| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{22, -7\}\), we follow these steps: 1. **Calculate the mean (average) of the set:** \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{22 - 7}{2} = \frac{15}{2} = 7.5 \] 2. **Compute the squared deviations from the mean for each data point:** - For 22: \[ (22 - 7.5)^2 = 14.5^2 = 210.25 \] - For -7: \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] 3. **Sum the squared deviations:** \[ 210.25 + 210.25 = 420.5 \] 4. **Divide the sum of squared deviations by \(n-1\) (where \(n\) is the number of data points):** \[ \text{Sample Variance} = \frac{420.5}{2-1} = \frac{420.5}{1} = 420.5 \] Thus, the sample variance is: \[ \boxed{420.5} \] |
\frac{841}{2} |
| deepseek-r1 | To compute the sample variance of the numbers \( \{22, -7\} \): 1. **Calculate the mean**: \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Find the deviations from the mean and square them**: \[ 22 - 7.5 = 14.5 \quad \text{and} \quad (14.5)^2 = 210.25 \] \[ -7 - 7.5 = -14.5 \quad \text{and} \quad (-14.5)^2 = 210.25 \] 3. **Sum the squared deviations**: \[ 210.25 + 210.25 = 420.5 \] 4. **Divide by \( n - 1 \) (where \( n = 2 \))**: \[ \text{Sample Variance} = \frac{420.5}{2 - 1} = 420.5 \] Thus, the sample variance of the data set is \(\boxed{420.5}\). |
\frac{841}{2} |
| gpto1 | To compute the sample variance of the data set \(\{22, -7\}\), we follow these steps: **1. Compute the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] **2. Compute the squared differences from the mean for each data point**: For 22: \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] For -7: \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] **3. Sum the squared differences**: \[ \text{Sum} = 210.25 + 210.25 = 420.5 \] **4. Divide by \(n - 1\) to find the sample variance**: Since the sample size \(n = 2\), we have: \[ s^2 = \frac{420.5}{2 - 1} = \frac{420.5}{1} = 420.5 \] **Answer**: \[ \boxed{420.5} \] |
\frac{841}{2} |
| gpt-4o-2024-08-06 | To compute the sample variance of the data set \(\{22, -7\}\), we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Calculate the squared deviations from the mean**: - For \(22\): \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] - For \(-7\): \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] 3. **Sum the squared deviations**: \[ 210.25 + 210.25 = 420.5 \] 4. **Divide by \(n-1\) to find the sample variance** (where \(n\) is the number of data points, \(n = 2\)): \[ s^2 = \frac{420.5}{2 - 1} = \frac{420.5}{1} = 420.5 \] Thus, the sample variance is \(\boxed{420.5}\). |
\frac{841}{2} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the data set \({22, -7}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Calculate the squared differences from the mean for each data point:** - For \(22\): \[ (22 - 7.5)^2 = 14.5^2 = 210.25 \] - For \(-7\): \[ (-7 - 7.5)^2 = -14.5^2 = 210.25 \] 3. **Sum the squared differences:** \[ 210.25 + 210.25 = 420.5 \] 4. **Divide the sum of the squared differences by the number of data points minus one (n-1):** \[ \text{Sample Variance} = \frac{420.5}{2 - 1} = 420.5 \] Thus, the sample variance of \({22, -7}\) is: \[ \boxed{420.5} \] |
\frac{841}{2} |
| 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 of the given data set. The sample mean is calculated by summing all the data points and then dividing by the number of data points. In this case, the data set is {22, -7}. The sample mean is (22 + (-7)) / 2 = 15 / 2 = 7.5. ## Step 2: Calculate the deviations from the mean Next, we calculate the deviations of each data point from the sample mean. For the data point 22, the deviation is 22 - 7.5 = 14.5. For the data point -7, the deviation is -7 - 7.5 = -14.5. ## Step 3: Square the deviations We then square each of these deviations. The squared deviation for 22 is (14.5)^2 = 210.25. The squared deviation for -7 is (-14.5)^2 = 210.25. ## Step 4: Calculate the sum of the squared deviations Now, we sum these squared deviations. The sum is 210.25 + 210.25 = 420.5. ## Step 5: Calculate the sample variance Finally, to find the sample variance, we divide the sum of the squared deviations by the number of data points minus one (n-1). In this case, n = 2, so we divide by 2 - 1 = 1. The sample variance is 420.5 / 1 = 420.5. The final answer is: $\boxed{420.5}$ |
\frac{841}{2} |
| 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. In this case, the values are 22 and -7. The sample mean is (22 + (-7)) / 2 = 15 / 2 = 7.5. ## Step 2: Calculate the deviations from the mean Next, we need to calculate the deviations from the mean for each value. The deviation for the first value is 22 - 7.5 = 14.5, and the deviation for the second value is -7 - 7.5 = -14.5. ## Step 3: Calculate the squared deviations Now, we need to calculate the squared deviations. The squared deviation for the first value is (14.5)^2 = 210.25, and the squared deviation for the second value is (-14.5)^2 = 210.25. ## Step 4: Calculate the sum of the squared deviations The sum of the squared deviations is 210.25 + 210.25 = 420.5. ## Step 5: Calculate the sample variance Finally, we can calculate the sample variance by dividing the sum of the squared deviations by the number of values minus 1. In this case, the number of values is 2, so we divide by 2 - 1 = 1. The sample variance is 420.5 / 1 = 420.5. The final answer is: $\boxed{420.5}$ |
\frac{841}{2} |
| Qwen2-72B-Instruct | To compute the sample variance of a set of numbers, we first need to calculate the mean (average) of those numbers, then find the squared differences from the mean for each number, and finally average those squared differences. Given the set ${22, -7}$: 1. Calculate the mean: \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. Find the squared differences from the mean for each number: \[ (22 - 7.5)^2 = 14.5^2 = 210.25 \] \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] 3. Average those squared differences (for sample variance, divide by \(n-1\), where \(n\) is the number of items): \[ \text{Sample Variance} = \frac{210.25 + 210.25}{2-1} = \frac{420.5}{1} = 420.5 \] Therefore, the sample variance of ${22, -7}$ is $\boxed{420.5}$. |
\frac{841}{2} |
| baichuan4 | To compute the sample variance of a set of numbers, we first need to calculate 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 numbers 22 and -7, let's follow these steps: 1. Calculate the mean: Mean = (22 + (-7)) / 2 = 15 / 2 = 7.5 2. For each number, subtract the mean and square the result: For 22: (22 - 7.5)^2 = 14.5^2 = 210.25 For -7: (-7 - 7.5)^2 = (-14.5)^2 = 210.25 3. Sum the squared differences: Sum = 210.25 + 210.25 = 420.5 4. Divide the sum by the number of elements minus one (since it's sample variance): Sample Variance = Sum / (Number of elements - 1) = 420.5 / (2 - 1) = 420.5 / 1 = 420.5 Therefore, the sample variance of the set {22, -7} is 420.5. Here is the final answer in a box: $$ \boxed{420.5} $$ |
\frac{841}{2} |
| doubao | The mean of the two numbers is $\frac{22+(-7)}{2}=\frac{15}{2}=7.5$. The squared deviations from the mean are: $(22-7.5)^2=(14.5)^2=210.25$ and $(-7-7.5)^2=(-14.5)^2=210.25$. The sum of the squared deviations is $210.25+210.25=420.5$. The sample variance is the sum of the squared deviations divided by the number of data points minus 1, so the sample variance is $\frac{420.5}{2-1}=420.5$. Putting it in the box: $\boxed{420.5}$. |
\frac{841}{2} |
| gpto1 | To compute the sample variance of the data set \(\{22, -7\}\), we follow these steps: **1. Compute the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] **2. Compute the squared differences from the mean for each data point**: For 22: \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] For -7: \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] **3. Sum the squared differences**: \[ \text{Sum} = 210.25 + 210.25 = 420.5 \] **4. Divide by \(n - 1\) to find the sample variance**: Since the sample size \(n = 2\), we have: \[ s^2 = \frac{420.5}{2 - 1} = \frac{420.5}{1} = 420.5 \] **Answer**: \[ \boxed{420.5} \] |
\frac{841}{2} |
| kimi | To compute the sample variance of the dataset \(\{22, -7\}\), we will follow these steps: 1. **Calculate the mean (average) of the dataset:** The mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{\sum x_i}{n} \] where \(x_i\) are the data points and \(n\) is the number of data points. For our dataset \(\{22, -7\}\): \[ \bar{x} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Calculate the squared differences from the mean:** For each data point \(x_i\), compute \((x_i - \bar{x})^2\). For \(x_1 = 22\): \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] For \(x_2 = -7\): \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] 3. **Calculate the sum of the squared differences:** Add the squared differences together: \[ \sum (x_i - \bar{x})^2 = 210.25 + 210.25 = 420.5 \] 4. **Calculate the sample variance:** The sample variance \(s^2\) is given by: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \] where \(n-1\) is the degrees of freedom. For our dataset with \(n = 2\): \[ s^2 = \frac{420.5}{2-1} = \frac{420.5}{1} = 420.5 \] Therefore, the sample variance of the dataset \(\{22, -7\}\) is \(\boxed{420.5}\). |
\frac{841}{2} |
| minimax | To compute the sample variance of the set ${22, -7}$, we 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 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: $$ \text{mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 $$ 2. Calculate the deviations from the mean: $$ 22 - 7.5 = 14.5 $$ $$ -7 - 7.5 = -14.5 $$ 3. Square each deviation: $$ (14.5)^2 = 210.25 $$ $$ (-14.5)^2 = 210.25 $$ 4. Sum the squared deviations: $$ 210.25 + 210.25 = 420.5 $$ 5. Divide |
\frac{841}{2} |
| wenxinyiyan | 首先，我们需要计算给定数据集${22, -7}$的均值。均值的计算公式为： $\\text{均值} = \\frac{\\text{数据之和}}{\\text{数据个数}}$ 对于数据集${22, -7}$，均值计算如下： $\\text{均值} = \\frac{22 + (-7)}{2} = \\frac{15}{2} = 7.5$ 接下来，我们计算每个数据与均值的差的平方。对于数据$22$和$-7$，这些差的平方分别为： $(22 - 7.5)^2 = 14.5^2 = 210.25$ $(-7 - 7.5)^2 = 14.5^2 = 210.25$ 然后，我们计算这些差的平方的均值，即样本方差。样本方差的计算公式为： $\\text{样本方差} = \\frac{\\text{差的平方之和}}{\\text{数据个数}}$ 对于数据集${22, -7}$，样本方差计算如下： $\\text{样本方差} = \\frac{210.25 + 210.25}{2} = \\frac{420.5}{2} = 210.25$ 但是，这里有一个小问题：我们在计算样本方差时通常使用的是无偏估计，这意味着我们需要将分母调整为数据个数减去1。因此，正确的样本方差计算应该是： $\\text{样本方差} = \\frac{210.25 + 210.25}{2 - 1} = \\frac{420.5}{1} = 420.5$ 所以，数据集${22, -7}$的样本方差为$\\boxed{420.5}$。 |
\frac{841}{2} |
| zhipu | 1. **Calculate the mean of the sample**: \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Compute each deviation from the mean, square it, and sum the squares**: - For $22$: \[ (22 - 7.5)^2 = 14.5^2 = 210.25 \] - For $-7$: \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] Sum of squared deviations: \[ 210.25 + 210.25 = 420.5 \] 3. **Calculate the sample variance** using the formula for a sample (dividing by $n-1$ where $n$ is the sample size): \[ s^2 = \frac{\sum (x_i - \text{mean})^2}{n-1} = \frac{420.5}{2-1} = 420.5 \] Thus, the sample variance is $420.5$. \[ \boxed{420.5} \] |
\frac{841}{2} |
| deepseek-v3 | To compute the **sample variance** of the data set \({22, -7}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Find the Deviations from the Mean and Square Them:** - For \(22\): \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] - For \(-7\): \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] 3. **Calculate the Sum of Squared Deviations:** \[ 210.25 + 210.25 = 420.5 \] 4. **Compute the Sample Variance (\(s^2\)):** Since it's a sample, we divide by \(n - 1\) (where \(n = 2\)): \[ s^2 = \frac{420.5}{2 - 1} = 420.5 \] Therefore, the sample variance is: \[ \boxed{420.5} \] |
\frac{841}{2} |