| 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 \(\{25, 10, 0, 9, -1, -14, -21\}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} \] \[ \text{Mean} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} \] \[ \text{Mean} = \frac{8}{7} \approx 1.142857 \] 2. **Calculate the squared differences from the mean for each data point:** \[ (25 - 1.142857)^2 = 23.857143^2 \approx 569.142857 \] \[ (10 - 1.142857)^2 = 8.857143^2 \approx 78.448980 \] \[ (0 - 1.142857)^2 = -1.142857^2 \approx 1.306122 \] \[ (9 - 1.142857)^2 = 7.857143^2 \approx 61.734694 \] \[ (-1 - 1.142857)^2 = -2.142857^2 \approx 4.591837 \] \[ (-14 - 1.142857)^2 = -15.142857^2 \approx 229.306122 \] \[ (-21 - 1.142857)^2 = -22.142857^2 \approx 490.326531 \] 3. **Sum the squared differences:** \[ \sum (x_i - \bar{x})^2 = 569.142857 + 78.448980 + 1.306122 + 61.734694 + 4.591837 + 229.306122 + 490.326531 \] \[ \sum (x_i - \bar{x})^2 = 1434.857143 \] 4. **Calculate the variance:** \[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{1434.857143}{6} \approx 239.142857 \] 5. **Calculate the sample standard deviation:** \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{239.142857} \approx 15.4642 \] Thus, the sample standard deviation is: \[ \boxed{15.4642} \] |
3 \sqrt{\frac{186}{7}} |
| deepseek-r1 | To compute the sample standard deviation of the data set \({25, 10, 0, 9, -1, -14, -21}\), follow these steps: 1. **Calculate the mean**: \[ \text{Mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{8}{7} \approx 1.142857 \] 2. **Find the deviations from the mean and square them**: \[ \begin{aligned} (25 - \frac{8}{7})^2 &= \left(\frac{167}{7}\right)^2 = \frac{27889}{49} \approx 569.163, \\ (10 - \frac{8}{7})^2 &= \left(\frac{62}{7}\right)^2 = \frac{3844}{49} \approx 78.449, \\ (0 - \frac{8}{7})^2 &= \left(\frac{-8}{7}\right)^2 = \frac{64}{49} \approx 1.306, \\ (9 - \frac{8}{7})^2 &= \left(\frac{55}{7}\right)^2 = \frac{3025}{49} \approx 61.735, \\ (-1 - \frac{8}{7})^2 &= \left(\frac{-15}{7}\right)^2 = \frac{225}{49} \approx 4.592, \\ (-14 - \frac{8}{7})^2 &= \left(\frac{-106}{7}\right)^2 = \frac{11236}{49} \approx 229.306, \\ (-21 - \frac{8}{7})^2 &= \left(\frac{-155}{7}\right)^2 = \frac{24025}{49} \approx 490.306. \end{aligned} \] 3. **Sum the squared deviations**: \[ \frac{27889}{49} + \frac{3844}{49} + \frac{64}{49} + \frac{3025}{49} + \frac{225}{49} + \frac{11236}{49} + \frac{24025}{49} = \frac{70308}{49} \approx 1434.857 \] 4. **Calculate the sample variance**: \[ \text{Variance} = \frac{1434.857}{7-1} = \frac{1434.857}{6} \approx 239.1428 \] 5. **Compute the sample standard deviation**: \[ \text{Standard Deviation} = \sqrt{239.1428} \approx 15.464 \] Rounding to two decimal places, the sample standard deviation is \(\boxed{15.46}\). |
3 \sqrt{\frac{186}{7}} |
| gpto1 | To compute the sample standard deviation of the data set \(\{25, 10, 0, 9, -1, -14, -21\}\), we'll follow these steps: **1. Calculate the mean (\(\bar{x}\)):** \[ \bar{x} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{8}{7} \] **2. Calculate each \((x_i - \bar{x})^2\):** | \(x_i\) | \(x_i - \bar{x}\) | \((x_i - \bar{x})^2\) | |---------|------------------------------|-----------------------------| | 25 | \(25 - \frac{8}{7} = \frac{167}{7}\) | \(\left(\frac{167}{7}\right)^2 = \frac{27889}{49}\) | | 10 | \(10 - \frac{8}{7} = \frac{62}{7}\) | \(\left(\frac{62}{7}\right)^2 = \frac{3844}{49}\) | | 0 | \(0 - \frac{8}{7} = -\frac{8}{7}\) | \(\left(-\frac{8}{7}\right)^2 = \frac{64}{49}\) | | 9 | \(9 - \frac{8}{7} = \frac{55}{7}\) | \(\left(\frac{55}{7}\right)^2 = \frac{3025}{49}\) | | -1 | \(-1 - \frac{8}{7} = -\frac{15}{7}\) | \(\left(-\frac{15}{7}\right)^2 = \frac{225}{49}\) | | -14 | \(-14 - \frac{8}{7} = -\frac{106}{7}\)| \(\left(-\frac{106}{7}\right)^2 = \frac{11236}{49}\)| | -21 | \(-21 - \frac{8}{7} = -\frac{155}{7}\)| \(\left(-\frac{155}{7}\right)^2 = \frac{24025}{49}\)| **3. Sum the squared differences:** \[ \text{Sum of squared differences} = \frac{27889 + 3844 + 64 + 3025 + 225 + 11236 + 24025}{49} = \frac{70308}{49} \] **4. Compute the variance (\(s^2\)):** \[ s^2 = \frac{\text{Sum of squared differences}}{n - 1} = \frac{\frac{70308}{49}}{6} = \frac{11718}{49} \] **5. Compute the standard deviation (\(s\)):** \[ s = \sqrt{s^2} = \sqrt{\frac{11718}{49}} = \frac{\sqrt{11718}}{7} \] We can factor \(11718\) to simplify: \[ 11718 = 2 \times 3^3 \times 7 \times 31 \] Since there are no square factors beyond \(3^2\): \[ s = \frac{3 \sqrt{1302}}{7} \] Thus, the exact value of the sample standard deviation is \(\frac{3\sqrt{1302}}{7}\). \(\boxed{\dfrac{3\sqrt{1302}}{7}}\) |
3 \sqrt{\frac{186}{7}} |
| gpt-4o-2024-08-06 | To compute the sample standard deviation, we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \( n \) is the number of data points. Here, \( n = 7 \). The data points are: \( 25, 10, 0, 9, -1, -14, -21 \). \[ \bar{x} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{8}{7} \approx 1.142857 \] 2. **Calculate the squared deviations from the mean**: \[ (x_i - \bar{x})^2 \] - For \( x_1 = 25 \): \((25 - 1.142857)^2 = 571.877551\) - For \( x_2 = 10 \): \((10 - 1.142857)^2 = 78.877551\) - For \( x_3 = 0 \): \((0 - 1.142857)^2 = 1.306122\) - For \( x_4 = 9 \): \((9 - 1.142857)^2 = 61.734694\) - For \( x_5 = -1 \): \((-1 - 1.142857)^2 = 4.591837\) - For \( x_6 = -14 \): \((-14 - 1.142857)^2 = 229.877551\) - For \( x_7 = -21 \): \((-21 - 1.142857)^2 = 489.020408\) 3. **Calculate the variance**: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] \[ s^2 = \frac{571.877551 + 78.877551 + 1.306122 + 61.734694 + 4.591837 + 229.877551 + 489.020408}{6} \] \[ s^2 = \frac{1437.285714}{6} = 239.547619 \] 4. **Calculate the sample standard deviation**: \[ s = \sqrt{s^2} = \sqrt{239.547619} \approx 15.4757 \] Thus, the sample standard deviation is \(\boxed{15.4757}\). |
3 \sqrt{\frac{186}{7}} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample standard deviation, we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} \] \[ \text{Sum} = 25 + 10 + 0 + 9 - 1 - 14 - 21 = 7 \] \[ \text{Mean} = \frac{7}{7} = 1 \] 2. **Calculate the variance:** For each data point, subtract the mean and square the result: \[ (25 - 1)^2 = 24^2 = 576 \] \[ (10 - 1)^2 = 9^2 = 81 \] \[ (0 - 1)^2 = (-1)^2 = 1 \] \[ (9 - 1)^2 = 8^2 = 64 \] \[ (-1 - 1)^2 = (-2)^2 = 4 \] \[ (-14 - 1)^2 = (-15)^2 = 225 \] \[ (-21 - 1)^2 = (-22)^2 = 484 \] Sum these squared differences: \[ \text{Sum of squared differences} = 576 + 81 + 1 + 64 + 4 + 225 + 484 = 1435 \] 3. **Calculate the variance:** \[ \text{Variance} = \frac{1435}{7} = 205 \] 4. **Calculate the standard deviation:** \[ \text{Standard Deviation} = \sqrt{205} \] Thus, the sample standard deviation is: \[ \boxed{\sqrt{205}} \] |
3 \sqrt{\frac{186}{7}} |
| 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 $25 + 10 + 0 + 9 - 1 - 14 - 21 = 8$. There are 7 numbers in total. Thus, the mean is $8 / 7 \approx 1.1429$. ## Step 2: Calculate the deviations from the mean for each data point. We subtract the mean from each data point to find the deviation. The deviations are $25 - 1.1429, 10 - 1.1429, 0 - 1.1429, 9 - 1.1429, -1 - 1.1429, -14 - 1.1429, -21 - 1.1429$. ## Step 3: Calculate the squared deviations. We square each deviation to ensure they are all positive and to weight them by magnitude. The squared deviations are $(25 - 1.1429)^2, (10 - 1.1429)^2, (0 - 1.1429)^2, (9 - 1.1429)^2, (-1 - 1.1429)^2, (-14 - 1.1429)^2, (-21 - 1.1429)^2$. ## Step 4: Calculate the sum of the squared deviations. Summing the squared deviations gives us $(23.8571)^2 + (8.8571)^2 + (-1.1429)^2 + (7.8571)^2 + (-2.1429)^2 + (-15.1429)^2 + (-22.1429)^2$. ## Step 5: Calculate the variance. To find the variance, we divide the sum of the squared deviations by the number of items minus one (for sample variance). The number of items is 7, so we divide by 6. ## Step 6: Calculate the sample standard deviation. The sample standard deviation is the square root of the variance. ## Step 7: Compute the numerical value of the sample standard deviation. Let's calculate the numerical value of each step: - The squared deviations are approximately $569.1429, 78.1429, 1.3056, 61.7143, 4.5929, 229.3056, 490.3056$. - The sum of the squared deviations is approximately $1434.509$. - The variance is $1434.509 / 6 \approx 239.0848$. - The sample standard deviation is the square root of $239.0848$, which is approximately $15.456$. The final answer is: $\boxed{15.456}$ |
3 \sqrt{\frac{186}{7}} |
| 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 up all the numbers and then dividing by the total count of numbers. The given data set is ${25, 10, 0, 9, -1, -14, -21}$. The sum of these numbers is $25 + 10 + 0 + 9 - 1 - 14 - 21 = 8$. There are $7$ numbers in the data set. Therefore, the mean is $8 / 7 \approx 1.143$. ## Step 2: Subtract the mean from each data point to find the deviations Next, we subtract the mean from each data point to find the deviations from the mean. The deviations are $(25 - 1.143), (10 - 1.143), (0 - 1.143), (9 - 1.143), (-1 - 1.143), (-14 - 1.143), (-21 - 1.143)$. This simplifies to $23.857, 8.857, -1.143, 7.857, -2.143, -15.143, -22.143$. ## Step 3: Square each deviation We then square each deviation. The squared deviations are $(23.857)^2, (8.857)^2, (-1.143)^2, (7.857)^2, (-2.143)^2, (-15.143)^2, (-22.143)^2$. This simplifies to $569.51, 78.51, 1.31, 61.57, 4.60, 229.31, 490.31$. ## Step 4: Calculate the sum of the squared deviations Now, we calculate the sum of the squared deviations. The sum is $569.51 + 78.51 + 1.31 + 61.57 + 4.60 + 229.31 + 490.31 = 1435.12$. ## Step 5: Divide the sum of the squared deviations by the number of data points minus one To calculate the sample variance, we divide the sum of the squared deviations by the number of data points minus one. In this case, we divide by $7 - 1 = 6$. Therefore, the sample variance is $1435.12 / 6 \approx 239.19$. ## Step 6: Take the square root of the sample variance to find the sample standard deviation Finally, we take the square root of the sample variance to find the sample standard deviation. The square root of $239.19$ is approximately $15.47$. The final answer is: $\boxed{15.47}$ |
3 \sqrt{\frac{186}{7}} |
| Qwen2-72B-Instruct | To compute the sample standard deviation 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. Find the average of these squared deviations (this is called the variance). 5. Take the square root of the variance to get the standard deviation. Let's calculate it step by step for the given set of numbers: ${25, 10, 0, 9, -1, -14, -21}$ Step 1: Calculate the mean \[ \text{Mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{-2}{7} \approx -0.2857 \] Step 2: Subtract the mean from each number \[ \begin{align*} 25 - (-0.2857) &= 25.2857 \\ 10 - (-0.2857) &= 10.2857 \\ 0 - (-0.2857) &= 0.2857 \\ 9 - (-0.2857) &= 9.2857 \\ -1 - (-0.2857) &= -0.7143 \\ -14 - (-0.2857) &= -13.7143 \\ -21 - (-0.2857) &= -20.7143 \\ \end{align*} \] Step 3: Square each deviation \[ \begin{align*} (25.2857)^2 &\approx 640.00 \\ (10.2857)^2 &\approx 105.79 \\ (0.2857)^2 &\approx 0.08 \\ (9.2857)^2 &\approx 86.21 \\ (-0.7143)^2 &\approx 0.51 \\ (-13.7143)^2 &\approx 188.10 \\ (-20.7143)^2 &\approx 429.10 \\ \end{align*} \] Step 4: Find the average of these squared deviations (variance) \[ \text{Variance} = \frac{640.00 + 105.79 + 0.08 + 86.21 + 0.51 + 188.10 + 429.10}{6} \] \[ \text{Variance} = \frac{1449.79}{6} \approx 241.63 \] Note that we divide by $n-1$ (which is 6 in this case) instead of $n$ because we're calculating the sample variance, not the population variance. Step 5: Take the square root of the variance to get the standard deviation \[ \text{Standard Deviation} = \sqrt{241.63} \approx 15.54 \] Therefore, the sample standard deviation of the given set of numbers is approximately $\boxed{15.54}$. |
3 \sqrt{\frac{186}{7}} |
| 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: $25, 10, 0, 9, -1, -14, -21$, let's follow these steps: 1. Calculate the mean: $$\overline{x} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{7}{7} = 1$$ 2. For each number, subtract the mean and square the result: $$ (25 - 1)^2 = 24^2 = 576 $$ $$ (10 - 1)^2 = 9^2 = 81 $$ $$ (0 - 1)^2 = (-1)^2 = 1 $$ $$ (9 - 1)^2 = 8^2 = 64 $$ $$ (-1 - 1)^2 = (-2)^2 = 4 $$ $$ (-14 - 1)^2 = (-15)^2 = 225 $$ $$ (-21 - 1)^2 = (-22)^2 = 484 $$ 3. Sum up these squared differences: $$ 576 + 81 + 1 + 64 + 4 + 225 + 484 = 1435 $$ 4. Divide by the number of observations minus one: $$ \frac{1435}{7 - 1} = \frac{1435}{6} $$ 5. Take the square root of that result to get the sample standard deviation: $$ s = \sqrt{\frac{1435}{6}} $$ Now, let's compute the square root: $$ s = \sqrt{\frac{1435}{6}} \approx \sqrt{239.1667} \approx 15.4641 $$ Therefore, the sample standard deviation of the given numbers is approximately $15.4641$. $$\boxed{15.4641}$$ |
3 \sqrt{\frac{186}{7}} |
| doubao | First find the mean: $(25+10+0+9+(-1)+(-14)+(-21)) \div 7$ $=(35+0+9-1-14-21) \div 7$ $=(35+9-1-14-21) \div 7$ $=(44-1-14-21) \div 7$ $=(43-14-21) \div 7$ $=(29-21) \div 7$ $=8 \div 7 \approx 1.143$. Then calculate the deviations from the mean for each number: $25-1.143=23.857$; $10-1.143=8.857$; $0-1.143=-1.143$; $9-1.143=7.857$; $-1-1.143=-2.143$; $-14-1.143=-15.143$; $-21-1.143=-22.143$. Square these deviations: $23.857^2 \approx 570.24$; $8.857^2 \approx 78.44$; $(-1.143)^2 \approx 1.31$; $7.857^2 \approx 61.74$; $(-2.143)^2 \approx 4.59$; $(-15.143)^2 \approx 229.31$; $(-22.143)^2 \approx 490.24$. Find the sum of these squares: $570.24+78.44+1.31+61.74+4.59+229.31+490.24=1435.87$. Divide by the number of items minus 1 (6) and take the square root: $\sqrt{\frac{1435.87}{6}} \approx 16.24$. So the sample standard deviation is $\boxed{16.24}$. |
3 \sqrt{\frac{186}{7}} |
| gpto1 | To compute the sample standard deviation of the data set \(\{25, 10, 0, 9, -1, -14, -21\}\), we'll follow these steps: **1. Calculate the mean (\(\bar{x}\)):** \[ \bar{x} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{8}{7} \] **2. Calculate each \((x_i - \bar{x})^2\):** | \(x_i\) | \(x_i - \bar{x}\) | \((x_i - \bar{x})^2\) | |---------|------------------------------|-----------------------------| | 25 | \(25 - \frac{8}{7} = \frac{167}{7}\) | \(\left(\frac{167}{7}\right)^2 = \frac{27889}{49}\) | | 10 | \(10 - \frac{8}{7} = \frac{62}{7}\) | \(\left(\frac{62}{7}\right)^2 = \frac{3844}{49}\) | | 0 | \(0 - \frac{8}{7} = -\frac{8}{7}\) | \(\left(-\frac{8}{7}\right)^2 = \frac{64}{49}\) | | 9 | \(9 - \frac{8}{7} = \frac{55}{7}\) | \(\left(\frac{55}{7}\right)^2 = \frac{3025}{49}\) | | -1 | \(-1 - \frac{8}{7} = -\frac{15}{7}\) | \(\left(-\frac{15}{7}\right)^2 = \frac{225}{49}\) | | -14 | \(-14 - \frac{8}{7} = -\frac{106}{7}\)| \(\left(-\frac{106}{7}\right)^2 = \frac{11236}{49}\)| | -21 | \(-21 - \frac{8}{7} = -\frac{155}{7}\)| \(\left(-\frac{155}{7}\right)^2 = \frac{24025}{49}\)| **3. Sum the squared differences:** \[ \text{Sum of squared differences} = \frac{27889 + 3844 + 64 + 3025 + 225 + 11236 + 24025}{49} = \frac{70308}{49} \] **4. Compute the variance (\(s^2\)):** \[ s^2 = \frac{\text{Sum of squared differences}}{n - 1} = \frac{\frac{70308}{49}}{6} = \frac{11718}{49} \] **5. Compute the standard deviation (\(s\)):** \[ s = \sqrt{s^2} = \sqrt{\frac{11718}{49}} = \frac{\sqrt{11718}}{7} \] We can factor \(11718\) to simplify: \[ 11718 = 2 \times 3^3 \times 7 \times 31 \] Since there are no square factors beyond \(3^2\): \[ s = \frac{3 \sqrt{1302}}{7} \] Thus, the exact value of the sample standard deviation is \(\frac{3\sqrt{1302}}{7}\). \(\boxed{\dfrac{3\sqrt{1302}}{7}}\) |
3 \sqrt{\frac{186}{7}} |
| kimi | To compute the sample standard deviation of the dataset \( \{25, 10, 0, 9, -1, -14, -21\} \), 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. \[ \bar{x} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} \] First, compute the sum: \[ 25 + 10 + 0 + 9 + (-1) + (-14) + (-21) = 25 + 10 + 0 + 9 - 1 - 14 - 21 = 8 \] Now, divide by the number of values (7): \[ \bar{x} = \frac{8}{7} \approx 1.1429 \] 2. **Calculate the squared differences from the mean:** For each value \(x_i\), compute \((x_i - \bar{x})^2\). \[ \begin{aligned} (25 - 1.1429)^2 &\approx (23.8571)^2 \approx 567.4286 \\ (10 - 1.1429)^2 &\approx (8.8571)^2 \approx 78.5714 \\ (0 - 1.1429)^2 &\approx (-1.1429)^2 \approx 1.3061 \\ (9 - 1.1429)^2 &\approx (7.8571)^2 \approx 61.6429 \\ (-1 - 1.1429)^2 &\approx (-2.1429)^2 \approx 4.5918 \\ (-14 - 1.1429)^2 &\approx (-15.1429)^2 \approx 229.4286 \\ (-21 - 1.1429)^2 &\approx (-22.1429)^2 \approx 488.5714 \\ \end{aligned} \] 3. **Calculate the sum of the squared differences:** Add all the squared differences together: \[ 567.4286 + 78.5714 + 1.3061 + 61.6429 + 4.5918 + 229.4286 + 488.5714 \approx 1455.0000 \] 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. \[ s^2 = \frac{1455}{7-1} = \frac{1455}{6} \approx 242.5 \] 5. **Calculate the sample standard deviation (\(s\)):** The sample standard deviation is the square root of the sample variance. \[ s = \sqrt{242.5} \approx 15.57 \] Therefore, the sample standard deviation of the dataset is \(\boxed{15.57}\). |
3 \sqrt{\frac{186}{7}} |
| minimax | To compute the sample standard deviation of the given data set ${25, 10, 0, 9, -1, -14, -21}$, 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: $$ \text{Mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} $$ $$ \text{Mean} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} $$ $$ \text{Mean} = \frac{8}{7} $$ $$ \text{Mean} = \frac{8}{7} \approx 1.14 $$ 2. Calculate the deviations from the mean: $$ 25 - \frac{8}{ |
3 \sqrt{\frac{186}{7}} |
| wenxinyiyan | 计算平均值： $\text{平均值} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{-2}{7}$ 计算方差 $s^2$： $s^2 = \frac{1}{6} \left[ (25 + \frac{2}{7})^2 + (10 + \frac{2}{7})^2 + (0 + \frac{2}{7})^2 + (9 + \frac{2}{7})^2 + (-1 + \frac{2}{7})^2 + (-14 + \frac{2}{7})^2 + (-21 + \frac{2}{7})^2 \right]$ $s^2 = \frac{1}{6} \left[ (25 + \frac{2}{7})^2 + (10 + \frac{2}{7})^2 + (0 + \frac{2}{7})^2 + (9 + \frac{2}{7})^2 + (-1 + \frac{2}{7})^2 + (-14 + \frac{2}{7})^2 + (-21 + \frac{2}{7})^2 \right]$ $s^2 = \frac{1}{6} \left[ 625 + \frac{100}{49} + 2 \cdot 25 \cdot \frac{2}{7} + 100 + \frac{100}{49} + 2 \cdot 10 \cdot \frac{2}{7} + 0 + \frac{4}{49} + 2 \cdot 0 \cdot \frac{2}{7} + 81 + \frac{36}{49} + 2 \cdot 9 \cdot \frac{2}{7} + 1 + \frac{4}{49} - 2 \cdot 1 \cdot \frac{2}{7} + 196 + \frac{100}{49} - 2 \cdot 14 \cdot \frac{2}{7} + 441 + \frac{100}{49} - 2 \cdot 21 \cdot \frac{2}{7} \right]$ $s^2 = \frac{1}{6} \left[ 625 + 100 + 0 + 81 + 1 + 196 + 441 + \frac{100 + 100 + 4 + 36 + 4 + 100}{49} + 2 \cdot \frac{50 + 20 + 18 - 2 - 28 - 42}{7} \right]$ $s^2 = \frac{1}{6} \left[ 1444 + \frac{344}{49} + 2 \cdot \frac{26}{7} \right]$ $s^2 = \frac{1}{6} \left[ 1444 + \frac{344}{49} + \frac{52}{7} \right]$ $s^2 = \frac{1}{6} \left[ 1444 + \frac{344 + 52 \cdot 7}{49} \right]$ $s^2 = \frac{1}{6} \left[ 1444 + \frac{344 + 364}{49} \right]$ $s^2 = \frac{1}{6} \left[ 1444 + \frac{708}{49} \right]$ $s^2 = \frac{1}{6} \left[ 1444 + 14.448979591836735 \right]$ $s^2 = \frac{1}{6} \left[ 1458.4489795918367 \right]$ $s^2 = 243.07482993197278$ 计算样本标准差： $\text{样本标准差} = \sqrt{243.07482993197278} = \boxed{15.590552147367417}$ |
3 \sqrt{\frac{186}{7}} |
| zhipu | To find the sample standard deviation of the set ${25, 10, 0, 9, -1, -14, -21}$, we follow these steps: 1. **Calculate the mean (average) of the sample:** \[ \text{Mean} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{8}{7} \approx 1.14 \] 2. **Calculate each deviation from the mean, square it, and sum these squared deviations:** \[ \begin{align*} (25 - 1.14)^2 & = 595.36 \\ (10 - 1.14)^2 & = 77.44 \\ (0 - 1.14)^2 & = 1.2996 \\ (9 - 1.14)^2 & = 53.7164 \\ (-1 - 1.14)^2 & = 5.3764 \\ (-14 - 1.14)^2 & = 222.7764 \\ (-21 - 1.14)^2 & = 485.7764 \\ \end{align*} \] Sum of squared deviations: \[ 595.36 + 77.44 + 1.2996 + 53.7164 + 5.3764 + 222.7764 + 485.7764 \approx 1451.37 \] 3. **Calculate the variance (average of the squared deviations):** Since this is a sample standard deviation, we divide by \( n - 1 \) (where \( n \) is the number of observations): \[ \text{Variance} = \frac{1451.37}{7 - 1} = \frac{1451.37}{6} \approx 241.895 \] 4. **Take the square root of the variance to get the standard deviation:** \[ \text{Standard Deviation} = \sqrt{241.895} \approx 15.55 \] Thus, the sample standard deviation of the set is approximately \( 15.55 \). \[ \boxed{15.55} \] |
3 \sqrt{\frac{186}{7}} |
| deepseek-v3 | To compute the **sample standard deviation** of the dataset \(\{25, 10, 0, 9, -1, -14, -21\}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\))**: \[ \bar{x} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{8}{7} \approx 1.1429 \] 2. **Find the Squared Deviations from the Mean**: \[ \begin{align*} (25 - 1.1429)^2 &\approx 568.3265 \\ (10 - 1.1429)^2 &\approx 78.5510 \\ (0 - 1.1429)^2 &\approx 1.3061 \\ (9 - 1.1429)^2 &\approx 61.7347 \\ (-1 - 1.1429)^2 &\approx 4.5918 \\ (-14 - 1.1429)^2 &\approx 229.5918 \\ (-21 - 1.1429)^2 &\approx 488.5918 \\ \end{align*} \] Sum of squared deviations: \[ 568.3265 + 78.5510 + 1.3061 + 61.7347 + 4.5918 + 229.5918 + 488.5918 = 1432.6937 \] 3. **Compute the Sample Variance (\(s^2\))**: \[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{1432.6937}{6} \approx 238.7823 \] 4. **Calculate the Sample Standard Deviation (\(s\))**: \[ s = \sqrt{s^2} = \sqrt{238.7823} \approx 15.45 \] **Final Answer**: \[ \boxed{15.45} \] |
3 \sqrt{\frac{186}{7}} |