| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample standard deviation of the set \(\{-22, 22, 11, -19, 8, 7\}\), we follow these steps: 1. **Calculate the mean (average) of the data set.** The data set is \(\{-22, 22, 11, -19, 8, 7\}\). First, sum all the numbers: \[ -22 + 22 + 11 + -19 + 8 + 7 = 7 \] Next, divide the sum by the number of data points (which is 6): \[ \text{Mean} = \frac{7}{6} \approx 1.1667 \] 2. **Calculate the squared difference from the mean for each data point.** For each data point \(x_i\), compute \((x_i - \text{Mean})^2\): \[ (-22 - 1.1667)^2 \approx (-23.1667)^2 \approx 536.6944 \] \[ (22 - 1.1667)^2 \approx (20.8333)^2 \approx 434.0278 \] \[ (11 - 1.1667)^2 \approx (9.8333)^2 \approx 96.6944 \] \[ (-19 - 1.1667)^2 \approx (-20.1667)^2 \approx 406.6944 \] \[ (8 - 1.1667)^2 \approx (6.8333)^2 \approx 46.6944 \] \[ (7 - 1.1667)^2 \approx (5.8333)^2 \approx 34.0278 \] 3. **Calculate the variance.** Sum all the squared differences and divide by \(n-1\) (where \(n\) is the number of data points): \[ \text{Sum of squared differences} = 536.6944 + 434.0278 + 96.6944 + 406.6944 + 46.6944 + 34.0278 = 1554.8332 \] \[ \text{Variance} = \frac{1554.8332}{6-1} = \frac{1554.8332}{5} \approx 310.9666 \] 4. **Calculate the sample standard deviation.** The sample standard deviation is the square root of the variance: \[ \text{Sample standard deviation} = \sqrt{310.9666} \approx 17.6342 \] Thus, the sample standard deviation of the set \(\{-22, 22, 11, -19, 8, 7\}\) is approximately: \[ \boxed{17.6342} \] |
\sqrt{\frac{9329}{30}} |
| deepseek-r1 | To compute the sample standard deviation of the data set \({-22, 22, 11, -19, 8, 7}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} = \frac{7}{6} \approx 1.1667 \] 2. **Subtract the mean from each data point and square the result:** \[ \begin{aligned} (-22 - \frac{7}{6})^2 &= \left(-\frac{139}{6}\right)^2 = \frac{19321}{36} \\ (22 - \frac{7}{6})^2 &= \left(\frac{125}{6}\right)^2 = \frac{15625}{36} \\ (11 - \frac{7}{6})^2 &= \left(\frac{59}{6}\right)^2 = \frac{3481}{36} \\ (-19 - \frac{7}{6})^2 &= \left(-\frac{121}{6}\right)^2 = \frac{14641}{36} \\ (8 - \frac{7}{6})^2 &= \left(\frac{41}{6}\right)^2 = \frac{1681}{36} \\ (7 - \frac{7}{6})^2 &= \left(\frac{35}{6}\right)^2 = \frac{1225}{36} \\ \end{aligned} \] 3. **Sum the squared differences:** \[ \frac{19321}{36} + \frac{15625}{36} + \frac{3481}{36} + \frac{14641}{36} + \frac{1681}{36} + \frac{1225}{36} = \frac{55974}{36} = 1554.8333 \] 4. **Calculate the variance (divide by \(n-1\)):** \[ \text{Variance} = \frac{1554.8333}{5} \approx 310.9667 \] 5. **Take the square root to get the standard deviation:** \[ \text{Standard Deviation} = \sqrt{310.9667} \approx 17.634 \] Rounding to two decimal places, the sample standard deviation is \(\boxed{17.63}\). |
\sqrt{\frac{9329}{30}} |
| gpto1 | To compute the sample standard deviation of the data set \({-22, 22, 11, -19, 8, 7}\), we'll follow these steps: 1. **Find the mean (\(\overline{x}\))**: Sum of the data: \[ -22 + 22 + 11 -19 + 8 + 7 = 7 \] Mean: \[ \overline{x} = \frac{7}{6} = \frac{7}{6} \] 2. **Compute the squared differences from the mean and sum them**: For each data point \(x_i\), compute \((x_i - \overline{x})^2\): \[ \begin{align*} (-22 - \frac{7}{6})^2 &= \left(-\frac{139}{6}\right)^2 = \frac{19321}{36} \\ (22 - \frac{7}{6})^2 &= \left(\frac{125}{6}\right)^2 = \frac{15625}{36} \\ (11 - \frac{7}{6})^2 &= \left(\frac{59}{6}\right)^2 = \frac{3481}{36} \\ (-19 - \frac{7}{6})^2 &= \left(-\frac{121}{6}\right)^2 = \frac{14641}{36} \\ (8 - \frac{7}{6})^2 &= \left(\frac{41}{6}\right)^2 = \frac{1681}{36} \\ (7 - \frac{7}{6})^2 &= \left(\frac{35}{6}\right)^2 = \frac{1225}{36} \end{align*} \] Sum of squared differences: \[ S = \frac{19321 + 15625 + 3481 + 14641 + 1681 + 1225}{36} = \frac{55974}{36} \] 3. **Calculate the sample variance**: Since there are \(n = 6\) data points, the sample variance \(s^2\) is: \[ s^2 = \frac{S}{n - 1} = \frac{\frac{55974}{36}}{5} = \frac{55974}{180} = \frac{9329}{30} \] 4. **Compute the sample standard deviation**: \[ s = \sqrt{\frac{9329}{30}} \] So the sample standard deviation is \(\sqrt{\dfrac{9329}{30}}\). This is the exact value of the standard deviation for the given data set. \(\boxed{\displaystyle \sqrt{\dfrac{9329}{30}}}\) |
\sqrt{\frac{9329}{30}} |
| gpt-4o-2024-08-06 | To compute the sample standard deviation, we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} = \frac{7}{6} \approx 1.1667 \] 2. **Calculate each deviation from the mean and square it**: \[ (-22 - 1.1667)^2 = (-23.1667)^2 \approx 536.0278 \] \[ (22 - 1.1667)^2 = (20.8333)^2 \approx 434.0278 \] \[ (11 - 1.1667)^2 = (9.8333)^2 \approx 96.6944 \] \[ (-19 - 1.1667)^2 = (-20.1667)^2 \approx 406.6944 \] \[ (8 - 1.1667)^2 = (6.8333)^2 \approx 46.6944 \] \[ (7 - 1.1667)^2 = (5.8333)^2 \approx 34.0278 \] 3. **Sum the squared deviations**: \[ 536.0278 + 434.0278 + 96.6944 + 406.6944 + 46.6944 + 34.0278 = 1554.1666 \] 4. **Divide by \( n - 1 \) (where \( n \) is the number of data points)**: \[ \text{Variance} = \frac{1554.1666}{5} = 310.8333 \] 5. **Take the square root to find the sample standard deviation**: \[ s = \sqrt{310.8333} \approx 17.625 \] Thus, the sample standard deviation is \(\boxed{17.625}\). |
\sqrt{\frac{9329}{30}} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample standard deviation of the set \({-22, 22, 11, -19, 8, 7}\), we will follow these steps: 1. **Calculate the mean (average) of the numbers:** \[ \text{Mean} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} \] \[ \text{Mean} = \frac{5}{6} \] 2. **Calculate the variance:** The variance is the average of the squared differences from the mean. \[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} \] where \(x_i\) are the individual numbers and \(n\) is the total number of numbers. First, compute the squared differences from the mean: \[ (-22 - \frac{5}{6})^2 = \left(-\frac{137}{6}\right)^2 = \frac{18769}{36} \] \[ (22 - \frac{5}{6})^2 = \left(\frac{127}{6}\right)^2 = \frac{16129}{36} \] \[ (11 - \frac{5}{6})^2 = \left(\frac{61}{6}\right)^2 = \frac{3721}{36} \] \[ (-19 - \frac{5}{6})^2 = \left(-\frac{119}{6}\right)^2 = \frac{14161}{36} \] \[ (8 - \frac{5}{6})^2 = \left(\frac{43}{6}\right)^2 = \frac{1849}{36} \] \[ (7 - \frac{5}{6})^2 = \left(\frac{37}{6}\right)^2 = \frac{1369}{36} \] Now, sum these squared differences: \[ \sum \left(x_i - \text{Mean}\right)^2 = \frac{18769 + 16129 + 3721 + 14161 + 1849 + 1369}{36} \] \[ \sum \left(x_i - \text{Mean}\right)^2 = \frac{55908}{36} = 1553 \] Finally, compute the variance: \[ \text{Variance} = \frac{1553}{6} = 258.8333\ldots \] 3. **Calculate the standard deviation:** The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{258.8333\ldots} = 16.09 \] Therefore, the sample standard deviation of the set \({-22, 22, 11, -19, 8, 7}\) is: \[ \boxed{16.09} \] |
\sqrt{\frac{9329}{30}} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Calculate the mean of the given data set. To find the mean, we sum up all the numbers and then divide by the total count of numbers. The sum of the given numbers is -22 + 22 + 11 - 19 + 8 + 7 = 7. There are 6 numbers in total. Therefore, the mean is 7 / 6. ## Step 2: Calculate the variance of the data set. To find the variance, we first need to find the deviation of each number from the mean, then square each deviation, sum them up, and finally divide by the total count of numbers minus one (for sample variance). The mean is 7 / 6 = 1.1667. The deviations from the mean are: (-22 - 1.1667), (22 - 1.1667), (11 - 1.1667), (-19 - 1.1667), (8 - 1.1667), (7 - 1.1667). This simplifies to: -23.1667, 20.8333, 9.8333, -20.1667, 6.8333, 5.8333. Squaring each deviation gives: 536.3611, 433.3611, 96.3611, 406.3611, 46.3611, 33.3611. Summing these up gives 1551.6667. Dividing by 5 (the total count of numbers minus one) gives a variance of 310.3333. ## Step 3: Calculate the sample standard deviation. The sample standard deviation is the square root of the variance. Therefore, we take the square root of 310.3333 to find the sample standard deviation. The final answer is: $\boxed{17.6}$ |
\sqrt{\frac{9329}{30}} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Calculate the mean of the given data set. To find the sample standard deviation, we first need to calculate the mean of the given data set. The mean is calculated by summing all the numbers and then dividing by the total count of numbers. The given data set is ${-22, 22, 11, -19, 8, 7}$. The sum of these numbers is $-22 + 22 + 11 - 19 + 8 + 7 = 7$. There are 6 numbers in the data set. Therefore, the mean is $7 / 6 = 1.1667$. ## Step 2: Subtract the mean from each number in the data set. Next, we subtract the mean from each number in the data set to find the deviations from the mean. The deviations are $-22 - 1.1667 = -23.1667$, $22 - 1.1667 = 20.8333$, $11 - 1.1667 = 9.8333$, $-19 - 1.1667 = -20.1667$, $8 - 1.1667 = 6.8333$, and $7 - 1.1667 = 5.8333$. ## Step 3: Square each deviation. Now, we square each deviation. The squared deviations are $(-23.1667)^2 = 537.0833$, $(20.8333)^2 = 433.0833$, $(9.8333)^2 = 96.75$, $(-20.1667)^2 = 406.0833$, $(6.8333)^2 = 46.75$, and $(5.8333)^2 = 34.0833$. ## Step 4: Calculate the sum of the squared deviations. We then calculate the sum of the squared deviations. The sum is $537.0833 + 433.0833 + 96.75 + 406.0833 + 46.75 + 34.0833 = 1553.8332$. ## Step 5: Divide the sum of the squared deviations by the total count of numbers minus one. To calculate the sample variance, we divide the sum of the squared deviations by the total count of numbers minus one. In this case, we divide by $6 - 1 = 5$. Therefore, the sample variance is $1553.8332 / 5 = 310.76664$. ## Step 6: Calculate the sample standard deviation. Finally, we calculate the sample standard deviation by taking the square root of the sample variance. The sample standard deviation is $\sqrt{310.76664} = 17.602$. The final answer is: $\boxed{17.602}$ |
\sqrt{\frac{9329}{30}} |
| Qwen2-72B-Instruct | To compute the sample standard deviation of a set of numbers, we 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. Find the average of these squared deviations (sum them up and divide by n-1 for sample standard deviation). 5. Take the square root of this average. Let's calculate it step by step for the given set of numbers: {-22, 22, 11, -19, 8, 7} Step 1: Calculate the mean \[ \text{Mean} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} = \frac{7}{6} \] Step 2: Subtract the mean from each number \[ \begin{align*} -22 - \frac{7}{6} &= -\frac{139}{6} \\ 22 - \frac{7}{6} &= \frac{125}{6} \\ 11 - \frac{7}{6} &= \frac{59}{6} \\ -19 - \frac{7}{6} &= -\frac{121}{6} \\ 8 - \frac{7}{6} &= \frac{41}{6} \\ 7 - \frac{7}{6} &= \frac{35}{6} \end{align*} \] Step 3: Square each deviation \[ \begin{align*} \left(-\frac{139}{6}\right)^2 &= \frac{19321}{36} \\ \left(\frac{125}{6}\right)^2 &= \frac{15625}{36} \\ \left(\frac{59}{6}\right)^2 &= \frac{3481}{36} \\ \left(-\frac{121}{6}\right)^2 &= \frac{14641}{36} \\ \left(\frac{41}{6}\right)^2 &= \frac{1681}{36} \\ \left(\frac{35}{6}\right)^2 &= \frac{1225}{36} \end{align*} \] Step 4: Find the average of these squared deviations \[ \text{Average} = \frac{\frac{19321}{36} + \frac{15625}{36} + \frac{3481}{36} + \frac{14641}{36} + \frac{1681}{36} + \frac{1225}{36}}{6-1} \] \[ = \frac{\frac{56074}{36}}{5} = \frac{56074}{180} = \frac{28037}{90} \] Step 5: Take the square root of this average \[ \text{Sample Standard Deviation} = \sqrt{\frac{28037}{90}} \approx 17.24 \] Therefore, the sample standard deviation is approximately $\boxed{17.24}$. |
\sqrt{\frac{9329}{30}} |
| baichuan4 | To compute the sample standard deviation of a set of numbers, we first need to calculate the mean (average) of the numbers. Then, for each number, we subtract the mean and square the result. We sum up these squared differences, divide by the number of observations minus one (since it's a sample standard deviation), and finally take the square root of that result. Given the numbers: ${-22, 22, 11, -19, 8, 7}$ 1. Calculate the mean: Mean = (-22 + 22 + 11 - 19 + 8 + 7) / 6 Mean = 0 / 6 Mean = 0 2. For each number, subtract the mean and square the result: (-22 - 0)^2 = (-22)^2 = 484 (22 - 0)^2 = 22^2 = 484 (11 - 0)^2 = 11^2 = 121 (-19 - 0)^2 = (-19)^2 = 361 (8 - 0)^2 = 8^2 = 64 (7 - 0)^2 = 7^2 = 49 3. Sum up these squared differences: 484 + 484 + 121 + 361 + 64 + 49 = 1563 4. Divide by the number of observations minus one (since it's a sample standard deviation): Variance = 1563 / (6 - 1) Variance = 1563 / 5 Variance = 312.6 5. Take the square root of the variance to get the standard deviation: Standard Deviation = sqrt(312.6) Standard Deviation ≈ 17.6939 Therefore, the sample standard deviation of the given numbers is approximately 17.6939. Final answer: $ \boxed{17.6939} $ |
\sqrt{\frac{9329}{30}} |
| doubao | First find the mean: Mean = $(-22+22+11-19+8+7)\div6$ $=(0+11-19+8+7)\div6$ $=(11-19+8+7)\div6$ $=(-8+8+7)\div6$ $=(0+7)\div6$ $=7\div6=\frac{7}{6}$. Then find the deviations from the mean: $(-22-\frac{7}{6})$, $(22-\frac{7}{6})$, $(11-\frac{7}{6})$, $(-19-\frac{7}{6})$, $(8-\frac{7}{6})$, $(7-\frac{7}{6})$. Squaring these deviations: $(-22-\frac{7}{6})^2$, $(22-\frac{7}{6})^2$, $(11-\frac{7}{6})^2$, $(-19-\frac{7}{6})^2$, $(8-\frac{7}{6})^2$, $(7-\frac{7}{6})^2$. Finding the sum of these squared deviations: Sum = $(-22-\frac{7}{6})^2+(22-\frac{7}{6})^2+(11-\frac{7}{6})^2+(-19-\frac{7}{6})^2+(8-\frac{7}{6})^2+(7-\frac{7}{6})^2$. Dividing the sum by the number of data points (6) and taking the square root gives the sample standard deviation: Sample standard deviation = $\sqrt{\frac{Sum}{6}}$. After calculating, the sample standard deviation is approximately 14.69, so the answer is $\boxed{14.69}$. |
\sqrt{\frac{9329}{30}} |
| gpto1 | To compute the sample standard deviation of the data set \({-22, 22, 11, -19, 8, 7}\), we'll follow these steps: 1. **Find the mean (\(\overline{x}\))**: Sum of the data: \[ -22 + 22 + 11 -19 + 8 + 7 = 7 \] Mean: \[ \overline{x} = \frac{7}{6} = \frac{7}{6} \] 2. **Compute the squared differences from the mean and sum them**: For each data point \(x_i\), compute \((x_i - \overline{x})^2\): \[ \begin{align*} (-22 - \frac{7}{6})^2 &= \left(-\frac{139}{6}\right)^2 = \frac{19321}{36} \\ (22 - \frac{7}{6})^2 &= \left(\frac{125}{6}\right)^2 = \frac{15625}{36} \\ (11 - \frac{7}{6})^2 &= \left(\frac{59}{6}\right)^2 = \frac{3481}{36} \\ (-19 - \frac{7}{6})^2 &= \left(-\frac{121}{6}\right)^2 = \frac{14641}{36} \\ (8 - \frac{7}{6})^2 &= \left(\frac{41}{6}\right)^2 = \frac{1681}{36} \\ (7 - \frac{7}{6})^2 &= \left(\frac{35}{6}\right)^2 = \frac{1225}{36} \end{align*} \] Sum of squared differences: \[ S = \frac{19321 + 15625 + 3481 + 14641 + 1681 + 1225}{36} = \frac{55974}{36} \] 3. **Calculate the sample variance**: Since there are \(n = 6\) data points, the sample variance \(s^2\) is: \[ s^2 = \frac{S}{n - 1} = \frac{\frac{55974}{36}}{5} = \frac{55974}{180} = \frac{9329}{30} \] 4. **Compute the sample standard deviation**: \[ s = \sqrt{\frac{9329}{30}} \] So the sample standard deviation is \(\sqrt{\dfrac{9329}{30}}\). This is the exact value of the standard deviation for the given data set. \(\boxed{\displaystyle \sqrt{\dfrac{9329}{30}}}\) |
\sqrt{\frac{9329}{30}} |
| kimi | To compute the sample standard deviation of the given dataset \(\{-22, 22, 11, -19, 8, 7\}\), we will follow these steps: 1. **Calculate the mean (\(\bar{x}\)) of the dataset:** The mean is the sum of all the values divided by the number of values. \[ \bar{x} = \frac{-22 + 22 + 11 + (-19) + 8 + 7}{6} \] Simplify the numerator: \[ -22 + 22 + 11 - 19 + 8 + 7 = 7 \] Now divide by the number of values (6): \[ \bar{x} = \frac{7}{6} \approx 1.1667 \] 2. **Calculate the squared differences from the mean for each value:** For each \(x_i\) in the dataset, compute \((x_i - \bar{x})^2\). \[ \begin{align*} (-22 - 1.1667)^2 &\approx (-23.1667)^2 \approx 536.74 \\ (22 - 1.1667)^2 &\approx (20.8333)^2 \approx 434.56 \\ (11 - 1.1667)^2 &\approx (9.8333)^2 \approx 96.74 \\ (-19 - 1.1667)^2 &\approx (-20.1667)^2 \approx 406.74 \\ (8 - 1.1667)^2 &\approx (6.8333)^2 \approx 46.74 \\ (7 - 1.1667)^2 &\approx (5.8333)^2 \approx 34.06 \\ \end{align*} \] 3. **Calculate the sum of these squared differences:** \[ 536.74 + 434.56 + 96.74 + 406.74 + 46.74 + 34.06 = 1565.54 \] 4. **Calculate the sample variance (\(s^2\)):** The sample variance is the sum of the squared differences divided by \(n-1\) (where \(n\) is the number of values). \[ s^2 = \frac{1565.54}{6-1} = \frac{1565.54}{5} \approx 313.11 \] 5. **Calculate the sample standard deviation (\(s\)):** The sample standard deviation is the square root of the sample variance. \[ s = \sqrt{313.11} \approx 17.69 \] Therefore, the sample standard deviation of the dataset is \(\boxed{17.69}\). |
\sqrt{\frac{9329}{30}} |
| minimax | To compute the sample standard deviation of the given data set ${-22, 22, 11, -19, 8, 7}$, 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 deviations. 3. Square each deviation. 4. Find the mean of these squared deviations. 5. Take the square root of the mean of the squared deviations to get the sample standard deviation. Let's go through these steps: 1. Calculate the mean: $$ \text{Mean} = \frac{\sum{x_i}}{n} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} = \frac{7}{6} \approx 1.1667 $$ 2. Calculate the deviations from the mean: $$ (-22 - 1.1667) = -23.1667 $$ $$ (22 - 1.1667) = 20.8333 $$ $$ (11 - 1.1667) = 9.83 |
\sqrt{\frac{9329}{30}} |
| wenxinyiyan | 计算过程如下： 首先，我们需要计算这组数据的平均值。 平均值 $\bar{x}$ 的计算公式为： $$ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} $$ 其中 $n$ 是数据的数量，$x_i$ 是每个数据点。 给定的数据是 ${-22, 22, 11, -19, 8, 7}$，所以 $n = 6$。 计算平均值： $$ \bar{x} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} = \frac{7}{6} \approx 1.1667 $$ 接下来，我们计算每个数据点与平均值的差的平方，然后求这些平方差的平均值。 方差 $s^2$ 的计算公式为： $$ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} $$ 计算方差： $$ s^2 = \frac{(-22 - 1.1667)^2 + (22 - 1.1667)^2 + (11 - 1.1667)^2 + (-19 - 1.1667)^2 + (8 - 1.1667)^2 + (7 - 1.1667)^2}{5} $$ $$ s^2 = \frac{(23.1667)^2 + (20.8333)^2 + (9.8333)^2 + (20.1667)^2 + (6.8333)^2 + (5.8333)^2}{5} $$ $$ s^2 = \frac{536.7222 + 434.0278 + 96.7028 + 406.7028 + 46.7028 + 34.0370}{5} $$ $$ s^2 = \frac{1555.8954}{5} \approx 311.1791 $$ 最后，我们计算方差的平方根，得到样本标准偏差。 样本标准偏差 $s$ 的计算公式为： $$ s = \sqrt{s^2} $$ 计算样本标准偏差： $$ s = \sqrt{311.1791} \approx 17.6404 $$ 所以，样本标准偏差为 $\boxed{17.6404}$。 |
\sqrt{\frac{9329}{30}} |
| zhipu | 1. **Calculate the mean of the data set**: \[ \text{Mean} = \frac{-22 + 22 + 11 - 19 + 8 + 7}{6} = \frac{7}{6} \] 2. **Calculate the squared deviations from the mean for each data point**: - For $-22$: $(-22 - \frac{7}{6})^2 = (-\frac{134}{6})^2 = \frac{17956}{36}$ - For $22$: $(22 - \frac{7}{6})^2 = (\frac{130}{6})^2 = \frac{16900}{36}$ - For $11$: $(11 - \frac{7}{6})^2 = (\frac{65}{6})^2 = \frac{4225}{36}$ - For $-19$: $(-19 - \frac{7}{6})^2 = (-\frac{121}{6})^2 = \frac{14641}{36}$ - For $8$: $(8 - \frac{7}{6})^2 = (\frac{41}{6})^2 = \frac{1681}{36}$ - For $7$: $(7 - \frac{7}{6})^2 = (\frac{35}{6})^2 = \frac{1225}{36}$ 3. **Sum the squared deviations**: \[ \text{Sum of squared deviations} = \frac{17956 + 16900 + 4225 + 14641 + 1681 + 1225}{36} = \frac{59628}{36} \] 4. **Calculate the variance**: \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{\frac{59628}{36}}{5} = \frac{59628}{180} = 331.8777\ldots \] 5. **Calculate the standard deviation**: \[ \text{Standard Deviation} = \sqrt{331.8777\ldots} \approx 18.204 \] However, keeping the exact value: \[ \text{Standard Deviation} = \sqrt{\frac{59628}{180}} \] Simplifying further: \[ \text{Standard Deviation} = \frac{\sqrt{59628}}{\sqrt{180}} = \frac{244}{\sqrt{180}} = \frac{244\sqrt{5}}{30} \] Thus, the sample standard deviation is: \[ \boxed{\frac{244\sqrt{5}}{30}} \] |
\sqrt{\frac{9329}{30}} |
| deepseek-v3 | To compute the **sample standard deviation** of the dataset \(\{-22, 22, 11, -19, 8, 7\}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{-22 + 22 + 11 + (-19) + 8 + 7}{6} = \frac{7}{6} \approx 1.1667 \] 2. **Find the Squared Deviations from the Mean:** \[ \begin{align*} (-22 - 1.1667)^2 &= (-23.1667)^2 \approx 536.69 \\ (22 - 1.1667)^2 &= 20.8333^2 \approx 434.03 \\ (11 - 1.1667)^2 &= 9.8333^2 \approx 96.69 \\ (-19 - 1.1667)^2 &= (-20.1667)^2 \approx 406.69 \\ (8 - 1.1667)^2 &= 6.8333^2 \approx 46.69 \\ (7 - 1.1667)^2 &= 5.8333^2 \approx 34.03 \\ \end{align*} \] **Total of Squared Deviations:** \[ 536.69 + 434.03 + 96.69 + 406.69 + 46.69 + 34.03 = 1554.82 \] 3. **Compute the Variance (\(s^2\)):** \[ s^2 = \frac{1554.82}{6 - 1} = \frac{1554.82}{5} = 310.964 \] 4. **Calculate the Sample Standard Deviation (\(s\)):** \[ s = \sqrt{310.964} \approx 17.63 \] Therefore, the sample standard deviation is: \[ \boxed{17.63} \] |
\sqrt{\frac{9329}{30}} |