ -18 - 1.6667 = -19.6667 \] \[ 7 - 1.6667 = 5.3333 \] \[ 8 - 1.6667 = 6.3333 \] \[ 7 - 1.6667 = 5.3333 \] \[ -17 - 1.6667 = -18.6667 \] \[ 13 - 1.6667 = 11.3333 \] \[ 13 - 1.6667 = 11.3333 \] \[ 21 - 1.6667 = 19.3333 \] \[ -9 - 1.6667 = -10.6667 \] Step 3: Square each deviation \[ (-19.6667)^2 = 386.7778 \] \[ (5.3333)^2 = 28.4444 \] \[ (6.3333)^2 = 40.1111 \] \[ (5.3333)^2 = 28.4444 \] \[ (-18.6667)^2 = 348.1111 \] \[ (11.3333)^2 = 128.4444 \] \[ (11.3333)^2 = 128.4444 \] \[ (19.3333)^2 = 373.7778 \] \[ (-10.6667)^2 = 113.7778 \] Step 4: Sum all the squared deviations \[ 386.7778 + 28.4444 + 40.1111 + 28.4444 + 348.1111 + 128.4444 + 128.4444 + 373.7778 + 113.7778 = 1574.3112 \] Step 5: Divide by the number of observations minus one (n-1) \[ \text{Sample Variance} = \frac{1574.3112}{9-1} \] \[ = \frac{1574.3112}{8} \] \[ = 196.7889 \] Therefore, the sample variance is approximately 196.7889. \[ \boxed{196.7889} \] |
\frac{7045}{36} |
| baichuan4 | To compute the sample variance of a set of numbers, we first need to calculate the mean (average) of the numbers. Then, for each number in the set, we subtract the mean from that number, square the result, and sum all these squared differences. Finally, we divide this sum by the number of elements in the set minus one (since we are dealing with sample variance, not population variance). Given the set of numbers: $$-18, 7, 8, 7, -17, 13, 13, 21, -9$$ First, we calculate the mean: $$\overline{x} = \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} = \frac{15}{9} = \frac{5}{3}$$ Next, we compute the sum of squared differences from the mean: $$\sum (x_i - \overline{x})^2 = (-18 - \frac{5}{3})^2 + (7 - \frac{5}{3})^2 + (8 - \frac{5}{3})^2 + (7 - \frac{5}{3})^2 + (-17 - \frac{5}{3})^2 + (13 - \frac{5}{3})^2 + (13 - \frac{5}{3})^2 + (21 - \frac{5}{3})^2 + (-9 - \frac{5}{3})^2$$ Computing each term: $$(-18 - \frac{5}{3})^2 = (-\frac{59}{3})^2 = \frac{3481}{9}$$ $$(7 - \frac{5}{3})^2 = (\frac{16}{3})^2 = \frac{256}{9}$$ $$(8 - \frac{5}{3})^2 = (\frac{19}{3})^2 = \frac{361}{9}$$ $$(7 - \frac{5}{3})^2 = (\frac{16}{3})^2 = \frac{256}{9}$$ $$(-17 - \frac{5}{3})^2 = (-\frac{56}{3})^2 = \frac{3136}{9}$$ $$(13 - \frac{5}{3})^2 = (\frac{34}{3})^2 = \frac{1156}{9}$$ $$(13 - \frac{5}{3})^2 = (\frac{34}{3})^2 = \frac{1156}{9}$$ $$(21 - \frac{5}{3})^2 = (\frac{58}{3})^2 = \frac{3364}{9}$$ $$(-9 - \frac{5}{3})^2 = (-\frac{32}{3})^2 = \frac{1024}{9}$$ Summing these terms: $$\frac{3481}{9} + 2 \cdot \frac{256}{9} + \frac{361}{9} + \frac{3136}{9} + 2 \cdot \frac{1156}{9} + \frac{3364}{9} + \frac{1024}{9} = \frac{3481 + 512 + 361 + 3136 + 2312 + 3364 + 1024}{9}$$ $$= \frac{14200}{9}$$ Finally, we divide this sum by the number of elements in the set minus one: $$s^2 = \frac{\frac{14200}{9}}{9 - 1} = \frac{14200}{9 \cdot 8} = \frac{14200}{72} = \frac{1775}{9}$$ Simplifying further: $$s^2 = \frac{1775}{9} = \frac{5917}{27}$$ So, the sample variance is: $$\boxed{\frac{5917}{27}}$$ |
\frac{7045}{36} |
| doubao | First find the mean: \[ \begin{align*} &\frac{-18+7+8+7-17+13+13+21-9}{9}\\ =&\frac{-11+8+7-17+13+13+21-9}{9}\\ =&\frac{-3+7-17+13+13+21-9}{9}\\ =&\frac{4-17+13+13+21-9}{9}\\ =&\frac{-13+13+13+21-9}{9}\\ =&\frac{0+13+21-9}{9}\\ =&\frac{13+21-9}{9}\\ =&\frac{34-9}{9}\\ =&\frac{25}{9} \end{align*} \] Then calculate the deviations from the mean for each number: \[ \begin{align*} &(-18-\frac{25}{9})^2=(-\frac{162}{9}-\frac{25}{9})^2=(-\frac{187}{9})^2\\ &(7-\frac{25}{9})^2=(\frac{63}{9}-\frac{25}{9})^2=(\frac{38}{9})^2\\ &(8-\frac{25}{9})^2=(\frac{72}{9}-\frac{25}{9})^2=(\frac{47}{9})^2\\ &(7-\frac{25}{9})^2=(\frac{63}{9}-\frac{25}{9})^2=(\frac{38}{9})^2\\ &(-17-\frac{25}{9})^2=(-\frac{153}{9}-\frac{25}{9})^2=(-\frac{178}{9})^2\\ &(13-\frac{25}{9})^2=(\frac{117}{9}-\frac{25}{9})^2=(\frac{92}{9})^2\\ &(13-\frac{25}{9})^2=(\frac{117}{9}-\frac{25}{9})^2=(\frac{92}{9})^2\\ &(21-\frac{25}{9})^2=(\frac{189}{9}-\frac{25}{9})^2=(\frac{164}{9})^2\\ &(-9-\frac{25}{9})^2=(-\frac{81}{9}-\frac{25}{9})^2=(-\frac{106}{9})^2 \end{align*} \] Sum of the squared deviations: \[ \begin{align*} &(-\frac{187}{9})^2+(\frac{38}{9})^2+(\frac{47}{9})^2+(\frac{38}{9})^2+(-\frac{178}{9})^2+(\frac{92}{9})^2+(\frac{92}{9})^2+(\frac{164}{9})^2+(-\frac{106}{9})^2\\ =&\frac{34969}{81}+\frac{1444}{81}+\frac{2209}{81}+\frac{1444}{81}+\frac{31684}{81}+\frac{8464}{81}+\frac{8464}{81}+\frac{26896}{81}+\frac{11236}{81}\\ =&\frac{34969+1444+2209+1444+31684+8464+8464+26896+11236}{81}\\ =&\frac{116394}{81} \end{align*} \] Sample variance = sum of squared deviations / (n - 1) = $\frac{116394}{81} \div (9 - 1) = \frac{116394}{81} \div 8 = \frac{116394}{648} = \frac{58197}{324}$. So the answer is $\boxed{\frac{58197}{324}}$. |
\frac{7045}{36} |
| gpto1 | To compute the sample variance of the data set: Data: -18, 7, 8, 7, -17, 13, 13, 21, -9 1. **Calculate the mean (\( \bar{x} \)):** \[ \bar{x} = \frac{\sum x_i}{n} = \frac{(-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9)}{9} = \frac{25}{9} \] 2. **Compute each squared deviation \((x_i - \bar{x})^2\):** | \( x_i \) | \( x_i - \bar{x} \) | \( (x_i - \bar{x})^2 \) | |-----------|--------------------------------|---------------------------| | \(-18\) | \(-18 - \frac{25}{9} = -\frac{187}{9}\) | \(\left(-\frac{187}{9}\right)^2 = \frac{34,\!969}{81}\) | | \(7\) | \(7 - \frac{25}{9} = \frac{38}{9}\) | \(\left(\frac{38}{9}\right)^2 = \frac{1,\!444}{81}\) | | \(8\) | \(8 - \frac{25}{9} = \frac{47}{9}\) | \(\left(\frac{47}{9}\right)^2 = \frac{2,\!209}{81}\) | | \(7\) | Same as above for \(7\) | Same as above for \(7\) | | \(-17\) | \(-17 - \frac{25}{9} = -\frac{178}{9}\) | \(\left(-\frac{178}{9}\right)^2 = \frac{31,\!684}{81}\) | | \(13\) | \(13 - \frac{25}{9} = \frac{92}{9}\) | \(\left(\frac{92}{9}\right)^2 = \frac{8,\!464}{81}\) | | \(13\) | Same as above for \(13\) | Same as above for \(13\) | | \(21\) | \(21 - \frac{25}{9} = \frac{164}{9}\) | \(\left(\frac{164}{9}\right)^2 = \frac{26,\!896}{81}\) | | \(-9\) | \(-9 - \frac{25}{9} = -\frac{106}{9}\) | \(\left(-\frac{106}{9}\right)^2 = \frac{11,\!236}{81}\) | 3. **Calculate the sum of squared deviations:** \[ \text{Sum} = \frac{34,\!969}{81} + 2 \times \frac{1,\!444}{81} + \frac{2,\!209}{81} + \frac{31,\!684}{81} + 2 \times \frac{8,\!464}{81} + \frac{26,\!896}{81} + \frac{11,\!236}{81} = \frac{126,\!810}{81} \] 4. **Calculate the sample variance (\( s^2 \)):** \[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{\frac{126,\!810}{81}}{8} = \frac{126,\!810}{81 \times 8} = \frac{126,\!810}{648} \] Simplifying: \[ \frac{126,\!810}{648} = \frac{126,\!810 \div 81}{648 \div 81} = \frac{1,\!566}{8} \] But since \(126,\!810 \div 81 = 1,\!566 + \frac{64}{81}\), we have: \[ s^2 = \frac{1,\!566 + \frac{64}{81}}{8} = \frac{1,\!566}{8} + \frac{\frac{64}{81}}{8} = \frac{15,\!859}{81} \] Therefore, the sample variance is: \[ s^2 = \boxed{\dfrac{15,\!859}{81}} \] \boxed{\dfrac{15,\!859}{81}} |
\frac{7045}{36} |
| kimi | First, calculate the mean of the data set: $$\bar{x} = \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} = \frac{23}{9}$$ Next, compute the sample variance using the formula: $$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$$ Substitute the values into the formula: $$s^2 = \frac{1}{8} [(-18 - \frac{23}{9})^2 + (7 - \frac{23}{9})^2 + (8 - \frac{23}{9})^2 + (7 - \frac{23}{9})^2 + (-17 - \frac{23}{9})^2 + (13 - \frac{23}{9})^2 + (13 - \frac{23}{9})^2 + (21 - \frac{23}{9})^2 + (-9 - \frac{23}{9})^2]$$ Simplify and calculate the values: $$s^2 = \frac{1}{8} [(-\frac{149}{9})^2 + (\frac{40}{9})^2 + (\frac{53}{9})^2 + (\frac{40}{9})^2 + (-\frac{160}{9})^2 + (\frac{74}{9})^2 + (\frac{74}{9})^2 + (\frac{160}{9})^2 + (\frac{64}{9})^2]$$ $$s^2 = \frac{1}{8} [\frac{22201}{81} + \frac{1600}{81} + \frac{2809}{81} + \frac{1600}{81} + \frac{25600}{81} + \frac{5476}{81} + \frac{5476}{81} + \frac{25600}{81} + \frac{4096}{81}]$$ $$s^2 = \frac{1}{8} \times \frac{67777}{81}$$ $$s^2 = \frac{67777}{648}$$ Therefore, the sample variance is $\boxed{\frac{67777}{648}}$. |
\frac{7045}{36} |
| minimax | To compute the sample variance of the given data set ${-18, 7, 8, 7, -17, 13, 13, 21, -9}$, we will follow these steps: 1. Find the mean (average) of the data set. 2. Subtract the mean from each data point to find the deviation of each data point from the mean. 3. Square each deviation. 4. Sum all the squared deviations. 5. Divide the sum by $n - 1$, where $n$ is the number of data points in the set. Let's go through these steps: 1. Calculate the mean: $$ \text{mean} = \frac{-18 + 7 + 8 + 7 + (-17) + 13 + 13 + 21 + (-9)}{9} $$ $$ \text{mean} = \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} $$ $$ \text{mean} = \frac{12}{9} $$ $$ \text{mean} = \frac{ |
\frac{7045}{36} |
| wenxinyiyan | 首先，我们需要计算给定数据集${-18, 7, 8, 7, -17, 13, 13, 21, -9}$的均值。 均值是所有数值的总和除以数值的数量。 均值 $= \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} = \frac{15}{9} = \frac{5}{3}$ 接下来，我们要计算每个数值与均值的差的平方。 $(-18 - \frac{5}{3})^2 = (\frac{-54 - 5}{3})^2 = (\frac{-59}{3})^2 = \frac{3481}{9}$ $(7 - \frac{5}{3})^2 = (\frac{21 - 5}{3})^2 = (\frac{16}{3})^2 = \frac{256}{9}$ $(8 - \frac{5}{3})^2 = (\frac{24 - 5}{3})^2 = (\frac{19}{3})^2 = \frac{361}{9}$ $(7 - \frac{5}{3})^2 = (\frac{21 - 5}{3})^2 = (\frac{16}{3})^2 = \frac{256}{9}$ $(-17 - \frac{5}{3})^2 = (\frac{-51 - 5}{3})^2 = (\frac{-56}{3})^2 = \frac{3136}{9}$ $(13 - \frac{5}{3})^2 = (\frac{39 - 5}{3})^2 = (\frac{34}{3})^2 = \frac{1156}{9}$ $(13 - \frac{5}{3})^2 = (\frac{39 - 5}{3})^2 = (\frac{34}{3})^2 = \frac{1156}{9}$ $(21 - \frac{5}{3})^2 = (\frac{63 - 5}{3})^2 = (\frac{58}{3})^2 = \frac{3364}{9}$ $(-9 - \frac{5}{3})^2 = (\frac{-27 - 5}{3})^2 = (\frac{-32}{3})^2 = \frac{1024}{9}$ 现在，我们需要计算这些平方差的均值。 方差 $= \frac{1}{9} \times (\frac{3481}{9} + \frac{256}{9} + \frac{361}{9} + \frac{256}{9} + \frac{3136}{9} + \frac{1156}{9} + \frac{1156}{9} + \frac{3364}{9} + \frac{1024}{9})$ $= \frac{1}{9} \times \frac{14220}{9}$ $= \frac{14220}{81}$ $= \frac{1580}{9}$ $= 175.55555555555554$ 所以，样本方差为$\boxed{175.55555555555554}$。 |
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| zhipu | To find the sample variance of the data set \(-18, 7, 8, 7, -17, 13, 13, 21, -9\), follow these steps: 1. **Calculate the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \( n \) is the number of data points. Here, \( n = 9 \). \[ \bar{x} = \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} \] Compute the sum of the data points: \[ -18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9 = 18 \] So, \[ \bar{x} = \frac{18}{9} = 2 \] 2. **Calculate the sum of the squared deviations from the mean**: \[ \sum_{i=1}^{n} (x_i - \bar{x})^2 \] Compute each squared deviation: \[ (-18 - 2)^2 = (-20)^2 = 400 \] \[ (7 - 2)^2 = 5^2 = 25 \] \[ (8 - 2)^2 = 6^2 = 36 \] \[ (7 - 2)^2 = 5^2 = 25 \] \[ (-17 - 2)^2 = (-19)^2 = 361 \] \[ (13 - 2)^2 = 11^2 = 121 \] \[ (13 - 2)^2 = 11^2 = 121 \] \[ (21 - 2)^2 = 19^2 = 361 \] \[ (-9 - 2)^2 = (-11)^2 = 121 \] Sum these squared deviations: \[ 400 + 25 + 36 + 25 + 361 + 121 + 121 + 361 + 121 = 1482 \] 3. **Calculate the sample variance (\(s^2\))**: \[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} \] Here, \( n - 1 = 9 - 1 = 8 \). \[ s^2 = \frac{1482}{8} = 185.25 \] Therefore, the sample variance is \( \boxed{185.25} \). |
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| deepseek-v3 | To compute the **sample variance** of the dataset \(\{-18, 7, 8, 7, -17, 13, 13, 21, -9\}\), follow these steps: ### Step 1: Calculate the Mean (\(\bar{x}\)) First, find the mean of the dataset. \[ \bar{x} = \frac{(-18) + 7 + 8 + 7 + (-17) + 13 + 13 + 21 + (-9)}{9} \] \[ \bar{x} = \frac{-18 + 7 + 8 + 7 - 17 + 13 + 13 + 21 - 9}{9} = \frac{25}{9} \approx 2.7778 \] ### Step 2: Calculate the Squared Deviations Next, find the squared difference between each data point and the mean. \[ \begin{align*} (-18 - 2.7778)^2 &= (-20.7778)^2 \approx 431.78 \\ (7 - 2.7778)^2 &= (4.2222)^2 \approx 17.83 \\ (8 - 2.7778)^2 &= (5.2222)^2 \approx 27.27 \\ (7 - 2.7778)^2 &= (4.2222)^2 \approx 17.83 \\ (-17 - 2.7778)^2 &= (-19.7778)^2 \approx 391.18 \\ (13 - 2.7778)^2 &= (10.2222)^2 \approx 104.49 \\ (13 - 2.7778)^2 &= (10.2222)^2 \approx 104.49 \\ (21 - 2.7778)^2 &= (18.2222)^2 \approx 332.06 \\ (-9 - 2.7778)^2 &= (-11.7778)^2 \approx 138.71 \\ \end{align*} \] ### Step 3: Sum the Squared Deviations Add up all the squared deviations. \[ 431.78 + 17.83 + 27.27 + 17.83 + 391.18 + 104.49 + 104.49 + 332.06 + 138.71 \approx 1564.54 \] ### Step 4: Compute the Sample Variance Divide the sum of squared deviations by \(n - 1\) (where \(n\) is the number of data points) to get the sample variance. \[ s^2 = \frac{1564.54}{9 - 1} = \frac{1564.54}{8} \approx 195.57 \] ### Final Answer \[ \boxed{195.57} \] |
\frac{7045}{36} |