| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{1, -21, -21, -18, 8\}\), we follow these steps: 1. **Calculate the mean (average) of the sample:** \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] \[ \text{Mean} = \frac{1 - 21 - 21 - 18 + 8}{5} \] \[ \text{Mean} = \frac{-51}{5} = -10.2 \] 2. **Calculate the squared deviations from the mean for each data point:** \[ (1 - (-10.2))^2 = (1 + 10.2)^2 = 11.2^2 = 125.44 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = -10.8^2 = 116.64 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = -10.8^2 = 116.64 \] \[ (-18 - (-10.2))^2 = (-18 + 10.2)^2 = -7.8^2 = 60.84 \] \[ (8 - (-10.2))^2 = (8 + 10.2)^2 = 18.2^2 = 331.24 \] 3. **Sum the squared deviations:** \[ \sum (x_i - \bar{x})^2 = 125.44 + 116.64 + 116.64 + 60.84 + 331.24 \] \[ \sum (x_i - \bar{x})^2 = 750.8 \] 4. **Divide the sum of squared deviations by \(n-1\) (where \(n\) is the number of data points):** \[ \text{Sample Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{750.8}{5-1} = \frac{750.8}{4} = 187.7 \] Thus, the sample variance is: \[ \boxed{187.7} \] |
\frac{1877}{10} |
| deepseek-r1 | To compute the sample variance of the dataset \(\{1, -21, -21, -18, 8\}\), we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Compute the squared differences from the mean**: \[ (1 - (-10.2))^2 = (11.2)^2 = 125.44 \] \[ (-21 - (-10.2))^2 = (-10.8)^2 = 116.64 \] \[ (-21 - (-10.2))^2 = (-10.8)^2 = 116.64 \] \[ (-18 - (-10.2))^2 = (-7.8)^2 = 60.84 \] \[ (8 - (-10.2))^2 = (18.2)^2 = 331.24 \] 3. **Sum the squared differences**: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] 4. **Divide by \(n - 1\) (where \(n = 5\))**: \[ s^2 = \frac{750.8}{4} = 187.7 \] Thus, the sample variance is \(\boxed{187.7}\). |
\frac{1877}{10} |
| gpto1 | To compute the sample variance of the data set \(\{1, -21, -21, -18, 8\}\), follow these steps: **Step 1: Compute the sample mean \(\bar{x}\)** \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] **Step 2: Compute each squared deviation \((x_i - \bar{x})^2\)** \[ \begin{align*} x_1 &= 1, \quad (1 - (-10.2))^2 = (11.2)^2 = 125.44 \\ x_2 &= -21, \quad (-21 - (-10.2))^2 = (-10.8)^2 = 116.64 \\ x_3 &= -21, \quad \text{Same as } x_2 = 116.64 \\ x_4 &= -18, \quad (-18 - (-10.2))^2 = (-7.8)^2 = 60.84 \\ x_5 &= 8, \quad (8 - (-10.2))^2 = (18.2)^2 = 331.24 \\ \end{align*} \] **Step 3: Sum the squared deviations** \[ \sum_{i=1}^{n} (x_i - \bar{x})^2 = 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] **Step 4: Compute the sample variance \(s^2\)** \[ s^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{5 - 1} \times 750.8 = \frac{750.8}{4} = 187.7 \] **Answer:** \(\boxed{187.7}\) |
\frac{1877}{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: \( \{1, -21, -21, -18, 8\} \). **Step 1: Calculate the sample mean.** The sample mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] Calculating the sum of the data: \[ 1 + (-21) + (-21) + (-18) + 8 = 1 - 21 - 21 - 18 + 8 = -51 \] Now, calculate the mean: \[ \bar{x} = \frac{-51}{5} = -10.2 \] **Step 2: Calculate the sample variance.** The sample variance \(s^2\) is given by: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] where \(n = 5\) is the number of data points. Calculate each squared deviation from the mean: - For \(x_1 = 1\): \[ (1 - (-10.2))^2 = (1 + 10.2)^2 = 11.2^2 = 125.44 \] - For \(x_2 = -21\): \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = (-10.8)^2 = 116.64 \] - For \(x_3 = -21\): \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = (-10.8)^2 = 116.64 \] - For \(x_4 = -18\): \[ (-18 - (-10.2))^2 = (-18 + 10.2)^2 = (-7.8)^2 = 60.84 \] - For \(x_5 = 8\): \[ (8 - (-10.2))^2 = (8 + 10.2)^2 = 18.2^2 = 331.24 \] Sum of squared deviations: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] Now, calculate the sample variance: \[ s^2 = \frac{750.8}{5-1} = \frac{750.8}{4} = 187.7 \] Thus, the sample variance is: \[ \boxed{187.7} \] |
\frac{1877}{10} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the list \({1, -21, -21, -18, 8}\), we follow these steps: 1. **Calculate the mean (average) of the data:** \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] \[ \text{Mean} = \frac{1 - 21 - 21 - 18 + 8}{5} \] \[ \text{Mean} = \frac{1 - 21 - 21 - 18 + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Calculate the squared differences from the mean:** \[ (1 - (-10.2))^2 = (1 + 10.2)^2 = 11.2^2 = 125.44 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = -10.8^2 = 116.64 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = -10.8^2 = 116.64 \] \[ (-18 - (-10.2))^2 = (-18 + 10.2)^2 = -7.8^2 = 60.84 \] \[ (8 - (-10.2))^2 = (8 + 10.2)^2 = 18.2^2 = 331.24 \] 3. **Sum the squared differences:** \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] 4. **Divide by the number of data points minus one (n-1):** \[ \text{Sample Variance} = \frac{750.8}{5 - 1} = \frac{750.8}{4} = 187.7 \] Thus, the sample variance is: \[ \boxed{187.7} \] |
\frac{1877}{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 ${1, -21, -21, -18, 8}$. We add up all the numbers and divide by the total count of numbers. The sum is $1 + (-21) + (-21) + (-18) + 8 = -51$. There are $5$ numbers in the dataset. So, the sample mean is $\frac{-51}{5} = -10.2$. ## Step 2: Calculate the deviations from the mean Next, we calculate the deviations of each data point from the sample mean. For each number in the dataset, we subtract the sample mean. The deviations are $(1 - (-10.2)), (-21 - (-10.2)), (-21 - (-10.2)), (-18 - (-10.2)), (8 - (-10.2))$ which simplifies to $11.2, -10.8, -10.8, -7.8, 18.2$. ## Step 3: Square each deviation Now, we square each deviation. Squaring the deviations gives us $11.2^2, (-10.8)^2, (-10.8)^2, (-7.8)^2, 18.2^2$ which equals $125.44, 116.64, 116.64, 60.84, 331.24$. ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of these squared deviations. The sum is $125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8$. ## Step 5: Calculate the sample variance Finally, to find the sample variance, we divide the sum of the squared deviations by the total count of numbers minus one (which is $5 - 1 = 4$ in this case). The sample variance is $\frac{750.8}{4} = 187.7$. The final answer is: $\boxed{187.7}$ |
\frac{1877}{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 ${1, -21, -21, -18, 8}$. The sum of these values is $1 + (-21) + (-21) + (-18) + 8 = -51$. There are $5$ values in total. Therefore, the sample mean is $-51 / 5 = -10.2$. ## 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 $(1 - (-10.2))$, $((-21) - (-10.2))$, $((-21) - (-10.2))$, $((-18) - (-10.2))$, and $(8 - (-10.2))$. This simplifies to $11.2$, $-10.8$, $-10.8$, $-7.8$, and $18.2$. ## Step 3: Square each deviation We then square each deviation. The squared deviations are $(11.2)^2$, $(-10.8)^2$, $(-10.8)^2$, $(-7.8)^2$, and $(18.2)^2$. This simplifies to $125.44$, $116.64$, $116.64$, $60.84$, and $331.24$. ## Step 4: Calculate the sum of the squared deviations Now, we calculate the sum of the squared deviations. The sum is $125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 751.8$. ## 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. The number of values is $5$, so we divide by $5 - 1 = 4$. The sample variance is $751.8 / 4 = 187.95$. The final answer is: $\boxed{187.95}$ |
\frac{1877}{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: ${1, -21, -21, -18, 8}$ First, let's find the mean: \[ \text{Mean} = \frac{\sum x_i}{n} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] Next, we calculate the squared differences from the mean for each number: \[ (1 - (-10.2))^2 = 11.2^2 = 125.44 \\ (-21 - (-10.2))^2 = (-10.8)^2 = 116.64 \\ (-21 - (-10.2))^2 = (-10.8)^2 = 116.64 \\ (-18 - (-10.2))^2 = (-7.8)^2 = 60.84 \\ (8 - (-10.2))^2 = 18.2^2 = 331.24 \] Now, we sum these squared differences and divide by \(n-1\) (since this is a sample variance): \[ \text{Sample Variance} = s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1} = \frac{125.44 + 116.64 + 116.64 + 60.84 + 331.24}{5-1} = \frac{750.8}{4} = 187.7 \] Therefore, the sample variance of the given set of numbers is $\boxed{187.7}$. |
\frac{1877}{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 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: ${1, -21, -21, -18, 8}$, let's calculate the sample variance step by step. 1. Calculate the mean: Mean = (1 + (-21) + (-21) + (-18) + 8) / 5 = (-51) / 5 = -10.2 2. For each number, subtract the mean and square the result: (1 - (-10.2))^2 = (1 + 10.2)^2 = 11.2^2 = 125.44 (-21 - (-10.2))^2 = (-21 + 10.2)^2 = (-10.8)^2 = 116.64 (-21 - (-10.2))^2 = (-21 + 10.2)^2 = (-10.8)^2 = 116.64 (-18 - (-10.2))^2 = (-18 + 10.2)^2 = (-7.8)^2 = 60.84 (8 - (-10.2))^2 = (8 + 10.2)^2 = 18.2^2 = 331.24 3. Sum the squared differences: 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 749.8 4. Divide the sum by the number of elements minus one (since it's sample variance): Sample Variance = 749.8 / (5 - 1) = 749.8 / 4 = 187.45 Therefore, the sample variance of the given set of numbers is $\boxed{187.45}$. |
\frac{1877}{10} |
| doubao | First find the mean: $(1 - 21 - 21 - 18 + 8) \div 5 = (-31 - 21 - 18 + 8) \div 5 = (-52 - 18 + 8) \div 5 = (-70 + 8) \div 5 = -62 \div 5 = -12.4$. Then subtract the mean from each number: $1 - (-12.4) = 13.4$, $-21 - (-12.4) = -8.6$, $-21 - (-12.4) = -8.6$, $-18 - (-12.4) = -5.6$, $8 - (-12.4) = 20.4$. Square these differences: $13.4^2 = 179.56$, $(-8.6)^2 = 73.96$, $(-8.6)^2 = 73.96$, $(-5.6)^2 = 31.36$, $20.4^2 = 416.16$. Find the average of these squared differences: $(179.56 + 73.96 + 73.96 + 31.36 + 416.16) \div 5 = (253.52 + 73.96 + 31.36 + 416.16) \div 5 = (327.48 + 31.36 + 416.16) \div 5 = (358.84 + 416.16) \div 5 = 775 \div 5 = 155$. So the sample variance is $\boxed{155}$. |
\frac{1877}{10} |
| gpto1 | To compute the sample variance of the data set \(\{1, -21, -21, -18, 8\}\), follow these steps: **Step 1: Compute the sample mean \(\bar{x}\)** \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] **Step 2: Compute each squared deviation \((x_i - \bar{x})^2\)** \[ \begin{align*} x_1 &= 1, \quad (1 - (-10.2))^2 = (11.2)^2 = 125.44 \\ x_2 &= -21, \quad (-21 - (-10.2))^2 = (-10.8)^2 = 116.64 \\ x_3 &= -21, \quad \text{Same as } x_2 = 116.64 \\ x_4 &= -18, \quad (-18 - (-10.2))^2 = (-7.8)^2 = 60.84 \\ x_5 &= 8, \quad (8 - (-10.2))^2 = (18.2)^2 = 331.24 \\ \end{align*} \] **Step 3: Sum the squared deviations** \[ \sum_{i=1}^{n} (x_i - \bar{x})^2 = 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] **Step 4: Compute the sample variance \(s^2\)** \[ s^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{5 - 1} \times 750.8 = \frac{750.8}{4} = 187.7 \] **Answer:** \(\boxed{187.7}\) |
\frac{1877}{10} |
| kimi | To compute the sample variance of the given data set \(\{1, -21, -21, -18, 8\}\), we will follow these steps: 1. **Calculate the mean (\(\bar{x}\)) of the data set:** The mean is the sum of all the values divided by the number of values. \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] First, compute the sum: \[ 1 + (-21) + (-21) + (-18) + 8 = 1 - 21 - 21 - 18 + 8 = -51 \] Now, divide by the number of values (5): \[ \bar{x} = \frac{-51}{5} = -10.2 \] 2. **Calculate the squared differences from the mean for each data point:** For each \(x_i\) in the data set, compute \((x_i - \bar{x})^2\). \[ (1 - (-10.2))^2 = (1 + 10.2)^2 = 11.2^2 = 125.44 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = (-10.8)^2 = 116.64 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = (-10.8)^2 = 116.64 \] \[ (-18 - (-10.2))^2 = (-18 + 10.2)^2 = (-7.8)^2 = 60.84 \] \[ (8 - (-10.2))^2 = (8 + 10.2)^2 = 18.2^2 = 331.24 \] 3. **Calculate the sum of these squared differences:** \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 746.8 \] 4. **Compute 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). \[ s^2 = \frac{746.8}{5 - 1} = \frac{746.8}{4} = 186.7 \] Therefore, the sample variance of the data set is \(\boxed{186.7}\). |
\frac{1877}{10} |
| minimax | To compute the sample variance of the data set ${1, -21, -21, -18, 8}$, 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 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: $$ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 $$ 2. Calculate the deviations from the mean: $$ 1 - (-10.2) = 1 + 10.2 = 11.2 $$ $$ -21 - (-10.2) = -21 + 10.2 = -10.8 $$ $$ -21 - (-10.2) = -21 + 10.2 = -10.8 $$ $$ -18 - (-1 |
\frac{1877}{10} |
| wenxinyiyan | 计算过程如下： 首先，我们需要计算样本的平均值。样本平均值的计算公式是： 平均值 = (样本1 + 样本2 + ... + 样本n) / n 对于给定的样本 {1, -21, -21, -18, 8}，我们有： 平均值 = (1 + (-21) + (-21) + (-18) + 8) / 5 = (-51) / 5 = -10.2 接下来，我们计算每个样本值与平均值的差的平方，然后求这些平方差的平均值。样本方差的计算公式是： 样本方差 = [(样本1 - 平均值)^2 + (样本2 - 平均值)^2 + ... + (样本n - 平均值)^2] / (n - 1) 对于给定的样本，我们有： 样本方差 = [(1 - (-10.2))^2 + ((-21) - (-10.2))^2 + ((-21) - (-10.2))^2 + ((-18) - (-10.2))^2 + (8 - (-10.2))^2] / (5 - 1) = [(11.2)^2 + (-10.8)^2 + (-10.8)^2 + (-7.8)^2 + (18.2)^2] / 4 = [125.44 + 116.64 + 116.64 + 60.84 + 331.24] / 4 = [750.8] / 4 = 187.7 所以，样本方差是 187.7。 最终答案： 样本方差 = 187.7 |
\frac{1877}{10} |
| zhipu | To compute the sample variance of the data set \(\{1, -21, -21, -18, 8\}\), follow these steps: 1. **Calculate the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{1 - 21 - 21 - 18 + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Calculate each deviation from the mean, square it, and sum the squared deviations**: \[ (1 - (-10.2))^2 = (1 + 10.2)^2 = 11.2^2 = 125.44 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = -10.8^2 = 116.64 \] \[ (-21 - (-10.2))^2 = (-21 + 10.2)^2 = -10.8^2 = 116.64 \] \[ (-18 - (-10.2))^2 = (-18 + 10.2)^2 = -7.8^2 = 60.84 \] \[ (8 - (-10.2))^2 = (8 + 10.2)^2 = 18.2^2 = 331.24 \] 3. **Sum the squared deviations**: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 751.8 \] 4. **Calculate the sample variance (\(s^2\))**: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{751.8}{5 - 1} = \frac{751.8}{4} = 187.95 \] Therefore, the sample variance is \(\boxed{187.95}\). |
\frac{1877}{10} |
| deepseek-v3 | To compute the **sample variance** of the dataset \(\{1, -21, -21, -18, 8\}\), follow these steps: ### Step 1: Calculate the Mean (\(\bar{x}\)) First, find the mean of the dataset: \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{1 - 21 - 21 - 18 + 8}{5} = \frac{-51}{5} = -10.2 \] ### Step 2: Compute the Squared Deviations from the Mean Next, calculate each data point's deviation from the mean and square it: \[ \begin{align*} (1 - (-10.2))^2 &= (11.2)^2 = 125.44 \\ (-21 - (-10.2))^2 &= (-10.8)^2 = 116.64 \\ (-21 - (-10.2))^2 &= (-10.8)^2 = 116.64 \\ (-18 - (-10.2))^2 &= (-7.8)^2 = 60.84 \\ (8 - (-10.2))^2 &= (18.2)^2 = 331.24 \\ \end{align*} \] ### Step 3: Sum the Squared Deviations Add up all the squared deviations: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] ### Step 4: Calculate the Sample Variance Finally, divide the sum of squared deviations by the number of data points minus one (since it's a sample variance): \[ s^2 = \frac{750.8}{5 - 1} = \frac{750.8}{4} = 187.7 \] ### Final Answer \[ \boxed{187.7} \] |
\frac{1877}{10} |