| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance 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:** \[ (25 - 1.142857)^2 \approx 23.857143^2 \approx 570.142857 \] \[ (10 - 1.142857)^2 \approx 8.857143^2 \approx 78.428571 \] \[ (0 - 1.142857)^2 \approx -1.142857^2 \approx 1.306122 \] \[ (9 - 1.142857)^2 \approx 7.857143^2 \approx 61.744898 \] \[ (-1 - 1.142857)^2 \approx -2.142857^2 \approx 4.591837 \] \[ (-14 - 1.142857)^2 \approx -15.142857^2 \approx 229.306122 \] \[ (-21 - 1.142857)^2 \approx -22.142857^2 \approx 490.306122 \] 3. **Sum the squared differences:** \[ \sum (x_i - \bar{x})^2 = 570.142857 + 78.428571 + 1.306122 + 61.744898 + 4.591837 + 229.306122 + 490.306122 \] \[ \sum (x_i - \bar{x})^2 = 1435.826531 \] 4. **Divide the sum of squared differences by \(n-1\) (where \(n\) is the number of data points):** \[ \text{Sample Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{1435.826531}{6} \approx 239.304422 \] Thus, the sample variance is: \[ \boxed{239.304422} \] |
\frac{1674}{7} |
| deepseek-r1 | To compute the sample variance of the dataset \(\{25, 10, 0, 9, -1, -14, -21\}\), we follow these steps: 1. **Calculate the mean**: \[ \bar{x} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{8}{7} \] 2. **Compute the squared differences from the mean**: \[ \begin{align*} (25 - \frac{8}{7})^2 &= \left(\frac{167}{7}\right)^2 = \frac{27889}{49}, \\ (10 - \frac{8}{7})^2 &= \left(\frac{62}{7}\right)^2 = \frac{3844}{49}, \\ (0 - \frac{8}{7})^2 &= \left(\frac{-8}{7}\right)^2 = \frac{64}{49}, \\ (9 - \frac{8}{7})^2 &= \left(\frac{55}{7}\right)^2 = \frac{3025}{49}, \\ (-1 - \frac{8}{7})^2 &= \left(\frac{-15}{7}\right)^2 = \frac{225}{49}, \\ (-14 - \frac{8}{7})^2 &= \left(\frac{-106}{7}\right)^2 = \frac{11236}{49}, \\ (-21 - \frac{8}{7})^2 &= \left(\frac{-155}{7}\right)^2 = \frac{24025}{49}. \end{align*} \] 3. **Sum the squared differences**: \[ \frac{27889}{49} + \frac{3844}{49} + \frac{64}{49} + \frac{3025}{49} + \frac{225}{49} + \frac{11236}{49} + \frac{24025}{49} = \frac{70308}{49} \] 4. **Divide by \(n - 1\)** (where \(n = 7\)): \[ s^2 = \frac{\frac{70308}{49}}{6} = \frac{70308}{294} = \frac{1674}{7} \] Thus, the sample variance is \(\boxed{\dfrac{1674}{7}}\). |
\frac{1674}{7} |
| gpto1 | To compute the sample variance of the data set \(\{25, 10, 0, 9, -1, -14, -21\}\), we'll follow these steps: **1. Compute the sample mean (\(\bar{x}\)):** First, find the sum of all data points and divide by the number of data points \(n = 7\). \[ \text{Sum} = 25 + 10 + 0 + 9 + (-1) + (-14) + (-21) = 8 \] \[ \bar{x} = \frac{\text{Sum}}{n} = \frac{8}{7} \] **2. Compute each squared difference \((x_i - \bar{x})^2\):** Calculate the difference between each data point and the mean, then square it. \[ \begin{align*} (25 - \frac{8}{7})^2 &= \left(\frac{167}{7}\right)^2 = \frac{27889}{49} \\ (10 - \frac{8}{7})^2 &= \left(\frac{62}{7}\right)^2 = \frac{3844}{49} \\ (0 - \frac{8}{7})^2 &= \left(-\frac{8}{7}\right)^2 = \frac{64}{49} \\ (9 - \frac{8}{7})^2 &= \left(\frac{55}{7}\right)^2 = \frac{3025}{49} \\ (-1 - \frac{8}{7})^2 &= \left(-\frac{15}{7}\right)^2 = \frac{225}{49} \\ (-14 - \frac{8}{7})^2 &= \left(-\frac{106}{7}\right)^2 = \frac{11236}{49} \\ (-21 - \frac{8}{7})^2 &= \left(-\frac{155}{7}\right)^2 = \frac{24025}{49} \\ \end{align*} \] **3. Sum the squared differences:** \[ S = \sum (x_i - \bar{x})^2 = \frac{27889}{49} + \frac{3844}{49} + \frac{64}{49} + \frac{3025}{49} + \frac{225}{49} + \frac{11236}{49} + \frac{24025}{49} = \frac{70308}{49} \] **4. Compute the sample variance (\(s^2\)):** Divide the sum of squared differences by \(n - 1\): \[ s^2 = \frac{S}{n - 1} = \frac{\frac{70308}{49}}{6} = \frac{70308}{294} = \frac{11718}{49} \] Simplify the fraction: \[ \frac{11718}{49} = \frac{1674}{7} \] So, the sample variance is: \[ s^2 = \boxed{\dfrac{1674}{7}} \] \(\boxed{\dfrac{1674}{7}}\) |
\frac{1674}{7} |
| gpt-4o-2024-08-06 | To compute the sample variance, 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, \( n = 7 \) and the data points are \( 25, 10, 0, 9, -1, -14, -21 \). \[ \bar{x} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{8}{7} \] 2. **Calculate the squared deviations from the mean**: \[ (x_i - \bar{x})^2 \] For each data point: - \( (25 - \frac{8}{7})^2 = \left(\frac{175}{7} - \frac{8}{7}\right)^2 = \left(\frac{167}{7}\right)^2 = \frac{27889}{49} \) - \( (10 - \frac{8}{7})^2 = \left(\frac{70}{7} - \frac{8}{7}\right)^2 = \left(\frac{62}{7}\right)^2 = \frac{3844}{49} \) - \( (0 - \frac{8}{7})^2 = \left(0 - \frac{8}{7}\right)^2 = \left(-\frac{8}{7}\right)^2 = \frac{64}{49} \) - \( (9 - \frac{8}{7})^2 = \left(\frac{63}{7} - \frac{8}{7}\right)^2 = \left(\frac{55}{7}\right)^2 = \frac{3025}{49} \) - \( (-1 - \frac{8}{7})^2 = \left(-\frac{7}{7} - \frac{8}{7}\right)^2 = \left(-\frac{15}{7}\right)^2 = \frac{225}{49} \) - \( (-14 - \frac{8}{7})^2 = \left(-\frac{98}{7} - \frac{8}{7}\right)^2 = \left(-\frac{106}{7}\right)^2 = \frac{11236}{49} \) - \( (-21 - \frac{8}{7})^2 = \left(-\frac{147}{7} - \frac{8}{7}\right)^2 = \left(-\frac{155}{7}\right)^2 = \frac{24025}{49} \) 3. **Sum the squared deviations**: \[ \sum (x_i - \bar{x})^2 = \frac{27889}{49} + \frac{3844}{49} + \frac{64}{49} + \frac{3025}{49} + \frac{225}{49} + \frac{11236}{49} + \frac{24025}{49} \] \[ = \frac{27889 + 3844 + 64 + 3025 + 225 + 11236 + 24025}{49} = \frac{70308}{49} \] 4. **Calculate the sample variance**: \[ s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2 \] \[ s^2 = \frac{1}{6} \times \frac{70308}{49} = \frac{70308}{294} \] 5. **Simplify the fraction**: \[ \frac{70308}{294} = 239 \] Thus, the sample variance is \(\boxed{239}\). |
\frac{1674}{7} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the given data set \({25, 10, 0, 9, -1, -14, -21}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} \] \[ \text{Sum} = 25 + 10 + 0 + 9 - 1 - 14 - 21 = 7 \] \[ \text{Mean} = \frac{7}{7} = 1 \] 2. **Calculate the squared differences from the mean:** \[ (25 - 1)^2 = 24^2 = 576 \] \[ (10 - 1)^2 = 9^2 = 81 \] \[ (0 - 1)^2 = (-1)^2 = 1 \] \[ (9 - 1)^2 = 8^2 = 64 \] \[ (-1 - 1)^2 = (-2)^2 = 4 \] \[ (-14 - 1)^2 = (-15)^2 = 225 \] \[ (-21 - 1)^2 = (-22)^2 = 484 \] 3. **Sum the squared differences:** \[ \text{Sum of squared differences} = 576 + 81 + 1 + 64 + 4 + 225 + 484 = 1435 \] 4. **Calculate the sample variance:** \[ \text{Sample variance} = \frac{1435}{7} = 205 \] Thus, the sample variance of the data set \({25, 10, 0, 9, -1, -14, -21}\) is: \[ \boxed{205} \] |
\frac{1674}{7} |
| 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 the numbers in the dataset is $25 + 10 + 0 + 9 - 1 - 14 - 21 = 8$. ## Step 3: Calculate the sample mean There are 7 numbers in the dataset. The sample mean is the sum of the numbers divided by the count, which is $8 / 7 \approx 1.1429$. ## Step 4: Calculate the deviations from the mean For each number in the dataset, we subtract the sample mean to find the deviation from the mean. ## Step 5: Calculate the squared deviations We then square each deviation from the mean. ## Step 6: Calculate the sum of the squared deviations 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 7: Calculate each squared deviation $(23.8571)^2 + (8.8571)^2 + (-1.1429)^2 + (7.8571)^2 + (-2.1429)^2 + (-15.1429)^2 + (-22.1429)^2$. ## Step 8: Perform the squaring $569.1429 + 78.1429 + 1.3056 + 61.7143 + 4.5929 + 229.3056 + 490.3056$. ## Step 9: Sum the squared deviations $1434.5098$. ## Step 10: Calculate the sample variance The sample variance is the sum of the squared deviations divided by the count of numbers minus one. In this case, it's $1434.5098 / (7 - 1)$. ## Step 11: Calculate the final sample variance $1434.5098 / 6 = 239.08496$. The final answer is: $\boxed{239.08496}$ |
\frac{1674}{7} |
| 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 ${25, 10, 0, 9, -1, -14, -21}$. The sum of these values is $25 + 10 + 0 + 9 - 1 - 14 - 21 = 8$. There are 7 values in total. Therefore, the sample mean is $8 / 7 \approx 1.143$. ## Step 2: Subtract the sample mean from each value Next, we subtract the sample mean from each value to find the deviations from the mean. The deviations are $(25 - 1.143), (10 - 1.143), (0 - 1.143), (9 - 1.143), (-1 - 1.143), (-14 - 1.143), (-21 - 1.143)$, which 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 $(23.857)^2, (8.857)^2, (-1.143)^2, (7.857)^2, (-2.143)^2, (-15.143)^2, (-22.143)^2$, which simplifies to $569.51, 78.51, 1.31, 61.57, 4.60, 229.11, 490.11$. ## Step 4: Calculate the sum of the squared deviations Now, we calculate the sum of the squared deviations. The sum is $569.51 + 78.51 + 1.31 + 61.57 + 4.60 + 229.11 + 490.11 = 1434.22$. ## 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 7, so we divide by 6. The sample variance is $1434.22 / 6 \approx 239.037$. The final answer is: $\boxed{239.037}$ |
\frac{1674}{7} |
| Qwen2-72B-Instruct | To compute the sample variance of a set of numbers, we first need to find the mean (average) of those numbers, then calculate the squared differences from the mean for each number, and finally take the average of those squared differences. Given the set of numbers: ${25, 10, 0, 9, -1, -14, -21}$ First, let's find the mean: \[ \text{Mean} = \frac{\sum x_i}{n} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} = \frac{-2}{7} \] Next, we calculate the squared differences from the mean for each number: - For $x_1 = 25$: $(25 - (-\frac{2}{7}))^2 = (25 + \frac{2}{7})^2 = (\frac{177}{7})^2$ - For $x_2 = 10$: $(10 - (-\frac{2}{7}))^2 = (10 + \frac{2}{7})^2 = (\frac{72}{7})^2$ - For $x_3 = 0$: $(0 - (-\frac{2}{7}))^2 = (\frac{2}{7})^2$ - For $x_4 = 9$: $(9 - (-\frac{2}{7}))^2 = (9 + \frac{2}{7})^2 = (\frac{65}{7})^2$ - For $x_5 = -1$: $(-1 - (-\frac{2}{7}))^2 = (-1 + \frac{2}{7})^2 = (-\frac{5}{7})^2$ - For $x_6 = -14$: $(-14 - (-\frac{2}{7}))^2 = (-14 + \frac{2}{7})^2 = (-\frac{96}{7})^2$ - For $x_7 = -21$: $(-21 - (-\frac{2}{7}))^2 = (-21 + \frac{2}{7})^2 = (-\frac{145}{7})^2$ Now, we sum these squared differences and divide by $n-1$ (since this is the sample variance formula): \[ s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1} = \frac{(\frac{177}{7})^2 + (\frac{72}{7})^2 + (\frac{2}{7})^2 + (\frac{65}{7})^2 + (-\frac{5}{7})^2 + (-\frac{96}{7})^2 + (-\frac{145}{7})^2}{7-1} \] \[ = \frac{\frac{31329}{49} + \frac{5184}{49} + \frac{4}{49} + \frac{4225}{49} + \frac{25}{49} + \frac{9216}{49} + \frac{21025}{49}}{6} \] \[ = \frac{\frac{71008}{49}}{6} = \frac{71008}{294} = \frac{35504}{147} \approx 241.5238 \] Thus, the sample variance is approximately $\boxed{241.5238}$. |
\frac{1674}{7} |
| baichuan4 | To compute the sample variance of a set of numbers, we first need to find the mean (average) of the numbers. Then, for each number, we subtract the mean and square the result. Finally, we sum these squared differences and divide by the number of observations minus one (since it's a sample variance). Given the numbers: $25, 10, 0, 9, -1, -14, -21$, let's first calculate their mean. The mean is: $$\overline{x} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{7}{7} = 1$$ Now, we calculate the squared differences from the mean: $$(25 - 1)^2 = 24^2 = 576$$ $$(10 - 1)^2 = 9^2 = 81$$ $$(0 - 1)^2 = (-1)^2 = 1$$ $$(9 - 1)^2 = 8^2 = 64$$ $$(-1 - 1)^2 = (-2)^2 = 4$$ $$(-14 - 1)^2 = (-15)^2 = 225$$ $$(-21 - 1)^2 = (-22)^2 = 484$$ Next, we sum these squared differences: $$576 + 81 + 1 + 64 + 4 + 225 + 484 = 1435$$ Finally, we divide this sum by the number of observations minus one (since it's a sample variance): $$s^2 = \frac{1435}{7 - 1} = \frac{1435}{6} = 239.166\ldots$$ Therefore, the sample variance is approximately $239.166\ldots$. However, since we are asked to put the final answer in a $\boxed{}$, we will round it to the nearest whole number. $$\boxed{239}$$ |
\frac{1674}{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 sum of the squared differences from the mean: \[ \begin{align*} &[(25-1.143)^2+(10-1.143)^2+(0-1.143)^2+(9-1.143)^2+(-1-1.143)^2+(-14-1.143)^2+(-21-1.143)^2]\\ =&(23.857^2+8.857^2+(-1.143)^2+7.857^2+(-2.143)^2+(-15.143)^2+(-22.143)^2)\\ =&(569.494+78.432+1.304+61.744+4.592+229.024+490.294)\\ =&(647.926+1.304+61.744+4.592+229.024+490.294)\\ =&(649.23+61.744+4.592+229.024+490.294)\\ =&(710.974+4.592+229.024+490.294)\\ =&(715.566+229.024+490.294)\\ =&(944.59+490.294)\\ =&1434.884 \end{align*} \] Finally divide by the number of items (7) to get the sample variance: $1434.884 \div 7 = 204.983$. So the answer is $\boxed{204.983}$. |
\frac{1674}{7} |
| gpto1 | To compute the sample variance of the data set \(\{25, 10, 0, 9, -1, -14, -21\}\), we'll follow these steps: **1. Compute the sample mean (\(\bar{x}\)):** First, find the sum of all data points and divide by the number of data points \(n = 7\). \[ \text{Sum} = 25 + 10 + 0 + 9 + (-1) + (-14) + (-21) = 8 \] \[ \bar{x} = \frac{\text{Sum}}{n} = \frac{8}{7} \] **2. Compute each squared difference \((x_i - \bar{x})^2\):** Calculate the difference between each data point and the mean, then square it. \[ \begin{align*} (25 - \frac{8}{7})^2 &= \left(\frac{167}{7}\right)^2 = \frac{27889}{49} \\ (10 - \frac{8}{7})^2 &= \left(\frac{62}{7}\right)^2 = \frac{3844}{49} \\ (0 - \frac{8}{7})^2 &= \left(-\frac{8}{7}\right)^2 = \frac{64}{49} \\ (9 - \frac{8}{7})^2 &= \left(\frac{55}{7}\right)^2 = \frac{3025}{49} \\ (-1 - \frac{8}{7})^2 &= \left(-\frac{15}{7}\right)^2 = \frac{225}{49} \\ (-14 - \frac{8}{7})^2 &= \left(-\frac{106}{7}\right)^2 = \frac{11236}{49} \\ (-21 - \frac{8}{7})^2 &= \left(-\frac{155}{7}\right)^2 = \frac{24025}{49} \\ \end{align*} \] **3. Sum the squared differences:** \[ S = \sum (x_i - \bar{x})^2 = \frac{27889}{49} + \frac{3844}{49} + \frac{64}{49} + \frac{3025}{49} + \frac{225}{49} + \frac{11236}{49} + \frac{24025}{49} = \frac{70308}{49} \] **4. Compute the sample variance (\(s^2\)):** Divide the sum of squared differences by \(n - 1\): \[ s^2 = \frac{S}{n - 1} = \frac{\frac{70308}{49}}{6} = \frac{70308}{294} = \frac{11718}{49} \] Simplify the fraction: \[ \frac{11718}{49} = \frac{1674}{7} \] So, the sample variance is: \[ s^2 = \boxed{\dfrac{1674}{7}} \] \(\boxed{\dfrac{1674}{7}}\) |
\frac{1674}{7} |
| kimi | The sample variance is calculated as follows: $$\frac{1}{7}[(25-7)^2+(10-7)^2+(0-7)^2+(9-7)^2+(-1-7)^2+(-14-7)^2+(-21-7)^2]$$ $$=\frac{1}{7}(324+9+49+4+16+441+324)$$ $$=\frac{1}{7}(1167)$$ $$=\boxed{166.7}$$ |
\frac{1674}{7} |
| minimax | To compute the sample variance 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. Sum all the squared deviations. 5. Divide by the number of data points minus one (since this is a sample variance). Let's go through these steps: 1. Calculate the mean: $$ \text{mean} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} $$ $$ \text{mean} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} $$ $$ \text{mean} = \frac{8}{7} $$ $$ \text{mean} \approx 1.14 $$ 2. Calculate the deviations from the mean: $$ 25 - 1.14 = 23.86 $$ $$ 1 |
\frac{1674}{7} |
| 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 $\bar{x}$ 是所有数据之和除以数据的个数。 $\bar{x} = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7}$ $\bar{x} = \frac{8}{7}$ 接下来，我们计算每个数据与平均值的差的平方。 $(25 - \frac{8}{7})^2, (10 - \frac{8}{7})^2, (0 - \frac{8}{7})^2, (9 - \frac{8}{7})^2, (-1 - \frac{8}{7})^2, (-14 - \frac{8}{7})^2, (-21 - \frac{8}{7})^2$ 计算这些平方差之后，我们得到： $(\frac{167}{7})^2, (\frac{62}{7})^2, (\frac{8}{7})^2, (\frac{55}{7})^2, (\frac{15}{7})^2, (\frac{106}{7})^2, (\frac{155}{7})^2$ 现在，我们计算这些平方差的平均值，即样本方差 $s^2$。 $s^2 = \frac{(\frac{167}{7})^2 + (\frac{62}{7})^2 + (\frac{8}{7})^2 + (\frac{55}{7})^2 + (\frac{15}{7})^2 + (\frac{106}{7})^2 + (\frac{155}{7})^2}{7}$ $s^2 = \frac{\frac{27889}{49} + \frac{3844}{49} + \frac{64}{49} + \frac{3025}{49} + \frac{225}{49} + \frac{11236}{49} + \frac{24025}{49}}{7}$ $s^2 = \frac{\frac{69288}{49}}{7}$ $s^2 = \frac{69288}{343}$ $s^2 = \frac{9898 \times 7 + 1}{343}$ $s^2 = \frac{69289}{343}$ $s^2 = 202.0087463556851$ 所以，样本方差为 $\boxed{202.0087463556851}$。 |
\frac{1674}{7} |
| zhipu | 1. **Calculate the mean of the data set**: The data set given is ${25, 10, 0, 9, -1, -14, -21}$. First, we find the mean ($\mu$) of these numbers: \[ \mu = \frac{25 + 10 + 0 + 9 - 1 - 14 - 21}{7} = \frac{8}{7} \] 2. **Calculate the squared deviations from the mean**: For each data point $x_i$, compute $(x_i - \mu)^2$: - For $25$: $(25 - \frac{8}{7})^2 = (\frac{163}{7})^2 = \frac{26569}{49}$ - For $10$: $(10 - \frac{8}{7})^2 = (\frac{62}{7})^2 = \frac{3844}{49}$ - For $0$: $(0 - \frac{8}{7})^2 = (\frac{-8}{7})^2 = \frac{64}{49}$ - For $9$: $(9 - \frac{8}{7})^2 = (\frac{55}{7})^2 = \frac{3025}{49}$ - For $-1$: $(-1 - \frac{8}{7})^2 = (\frac{-15}{7})^2 = \frac{225}{49}$ - For $-14$: $(-14 - \frac{8}{7})^2 = (\frac{-104}{7})^2 = \frac{10816}{49}$ - For $-21$: $(-21 - \frac{8}{7})^2 = (\frac{-157}{7})^2 = \frac{24649}{49}$ 3. **Sum the squared deviations**: \[ \text{Sum of squared deviations} = \frac{26569 + 3844 + 64 + 3025 + 225 + 10816 + 24649}{49} = \frac{73442}{49} \] 4. **Compute the sample variance**: The sample variance $s^2$ is the sum of squared deviations divided by the degrees of freedom ($n-1$ where $n$ is the number of observations): \[ s^2 = \frac{\frac{73442}{49}}{7-1} = \frac{73442}{343} = 214.9705 \] However, keeping the exact value as a fraction: \[ s^2 = \frac{73442}{343} \] Simplifying this fraction: \[ s^2 = \frac{2026}{49} \] 5. **Conclusion**: The sample variance of the data set is $\frac{2026}{49}$. Thus, the final answer is ${\frac{2026}{49}}$. The final answer is ${\boxed{\frac{2026}{49}}}$. |
\frac{1674}{7} |
| deepseek-v3 | To compute the **sample variance** of the dataset \(\{25, 10, 0, 9, -1, -14, -21\}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{25 + 10 + 0 + 9 + (-1) + (-14) + (-21)}{7} \] \[ \bar{x} = \frac{8}{7} \approx 1.1429 \] 2. **Find the Squared Deviations from the Mean:** \[ \begin{align*} (25 - 1.1429)^2 &\approx 568.306 \\ (10 - 1.1429)^2 &\approx 79.510 \\ (0 - 1.1429)^2 &\approx 1.306 \\ (9 - 1.1429)^2 &\approx 61.224 \\ (-1 - 1.1429)^2 &\approx 4.592 \\ (-14 - 1.1429)^2 &\approx 230.592 \\ (-21 - 1.1429)^2 &\approx 488.800 \\ \end{align*} \] **Total of squared deviations:** \[ 568.306 + 79.510 + 1.306 + 61.224 + 4.592 + 230.592 + 488.800 = 1434.33 \] 3. **Compute the Sample Variance (\(s^2\)):** \[ s^2 = \frac{\text{Total of squared deviations}}{n - 1} = \frac{1434.33}{6} \approx 239.055 \] Rounding to two decimal places: \[ s^2 \approx 239.06 \] **Final Answer:** \[ \boxed{239.06} \] |
\frac{1674}{7} |