| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{4, -19\}\), we follow these steps: 1. **Calculate the mean (average) of the set:** \[ \text{Mean} = \frac{4 + (-19)}{2} = \frac{4 - 19}{2} = \frac{-15}{2} = -7.5 \] 2. **Compute the squared deviations from the mean for each data point:** - For the first data point, \(4\): \[ (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 \] - For the second data point, \(-19\): \[ (-19 - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25 \] 3. **Calculate the sum of these squared deviations:** \[ 132.25 + 132.25 = 264.5 \] 4. **Divide the sum of squared deviations by the number of data points minus one (since it's a sample variance):** \[ \text{Sample Variance} = \frac{264.5}{2 - 1} = \frac{264.5}{1} = 264.5 \] Thus, the sample variance of the set \(\{4, -19\}\) is: \[ \boxed{264.5} \] |
\frac{529}{2} |
| deepseek-r1 | To compute the sample variance of the dataset \(\{4, -19\}\): 1. **Calculate the mean**: \[ \bar{x} = \frac{4 + (-19)}{2} = \frac{-15}{2} = -7.5 \] 2. **Find the squared deviations from the mean**: - For the first data point (4): \[ (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 \] - For the second data point (-19): \[ (-19 - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25 \] 3. **Sum the squared deviations**: \[ 132.25 + 132.25 = 264.5 \] 4. **Divide by \(n - 1\)** (where \(n = 2\)): \[ \frac{264.5}{2 - 1} = \frac{264.5}{1} = 264.5 \] Thus, the sample variance of the dataset \(\{4, -19\}\) is \(\boxed{264.5}\). |
\frac{529}{2} |
| gpto1 | To compute the sample variance \( s^2 \) of the data set \(\{4, -19\}\), follow these steps: **1. Find the sample mean \(\overline{x}\):** \[ \overline{x} = \frac{4 + (-19)}{2} = \frac{-15}{2} = -7.5 \] **2. Compute each squared deviation from the mean:** For \( x_1 = 4 \): \[ (4 - (-7.5))^2 = (11.5)^2 = 132.25 \] For \( x_2 = -19 \): \[ (-19 - (-7.5))^2 = (-11.5)^2 = 132.25 \] **3. Sum the squared deviations:** \[ \sum (x_i - \overline{x})^2 = 132.25 + 132.25 = 264.5 \] **4. Calculate the sample variance:** Since there are \( n = 2 \) observations, the sample variance is: \[ s^2 = \frac{1}{n - 1} \sum (x_i - \overline{x})^2 = \frac{1}{1} \times 264.5 = 264.5 \] **Final Answer:** \[ \boxed{264.5} \] \(\boxed{264.5}\) |
\frac{529}{2} |
| gpt-4o-2024-08-06 | To compute the sample variance of the data set \(\{4, -19\}\), we follow these steps: 1. **Calculate the sample mean**: \[ \bar{x} = \frac{4 + (-19)}{2} = \frac{4 - 19}{2} = \frac{-15}{2} = -7.5 \] 2. **Calculate each deviation from the mean**: - For \(x_1 = 4\), the deviation is \(4 - (-7.5) = 4 + 7.5 = 11.5\). - For \(x_2 = -19\), the deviation is \(-19 - (-7.5) = -19 + 7.5 = -11.5\). 3. **Square each deviation**: - The square of the deviation for \(x_1 = 4\) is \((11.5)^2 = 132.25\). - The square of the deviation for \(x_2 = -19\) is \((-11.5)^2 = 132.25\). 4. **Calculate the sample variance**: The sample variance \(s^2\) is given by the formula: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] where \(n\) is the number of data points. Here, \(n = 2\). \[ s^2 = \frac{1}{2-1} \left( 132.25 + 132.25 \right) = \frac{1}{1} \times 264.5 = 264.5 \] Thus, the sample variance of the data set \(\{4, -19\}\) is \(\boxed{264.5}\). |
\frac{529}{2} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the data set \({4, -19}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{4 + (-19)}{2} = \frac{4 - 19}{2} = \frac{-15}{2} = -7.5 \] 2. **Calculate the squared differences from the mean for each data point:** - For \(4\): \[ (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 \] - For \(-19\): \[ (-19 - (-7.5))^2 = (-19 + 7.5)^2 = -11.5^2 = 132.25 \] 3. **Sum the squared differences:** \[ 132.25 + 132.25 = 264.5 \] 4. **Divide the sum of the squared differences by the number of data points minus 1 (n-1):** \[ \text{Sample Variance} = \frac{264.5}{2 - 1} = \frac{264.5}{1} = 264.5 \] Thus, the sample variance of \({4, -19}\) is \(\boxed{264.5}\). |
\frac{529}{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 {4, -19}. The sample mean is (4 + (-19))/2 = -7.5. ## Step 2: Calculate the deviations from the mean Next, we need to find the deviations of each data point from the sample mean. For the data point 4, the deviation is 4 - (-7.5) = 11.5. For the data point -19, the deviation is -19 - (-7.5) = -11.5. ## Step 3: Square the deviations After finding the deviations, we need to square each of them. The squared deviation for the data point 4 is (11.5)^2 = 132.25. The squared deviation for the data point -19 is (-11.5)^2 = 132.25. ## Step 4: Calculate the sum of the squared deviations Now, we sum the squared deviations. The sum is 132.25 + 132.25 = 264.5. ## Step 5: Calculate the sample variance Finally, we calculate the sample variance by dividing 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 264.5/1 = 264.5. The final answer is: $\boxed{264.5}$ |
\frac{529}{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 4 and -19. The sample mean is (4 + (-19)) / 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 4 - (-7.5) = 11.5, and the deviation for the second value is -19 - (-7.5) = -11.5. ## Step 3: Calculate the squared deviations Now, we need to calculate the squared deviations. The squared deviation for the first value is (11.5)^2 = 132.25, and the squared deviation for the second value is (-11.5)^2 = 132.25. ## Step 4: Calculate the sum of the squared deviations The sum of the squared deviations is 132.25 + 132.25 = 264.5. ## 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 1. The sample variance is 264.5 / 1 = 264.5. The final answer is: $\boxed{264.5}$ |
\frac{529}{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 ${4, -19}$: 1. Calculate the mean: \[ \text{Mean} = \frac{4 + (-19)}{2} = \frac{-15}{2} = -7.5 \] 2. Find the squared differences from the mean for each number: \[ (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 \] \[ (-19 - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25 \] 3. Average those squared differences (since there are only two numbers, this is just the mean of these two values): \[ \text{Sample Variance} = \frac{132.25 + 132.25}{2-1} = \frac{264.5}{1} = 264.5 \] Therefore, the sample variance of ${4, -19}$ is $\boxed{264.5}$. |
\frac{529}{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 set of numbers ${4, -19}$, let's follow these steps: 1. Calculate the mean: Mean = (4 + (-19)) / 2 = -15 / 2 = -7.5 2. For each number, subtract the mean and square the result: (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 (-19 - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25 3. Sum the squared differences: 132.25 + 132.25 = 264.5 4. Divide the sum by the number of elements minus one (since it's sample variance): Variance = 264.5 / (2 - 1) = 264.5 / 1 = 264.5 Therefore, the sample variance of the set ${4, -19}$ is $\boxed{264.5}$. |
\frac{529}{2} |
| doubao | First find the mean: $(4 + (-19)) \div 2 = (-15) \div 2 = -7.5$. Then find the squared differences from the mean: $(4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25$ and $((-19) - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25$. The sum of the squared differences is $132.25 + 132.25 = 264.5$. The sample variance is the sum of the squared differences divided by the number of data points minus 1, so it is $264.5 \div (2 - 1) = 264.5$. Putting it in the box: $\boxed{264.5}$. |
\frac{529}{2} |
| gpto1 | To compute the sample variance \( s^2 \) of the data set \(\{4, -19\}\), follow these steps: **1. Find the sample mean \(\overline{x}\):** \[ \overline{x} = \frac{4 + (-19)}{2} = \frac{-15}{2} = -7.5 \] **2. Compute each squared deviation from the mean:** For \( x_1 = 4 \): \[ (4 - (-7.5))^2 = (11.5)^2 = 132.25 \] For \( x_2 = -19 \): \[ (-19 - (-7.5))^2 = (-11.5)^2 = 132.25 \] **3. Sum the squared deviations:** \[ \sum (x_i - \overline{x})^2 = 132.25 + 132.25 = 264.5 \] **4. Calculate the sample variance:** Since there are \( n = 2 \) observations, the sample variance is: \[ s^2 = \frac{1}{n - 1} \sum (x_i - \overline{x})^2 = \frac{1}{1} \times 264.5 = 264.5 \] **Final Answer:** \[ \boxed{264.5} \] \(\boxed{264.5}\) |
\frac{529}{2} |
| kimi | To compute the sample variance of the set \(\{4, -19\}\), we will follow these steps: 1. **Calculate the mean (average) of the data set:** The mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \] where \(x_i\) are the data points and \(n\) is the number of data points. For our data set \(\{4, -19\}\): \[ \bar{x} = \frac{4 + (-19)}{2} = \frac{4 - 19}{2} = \frac{-15}{2} = -7.5 \] 2. **Calculate the squared differences from the mean:** For each data point \(x_i\), we find \((x_i - \bar{x})^2\). For \(x_1 = 4\): \[ (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 \] For \(x_2 = -19\): \[ (-19 - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25 \] 3. **Calculate the sum of the squared differences:** Add the squared differences together: \[ \sum_{i=1}^n (x_i - \bar{x})^2 = 132.25 + 132.25 = 264.5 \] 4. **Divide by the number of data points minus one (since it's a sample variance):** The sample variance \(s^2\) is given by: \[ s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1} \] For our data set with \(n = 2\): \[ s^2 = \frac{264.5}{2-1} = \frac{264.5}{1} = 264.5 \] Therefore, the sample variance of the set \(\{4, -19\}\) is \(\boxed{264.5}\). |
\frac{529}{2} |
| minimax | To compute the sample variance of the set ${4, -19}$, 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. The mean (average) of the data set is: $$ \text{mean} = \frac{4 + (-19)}{2} = \frac{-15}{2} = -7.5 $$ 2. The deviations from the mean are: $$ 4 - (-7.5) = 4 + 7.5 = 11.5 $$ $$ -19 - (-7.5) = -19 + 7.5 = -11.5 $$ 3. The squared deviations are: $$ (11.5)^2 = 132.25 $$ $$ (-11.5)^2 = 132.25 $$ 4. The sum of the squared deviations |
\frac{529}{2} |
| wenxinyiyan | 首先，我们需要计算给定数据集${4, -19}$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${4, -19}$，均值计算如下： $\\text{均值} = \\frac{4 + (-19)}{2} = \\frac{-15}{2} = -7.5$ 接下来，我们计算每个数据与均值的差的平方。对于数据集中的每个数$x$，我们计算$(x - \\text{均值})^2$： $(4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25$ $(-19 - (-7.5))^2 = (-19 + 7.5)^2 = -11.5^2 = 132.25$ 最后，我们计算这些平方差的均值，即样本方差。样本方差的计算公式为： $\\text{样本方差} = \\frac{\\text{平方差之和}}{\\text{数据个数}}$ 对于数据集${4, -19}$，样本方差计算如下： $\\text{样本方差} = \\frac{132.25 + 132.25}{2} = \\frac{264.5}{2} = 132.25$ 所以，数据集${4, -19}$的样本方差为$132.25$。 故答案为：$\\boxed{132.25}$。 |
\frac{529}{2} |
| zhipu | To compute the sample variance of the data set \(\{4, -19\}\), follow these steps: 1. **Calculate the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{4 + (-19)}{2} = \frac{4 - 19}{2} = \frac{-15}{2} = -7.5 \] 2. **Compute the squared deviations from the mean**: - For \(4\): \[ (4 - (-7.5))^2 = (4 + 7.5)^2 = 11.5^2 = 132.25 \] - For \(-19\): \[ (-19 - (-7.5))^2 = (-19 + 7.5)^2 = (-11.5)^2 = 132.25 \] 3. **Sum the squared deviations**: \[ 132.25 + 132.25 = 264.5 \] 4. **Calculate the sample variance (\(s^2\))**: The formula for sample variance is: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \] where \(n\) is the number of observations. Here, \(n = 2\): \[ s^2 = \frac{264.5}{2 - 1} = \frac{264.5}{1} = 264.5 \] Therefore, the sample variance of the data set \(\{4, -19\}\) is \(\boxed{264.5}\). |
\frac{529}{2} |
| deepseek-v3 | To compute the **sample variance** of the dataset \(\{4, -19\}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{4 + (-19)}{2} = \frac{-15}{2} = -7.5 \] 2. **Find the Deviations from the Mean:** \[ 4 - (-7.5) = 11.5 \\ -19 - (-7.5) = -11.5 \] 3. **Square the Deviations:** \[ (11.5)^2 = 132.25 \\ (-11.5)^2 = 132.25 \] 4. **Calculate the Sum of Squared Deviations:** \[ 132.25 + 132.25 = 264.5 \] 5. **Compute the Sample Variance (\(s^2\)):** \[ s^2 = \frac{\text{Sum of Squared Deviations}}{n - 1} = \frac{264.5}{2 - 1} = 264.5 \] Therefore, the sample variance is: \[ \boxed{264.5} \] |
\frac{529}{2} |