| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{10, 18, 15, -5, -14, -21\}\), we follow these steps: 1. **Calculate the mean (average) of the sample:** \[ \text{Mean} = \frac{10 + 18 + 15 + (-5) + (-14) + (-21)}{6} \] \[ \text{Mean} = \frac{10 + 18 + 15 - 5 - 14 - 21}{6} \] \[ \text{Mean} = \frac{6}{6} = 1 \] 2. **Calculate the squared differences from the mean for each data point:** \[ (10 - 1)^2 = 9^2 = 81 \] \[ (18 - 1)^2 = 17^2 = 289 \] \[ (15 - 1)^2 = 14^2 = 196 \] \[ (-5 - 1)^2 = (-6)^2 = 36 \] \[ (-14 - 1)^2 = (-15)^2 = 225 \] \[ (-21 - 1)^2 = (-22)^2 = 484 \] 3. **Sum the squared differences:** \[ \sum (x_i - \bar{x})^2 = 81 + 289 + 196 + 36 + 225 + 484 \] \[ \sum (x_i - \bar{x})^2 = 1311 \] 4. **Calculate the sample variance:** \[ \text{Sample Variance} = \frac{\sum (x_i - \bar{x})^2}{n - 1} \] where \(n\) is the number of data points. Here, \(n = 6\): \[ \text{Sample Variance} = \frac{1311}{6 - 1} = \frac{1311}{5} = 262.2 \] Thus, the sample variance is: \[ \boxed{262.2} \] |
\frac{2619}{10} |
| deepseek-r1 | To compute the sample variance of the dataset \({10, 18, 15, -5, -14, -21}\), follow these steps: 1. **Calculate the mean**: \[ \text{Mean} = \frac{10 + 18 + 15 - 5 - 14 - 21}{6} = \frac{3}{6} = 0.5 \] 2. **Find the squared differences from the mean**: \[ \begin{align*} (10 - 0.5)^2 &= 9.5^2 = 90.25 \\ (18 - 0.5)^2 &= 17.5^2 = 306.25 \\ (15 - 0.5)^2 &= 14.5^2 = 210.25 \\ (-5 - 0.5)^2 &= (-5.5)^2 = 30.25 \\ (-14 - 0.5)^2 &= (-14.5)^2 = 210.25 \\ (-21 - 0.5)^2 &= (-21.5)^2 = 462.25 \\ \end{align*} \] 3. **Sum the squared differences**: \[ 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1309.5 \] 4. **Divide by \(n - 1\)** (where \(n = 6\)): \[ \text{Sample Variance} = \frac{1309.5}{6 - 1} = \frac{1309.5}{5} = 261.9 \] Thus, the sample variance of the dataset is \(\boxed{261.9}\). |
\frac{2619}{10} |
| gpto1 | To compute the sample variance of the dataset \(\{10, 18, 15, -5, -14, -21\}\), we'll follow these steps: **Step 1: Calculate the sample mean (\(\bar{x}\))** Add up all the data points and divide by the number of data points (which is 6 in this case): \[ \bar{x} = \frac{10 + 18 + 15 + (-5) + (-14) + (-21)}{6} = \frac{3}{6} = 0.5 \] **Step 2: Calculate each squared deviation from the mean** Compute \((x_i - \bar{x})^2\) for each data point: 1. \( (10 - 0.5)^2 = (9.5)^2 = 90.25 \) 2. \( (18 - 0.5)^2 = (17.5)^2 = 306.25 \) 3. \( (15 - 0.5)^2 = (14.5)^2 = 210.25 \) 4. \( (-5 - 0.5)^2 = (-5.5)^2 = 30.25 \) 5. \( (-14 - 0.5)^2 = (-14.5)^2 = 210.25 \) 6. \( (-21 - 0.5)^2 = (-21.5)^2 = 462.25 \) **Step 3: Sum up the squared deviations** \[ \text{Total sum of squared deviations} = 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1309.5 \] **Step 4: Calculate the sample variance (\(s^2\))** Divide the total sum of squared deviations by \(n - 1\) (since it's a sample variance): \[ s^2 = \frac{1309.5}{6 - 1} = \frac{1309.5}{5} = 261.9 \] **Final Answer:** \(\boxed{261.9}\) |
\frac{2619}{10} |
| 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: \( \{10, 18, 15, -5, -14, -21\} \). **Step 1: Calculate the sample mean.** The sample mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \( n \) is the number of data points. Here, \( n = 6 \). \[ \bar{x} = \frac{10 + 18 + 15 - 5 - 14 - 21}{6} \] Calculate the sum of the data points: \[ 10 + 18 + 15 - 5 - 14 - 21 = 3 \] Now, calculate the mean: \[ \bar{x} = \frac{3}{6} = 0.5 \] **Step 2: Calculate the sample variance.** The sample variance \( s^2 \) is given by: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] Substitute \( \bar{x} = 0.5 \) and \( n = 6 \): \[ s^2 = \frac{1}{5} \left( (10 - 0.5)^2 + (18 - 0.5)^2 + (15 - 0.5)^2 + (-5 - 0.5)^2 + (-14 - 0.5)^2 + (-21 - 0.5)^2 \right) \] Calculate each squared deviation: \[ (10 - 0.5)^2 = 9.5^2 = 90.25 \] \[ (18 - 0.5)^2 = 17.5^2 = 306.25 \] \[ (15 - 0.5)^2 = 14.5^2 = 210.25 \] \[ (-5 - 0.5)^2 = (-5.5)^2 = 30.25 \] \[ (-14 - 0.5)^2 = (-14.5)^2 = 210.25 \] \[ (-21 - 0.5)^2 = (-21.5)^2 = 462.25 \] Sum these squared deviations: \[ 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1309.5 \] Now, calculate the sample variance: \[ s^2 = \frac{1309.5}{5} = 261.9 \] Thus, the sample variance is: \[ \boxed{261.9} \] |
\frac{2619}{10} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the given data set \({10, 18, 15, -5, -14, -21}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{10 + 18 + 15 + (-5) + (-14) + (-21)}{6} \] \[ \text{Sum} = 10 + 18 + 15 - 5 - 14 - 21 = 3 \] \[ \text{Mean} = \frac{3}{6} = 0.5 \] 2. **Calculate the squared differences from the mean:** \[ (10 - 0.5)^2 = 9.5^2 = 90.25 \] \[ (18 - 0.5)^2 = 17.5^2 = 306.25 \] \[ (15 - 0.5)^2 = 14.5^2 = 210.25 \] \[ (-5 - 0.5)^2 = -5.5^2 = 30.25 \] \[ (-14 - 0.5)^2 = -14.5^2 = 210.25 \] \[ (-21 - 0.5)^2 = -21.5^2 = 462.25 \] 3. **Sum the squared differences:** \[ \text{Sum of squared differences} = 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 \] \[ \text{Sum of squared differences} = 1009.5 \] 4. **Calculate the sample variance:** \[ \text{Sample variance} = \frac{1009.5}{6} = 168.25 \] Thus, the sample variance of the data set \({10, 18, 15, -5, -14, -21}\) is: \[ \boxed{168.25} \] |
\frac{2619}{10} |
| 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. The dataset is ${10, 18, 15, -5, -14, -21}$. We add up all the numbers and divide by the total count of numbers. ## Step 2: Sum up all the numbers in the dataset $10 + 18 + 15 - 5 - 14 - 21 = 3$. ## Step 3: Calculate the sample mean The total count of numbers is 6. So, the sample mean is $3 / 6 = 0.5$. ## Step 4: Calculate the deviations from the mean Next, we need to find the deviations of each number from the sample mean. This is done by subtracting the sample mean from each number in the dataset. ## Step 5: Calculate the squared deviations After finding the deviations, we need to square each deviation. The squared deviations are $(10-0.5)^2, (18-0.5)^2, (15-0.5)^2, (-5-0.5)^2, (-14-0.5)^2, (-21-0.5)^2$. ## Step 6: Calculate the squared deviations $(10-0.5)^2 = 9.5^2 = 90.25$, $(18-0.5)^2 = 17.5^2 = 306.25$, $(15-0.5)^2 = 14.5^2 = 210.25$, $(-5-0.5)^2 = -5.5^2 = 30.25$, $(-14-0.5)^2 = -14.5^2 = 210.25$, $(-21-0.5)^2 = -21.5^2 = 462.25$. ## Step 7: Sum up the squared deviations $90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1309$. ## Step 8: Calculate the sample variance The sample variance is the sum of the squared deviations divided by the total count of numbers minus 1. So, the sample variance is $1309 / (6 - 1) = 1309 / 5 = 261.8$. The final answer is: $\boxed{261.8}$ |
\frac{2619}{10} |
| 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 ${10, 18, 15, -5, -14, -21}$. The sum of these values is $10 + 18 + 15 - 5 - 14 - 21 = 3$. There are 6 values in total. Therefore, the sample mean is $3 / 6 = 0.5$. ## 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 $(10 - 0.5), (18 - 0.5), (15 - 0.5), (-5 - 0.5), (-14 - 0.5), (-21 - 0.5)$. This simplifies to $9.5, 17.5, 14.5, -5.5, -14.5, -21.5$. ## Step 3: Square each deviation Now, we square each deviation. The squared deviations are $9.5^2, 17.5^2, 14.5^2, (-5.5)^2, (-14.5)^2, (-21.5)^2$. This simplifies to $90.25, 306.25, 210.25, 30.25, 210.25, 462.25$. ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of these squared deviations. The sum is $90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1309.5$. ## Step 5: Calculate the sample variance Finally, to find the sample variance, we divide the sum of the squared deviations by the number of values minus one (which is $6 - 1 = 5$). Therefore, the sample variance is $1309.5 / 5 = 261.9$. The final answer is: $\boxed{261.9}$ |
\frac{2619}{10} |
| 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 average those squared differences. Given the set of numbers: ${10, 18, 15, -5, -14, -21}$ First, let's find the mean: \[ \text{Mean} = \frac{\sum x_i}{n} = \frac{10 + 18 + 15 + (-5) + (-14) + (-21)}{6} = \frac{(-7)}{6} = -1.1667 \] Next, we calculate the squared differences from the mean for each number: \[ (10 - (-1.1667))^2 = 11.1667^2 = 124.6944 \] \[ (18 - (-1.1667))^2 = 19.1667^2 = 367.3111 \] \[ (15 - (-1.1667))^2 = 16.1667^2 = 261.3611 \] \[ (-5 - (-1.1667))^2 = -3.8333^2 = 14.6778 \] \[ (-14 - (-1.1667))^2 = -12.8333^2 = 164.6778 \] \[ (-21 - (-1.1667))^2 = -19.8333^2 = 393.3333 \] Finally, we average these squared differences to get the sample variance: \[ \text{Sample Variance} = \frac{\sum(x_i - \bar{x})^2}{n-1} = \frac{124.6944 + 367.3111 + 261.3611 + 14.6778 + 164.6778 + 393.3333}{6-1} \] \[ = \frac{1326.0555}{5} = 265.2111 \] Therefore, the sample variance is $\boxed{265.2111}$. |
\frac{2619}{10} |
| 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 up these squared differences and divide by the number of observations minus one (since it's a sample variance). Given the numbers: $10, 18, 15, -5, -14, -21$, let's calculate the sample variance step by step. 1. **Calculate the Mean:** $$ \text{Mean} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1}{6} (10 + 18 + 15 - 5 - 14 - 21) = \frac{1}{6} (13) = 2.1667 $$ 2. **Calculate the Squared Differences from the Mean:** $$ \begin{align*} (10 - 2.1667)^2 &= 57.7778 \\ (18 - 2.1667)^2 &= 257.7778 \\ (15 - 2.1667)^2 &= 187.7778 \\ (-5 - 2.1667)^2 &= 48.7778 \\ (-14 - 2.1667)^2 &= 227.7778 \\ (-21 - 2.1667)^2 &= 467.7778 \\ \end{align*} $$ 3. **Sum the Squared Differences:** $$ \text{Sum of Squared Differences} = 57.7778 + 257.7778 + 187.7778 + 48.7778 + 227.7778 + 467.7778 = 1247.6666 $$ 4. **Divide by the Number of Observations Minus One:** $$ \text{Sample Variance} = \frac{\text{Sum of Squared Differences}}{n - 1} = \frac{1247.6666}{6 - 1} = \frac{1247.6666}{5} = 249.5333 $$ Therefore, the sample variance of the given numbers is approximately $249.5333$. $$ \boxed{249.5333} $$ |
\frac{2619}{10} |
| doubao | First find the mean: $\overline{x} = (10+18+15-5-14-21)\div6 = (28+15-5-14-21)\div6 = (43-5-14-21)\div6 = (38-14-21)\div6 = (24-21)\div6 = 3\div6 = 0.5$. Then calculate the deviations from the mean: $(10-0.5)^2 = 9.5^2 = 90.25$, $(18-0.5)^2 = 17.5^2 = 306.25$, $(15-0.5)^2 = 14.5^2 = 210.25$, $(-5-0.5)^2 = (-5.5)^2 = 30.25$, $(-14-0.5)^2 = (-14.5)^2 = 210.25$, $(-21-0.5)^2 = (-21.5)^2 = 462.25$. The sum of the squared deviations is $90.25+306.25+210.25+30.25+210.25+462.25 = 1309.5$. The sample variance is the sum of the squared deviations divided by $n-1$ (where $n$ is the number of data points), so the sample variance is $1309.5\div(6-1) = 1309.5\div5 = 261.9$. Put the answer in the box: $\boxed{261.9}$. |
\frac{2619}{10} |
| gpto1 | To compute the sample variance of the dataset \(\{10, 18, 15, -5, -14, -21\}\), we'll follow these steps: **Step 1: Calculate the sample mean (\(\bar{x}\))** Add up all the data points and divide by the number of data points (which is 6 in this case): \[ \bar{x} = \frac{10 + 18 + 15 + (-5) + (-14) + (-21)}{6} = \frac{3}{6} = 0.5 \] **Step 2: Calculate each squared deviation from the mean** Compute \((x_i - \bar{x})^2\) for each data point: 1. \( (10 - 0.5)^2 = (9.5)^2 = 90.25 \) 2. \( (18 - 0.5)^2 = (17.5)^2 = 306.25 \) 3. \( (15 - 0.5)^2 = (14.5)^2 = 210.25 \) 4. \( (-5 - 0.5)^2 = (-5.5)^2 = 30.25 \) 5. \( (-14 - 0.5)^2 = (-14.5)^2 = 210.25 \) 6. \( (-21 - 0.5)^2 = (-21.5)^2 = 462.25 \) **Step 3: Sum up the squared deviations** \[ \text{Total sum of squared deviations} = 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1309.5 \] **Step 4: Calculate the sample variance (\(s^2\))** Divide the total sum of squared deviations by \(n - 1\) (since it's a sample variance): \[ s^2 = \frac{1309.5}{6 - 1} = \frac{1309.5}{5} = 261.9 \] **Final Answer:** \(\boxed{261.9}\) |
\frac{2619}{10} |
| kimi | To compute the sample variance of the given dataset \(\{10, 18, 15, -5, -14, -21\}\), 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. \[ \bar{x} = \frac{10 + 18 + 15 + (-5) + (-14) + (-21)}{6} \] First, compute the sum: \[ 10 + 18 + 15 - 5 - 14 - 21 = 3 \] Now, divide by the number of values (6): \[ \bar{x} = \frac{3}{6} = 0.5 \] 2. **Calculate the squared differences from the mean for each value:** For each value \(x_i\) in the dataset, compute \((x_i - \bar{x})^2\). \[ (10 - 0.5)^2 = 9.5^2 = 90.25 \] \[ (18 - 0.5)^2 = 17.5^2 = 306.25 \] \[ (15 - 0.5)^2 = 14.5^2 = 210.25 \] \[ (-5 - 0.5)^2 = (-5.5)^2 = 30.25 \] \[ (-14 - 0.5)^2 = (-14.5)^2 = 210.25 \] \[ (-21 - 0.5)^2 = (-21.5)^2 = 462.25 \] 3. **Calculate the sum of these squared differences:** Add up all the squared differences: \[ 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1310 \] 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 in the dataset. \[ s^2 = \frac{1310}{6 - 1} = \frac{1310}{5} = 262 \] Therefore, the sample variance of the dataset is \(\boxed{262}\). |
\frac{2619}{10} |
| minimax | To compute the sample variance of the given data set ${10, 18, 15, -5, -14, -21}$, 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 deviation of each data point from the mean. 3. Square each deviation. 4. Sum all the squared deviations. 5. Divide the sum by the number of data points minus one (since this is a sample variance). Let's go through these steps: 1. Calculate the mean: $$ \text{Mean} = \frac{10 + 18 + 15 - 5 - 14 - 21}{6} = \frac{23}{6} \approx 3.8333 $$ 2. Calculate the deviations from the mean: $$ 10 - 3.8333 \approx 6.1667 $$ $$ 18 - 3.8333 \approx 14.1667 $$ $$ 15 - 3.8333 \approx 11.1667 $$ |
\frac{2619}{10} |
| wenxinyiyan | 首先，我们需要计算给定数据集${10, 18, 15, -5, -14, -21}$的均值。均值的计算公式为： $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ 其中，$n$是数据点的数量，$x_i$是各个数据点。 对于给定的数据集，均值为： $\bar{x} = \frac{1}{6} (10 + 18 + 15 - 5 - 14 - 21) = \frac{1}{6} \times 3 = \frac{1}{2}$ 接下来，我们计算每个数据点与均值的差的平方。这些平方差分别是： $(10 - \frac{1}{2})^2 = (\frac{19}{2})^2 = \frac{361}{4}$ $(18 - \frac{1}{2})^2 = (\frac{35}{2})^2 = \frac{1225}{4}$ $(15 - \frac{1}{2})^2 = (\frac{29}{2})^2 = \frac{841}{4}$ $(-5 - \frac{1}{2})^2 = (-\frac{11}{2})^2 = \frac{121}{4}$ $(-14 - \frac{1}{2})^2 = (-\frac{29}{2})^2 = \frac{841}{4}$ $(-21 - \frac{1}{2})^2 = (-\frac{43}{2})^2 = \frac{1849}{4}$ 最后，我们计算这些平方差的均值，得到样本方差$s^2$： $s^2 = \frac{1}{6-1} \left[ \frac{361}{4} + \frac{1225}{4} + \frac{841}{4} + \frac{121}{4} + \frac{841}{4} + \frac{1849}{4} \right]$ $= \frac{1}{5} \left[ \frac{5248}{4} \right] = \frac{1312}{5} = \boxed{262.4}$ 所以，给定数据集的样本方差是262.4。 |
\frac{2619}{10} |
| zhipu | To find the sample variance of the data set \(\{10, 18, 15, -5, -14, -21\}\), we follow these steps: 1. **Calculate the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \(n\) is the number of data points. Here, \(n = 6\): \[ \bar{x} = \frac{10 + 18 + 15 - 5 - 14 - 21}{6} = \frac{3}{6} = 0.5 \] 2. **Calculate the squared deviations from the mean**: For each data point \(x_i\), compute \((x_i - \bar{x})^2\): \[ (10 - 0.5)^2 = 9.5^2 = 90.25 \] \[ (18 - 0.5)^2 = 17.5^2 = 306.25 \] \[ (15 - 0.5)^2 = 14.5^2 = 210.25 \] \[ (-5 - 0.5)^2 = (-5.5)^2 = 30.25 \] \[ (-14 - 0.5)^2 = (-14.5)^2 = 210.25 \] \[ (-21 - 0.5)^2 = (-21.5)^2 = 462.25 \] 3. **Sum the squared deviations**: \[ \sum_{i=1}^{n} (x_i - \bar{x})^2 = 90.25 + 306.25 + 210.25 + 30.25 + 210.25 + 462.25 = 1319 \] 4. **Calculate the sample variance (\(s^2\))**: The formula for sample variance is: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] Here, \(n - 1 = 6 - 1 = 5\): \[ s^2 = \frac{1319}{5} = 263.8 \] Therefore, the sample variance of the data set is \(\boxed{263.8}\). |
\frac{2619}{10} |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-bed0af60-07c6-961a-af4a-e71f45b1a342', 'request_id': 'bed0af60-07c6-961a-af4a-e71f45b1a342'} | \frac{2619}{10} |