| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample standard deviation 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 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. **Calculate the variance by averaging the squared differences:** \[ \text{Variance} = \frac{210.25 + 210.25}{2} = \frac{420.5}{2} = 210.25 \] 4. **Compute the sample standard deviation by taking the square root of the variance:** \[ \text{Standard Deviation} = \sqrt{210.25} = 14.5 \] Thus, the sample standard deviation of \(\{22, -7\}\) is: \[ \boxed{14.5} \] |
\frac{29}{\sqrt{2}} |
| deepseek-r1 | To compute the sample standard deviation of the data set \(\{22, -7\}\): 1. **Calculate the mean**: \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Find the deviations from the mean**: \[ 22 - 7.5 = 14.5 \quad \text{and} \quad -7 - 7.5 = -14.5 \] 3. **Square each deviation**: \[ (14.5)^2 = 210.25 \quad \text{and} \quad (-14.5)^2 = 210.25 \] 4. **Sum the squared deviations**: \[ 210.25 + 210.25 = 420.5 \] 5. **Calculate the sample variance** (divide by \(n - 1\), where \(n = 2\)): \[ \text{Variance} = \frac{420.5}{2 - 1} = 420.5 \] 6. **Take the square root of the variance to get the sample standard deviation**: \[ \text{Standard Deviation} = \sqrt{420.5} \] Recognizing that \(420.5 = \frac{841}{2}\), we can simplify the standard deviation: \[ \sqrt{\frac{841}{2}} = \frac{\sqrt{841}}{\sqrt{2}} = \frac{29}{\sqrt{2}} = \frac{29\sqrt{2}}{2} \] Thus, the sample standard deviation of the data set is \(\boxed{\dfrac{29\sqrt{2}}{2}}\). |
\frac{29}{\sqrt{2}} |
| gpto1 | To compute the sample standard deviation of the data set \(\{22, -7\}\): **Step 1: Find the sample mean \(\overline{x}\).** \[ \overline{x} = \frac{(22) + (-7)}{2} = \frac{15}{2} = 7.5 \] **Step 2: Calculate the squared deviations from the mean.** For each data point \(x_i\): \[ (x_i - \overline{x})^2 \] Compute for each \(x_i\): 1. For \(x_1 = 22\): \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] 2. For \(x_2 = -7\): \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] **Step 3: Sum the squared deviations.** \[ \sum (x_i - \overline{x})^2 = 210.25 + 210.25 = 420.5 \] **Step 4: Compute the sample variance \(s^2\).** Since \(n = 2\), \(n - 1 = 1\): \[ s^2 = \frac{\sum (x_i - \overline{x})^2}{n - 1} = \frac{420.5}{1} = 420.5 \] **Step 5: Compute the sample standard deviation \(s\).** \[ s = \sqrt{s^2} = \sqrt{420.5} \] Since \(420.5 = \frac{841}{2}\) and \(\sqrt{841} = 29\): \[ s = \sqrt{\frac{841}{2}} = \frac{29}{\sqrt{2}} = \frac{29\sqrt{2}}{2} \] **Final Answer:** \[ \boxed{\dfrac{29\sqrt{2}}{2}} \] |
\frac{29}{\sqrt{2}} |
| gpt-4o-2024-08-06 | To compute the sample standard deviation 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. **Calculate the sample variance**: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{2-1} (210.25 + 210.25) = 420.5 \] 4. **Calculate the sample standard deviation**: \[ s = \sqrt{s^2} = \sqrt{420.5} \] 5. **Compute the final value**: \[ \sqrt{420.5} \approx 20.515 \] Thus, the sample standard deviation is \(\boxed{20.515}\). |
\frac{29}{\sqrt{2}} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample standard deviation 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 variance:** The variance is the average of the squared differences from the mean. We first find the squared differences from the mean: \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] Now, we average these squared differences: \[ \text{Variance} = \frac{210.25 + 210.25}{2} = \frac{420.5}{2} = 210.25 \] 3. **Calculate the standard deviation:** The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{210.25} = 14.5 \] Thus, the sample standard deviation of \(\{22, -7\}\) is \(\boxed{14.5}\). |
\frac{29}{\sqrt{2}} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Calculate the mean of the given numbers To find the sample standard deviation, we first need to calculate the mean of the given numbers. The mean is calculated by adding all the numbers together and then dividing by the total count of numbers. In this case, the numbers are 22 and -7. So, the mean is (22 + (-7)) / 2 = 15 / 2 = 7.5. ## Step 2: Calculate the deviations from the mean Next, we need to find the deviations of each number from the mean. This is done by subtracting the mean from each number. So, the deviations are (22 - 7.5) and (-7 - 7.5), which equals 14.5 and -14.5. ## Step 3: Square each deviation After finding the deviations, we need to square each one. Squaring the deviations gives us (14.5)^2 and (-14.5)^2, which equals 210.25 and 210.25. ## Step 4: Calculate the sum of the squared deviations Now, we add the squared deviations together. The sum is 210.25 + 210.25 = 420.5. ## Step 5: Divide the sum of squared deviations by the number of items minus one Since we are calculating the sample standard deviation, we divide the sum of squared deviations by the number of items minus one. In this case, we divide by 2 - 1 = 1. So, 420.5 / 1 = 420.5. ## Step 6: Take the square root of the result Finally, we take the square root of 420.5 to find the sample standard deviation. The final answer is: $\boxed{20.5}$ |
\frac{29}{\sqrt{2}} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Calculate the mean of the given numbers To find the sample standard deviation, we first need to calculate the mean of the given numbers. The mean is calculated by adding up all the numbers and then dividing by the total count of numbers. In this case, the numbers are 22 and -7. So, the 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 number. This is done by subtracting the mean from each number. For 22, the deviation is 22 - 7.5 = 14.5. For -7, the deviation is -7 - 7.5 = -14.5. ## Step 3: Calculate the squared deviations Now, we need to calculate the squared deviations. This is done by squaring each deviation. For 14.5, the squared deviation is 14.5^2 = 210.25. For -14.5, the squared deviation is (-14.5)^2 = 210.25. ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of the squared deviations. This is 210.25 + 210.25 = 420.5. ## Step 5: Calculate the variance The variance is calculated by dividing the sum of the squared deviations by the total count of numbers minus 1 (since this is a sample standard deviation). In this case, the variance is 420.5 / (2 - 1) = 420.5 / 1 = 420.5. ## Step 6: Calculate the sample standard deviation Finally, we calculate the sample standard deviation by taking the square root of the variance. The sample standard deviation is sqrt(420.5) = 20.5. The final answer is: $\boxed{20.5}$ |
\frac{29}{\sqrt{2}} |
| Qwen2-72B-Instruct | To compute the sample standard deviation of a set of numbers, we follow these steps: 1. Calculate the mean (average) of the numbers. 2. Subtract the mean from each number to find the deviation of each number. 3. Square each deviation. 4. Find the average of these squared deviations (this is called the variance). 5. Take the square root of the variance to get the standard deviation. Given the set of numbers ${22, -7}$: 1. Calculate the mean: \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. Calculate the deviations: \[ \text{Deviation of } 22 = 22 - 7.5 = 14.5 \\ \text{Deviation of } -7 = -7 - 7.5 = -14.5 \] 3. Square the deviations: \[ (14.5)^2 = 210.25 \\ (-14.5)^2 = 210.25 \] 4. Calculate the variance (average of squared deviations): \[ \text{Variance} = \frac{210.25 + 210.25}{2} = \frac{420.5}{2} = 210.25 \] 5. Calculate the standard deviation (square root of variance): \[ \text{Standard Deviation} = \sqrt{210.25} \approx 14.5 \] Therefore, the sample standard deviation of ${22, -7}$ is approximately $\boxed{14.5}$. |
\frac{29}{\sqrt{2}} |
| baichuan4 | To compute the sample standard deviation of a set of numbers, we first need to calculate the mean (average) of the numbers, then find the squared differences between each number and the mean, take the average of those squared differences, and finally take the square root of that average. For the given set of numbers ${22, -7}$, we can follow these steps: 1. Calculate the mean: Mean = (22 + (-7)) / 2 = 15 / 2 = 7.5 2. Find the squared differences between each number and 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. Take the average of the squared differences: Average of squared differences = (210.25 + 210.25) / 2 = 420.5 / 2 = 210.25 4. Take the square root of the average of the squared differences to get the sample standard deviation: Sample standard deviation = √210.25 ≈ 14.4967 Therefore, the sample standard deviation of the set ${22, -7}$ is approximately $14.4967$. We can put this answer in a box as requested: $$ \boxed{14.4967} $$ |
\frac{29}{\sqrt{2}} |
| doubao | First find the mean: $(22 + (-7)) \div 2 = (22 - 7) \div 2 = 15 \div 2 = 7.5$. Then for each number, subtract the mean and square the result: $(22 - 7.5)^2 = 14.5^2 = 210.25$ and $(-7 - 7.5)^2 = (-14.5)^2 = 210.25$. Find the average of these squared differences: $(210.25 + 210.25) \div 2 = 420.5 \div 2 = 210.25$. The sample standard deviation is the square root of this average: $\sqrt{210.25} = 14.5$. So the answer is $\boxed{14.5}$. |
\frac{29}{\sqrt{2}} |
| gpto1 | To compute the sample standard deviation of the data set \(\{22, -7\}\): **Step 1: Find the sample mean \(\overline{x}\).** \[ \overline{x} = \frac{(22) + (-7)}{2} = \frac{15}{2} = 7.5 \] **Step 2: Calculate the squared deviations from the mean.** For each data point \(x_i\): \[ (x_i - \overline{x})^2 \] Compute for each \(x_i\): 1. For \(x_1 = 22\): \[ (22 - 7.5)^2 = (14.5)^2 = 210.25 \] 2. For \(x_2 = -7\): \[ (-7 - 7.5)^2 = (-14.5)^2 = 210.25 \] **Step 3: Sum the squared deviations.** \[ \sum (x_i - \overline{x})^2 = 210.25 + 210.25 = 420.5 \] **Step 4: Compute the sample variance \(s^2\).** Since \(n = 2\), \(n - 1 = 1\): \[ s^2 = \frac{\sum (x_i - \overline{x})^2}{n - 1} = \frac{420.5}{1} = 420.5 \] **Step 5: Compute the sample standard deviation \(s\).** \[ s = \sqrt{s^2} = \sqrt{420.5} \] Since \(420.5 = \frac{841}{2}\) and \(\sqrt{841} = 29\): \[ s = \sqrt{\frac{841}{2}} = \frac{29}{\sqrt{2}} = \frac{29\sqrt{2}}{2} \] **Final Answer:** \[ \boxed{\dfrac{29\sqrt{2}}{2}} \] |
\frac{29}{\sqrt{2}} |
| kimi | To compute the sample standard deviation of the set ${22, -7}$, we follow these steps: 1. **Calculate the mean (average) of the data set:** The mean $\mu$ is given by the formula: \[ \mu = \frac{1}{n} \sum_{i=1}^{n} x_i \] where $n$ is the number of data points, and $x_i$ are the data points themselves. For our data set ${22, -7}$: \[ \mu = \frac{1}{2} (22 + (-7)) = \frac{1}{2} \times 15 = 7.5 \] 2. **Calculate the variance:** The variance $\sigma^2$ is given by: \[ \sigma^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \mu)^2 \] where $n$ is the number of data points, and $\mu$ is the mean. For our data set: \[ \sigma^2 = \frac{1}{2-1} [(22 - 7.5)^2 + ((-7) - 7.5)^2] \] \[ = \frac{1}{1} [(14.5)^2 + (-14.5)^2] \] \[ = 14.5^2 + 14.5^2 \] \[ = 210.25 + 210.25 \] \[ = 420.5 \] 3. **Calculate the standard deviation:** The standard deviation $\sigma$ is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{420.5} \] 4. **Simplify the square root:** \[ \sigma = \sqrt{420.5} \approx 20.51 \] Thus, the sample standard deviation of the set ${22, -7}$ is $\boxed{20.51}$. |
\frac{29}{\sqrt{2}} |
| minimax | To compute the sample standard deviation of the set ${22, -7}$, we will follow these steps: 1. Find the mean (average) of the numbers. 2. Subtract the mean from each number to find the deviation of each number from the mean. 3. Square each deviation. 4. Find the average of these squared deviations. 5. Take the square root of the average squared deviation to get the standard deviation. 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. Calculate the average of these squared deviations (the variance for a sample): $$ \text{variance} = \frac{210.25 + 21 |
\frac{29}{\sqrt{2}} |
| wenxinyiyan | 首先，我们需要计算样本的平均值（均值）。样本均值是所有数值的和除以数值的数量。 对于给定的数据集${22, -7}$，均值的计算公式为： $\text{均值} = \frac{1}{n}\sum_{i=1}^{n}x_i$ 其中，$n$是样本数量，$x_i$是样本中的每一个数值。 将数值代入公式，我们得到： $\text{均值} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5$ 接下来，我们计算每个数值与均值的差的平方，然后求这些平方差的均值，最后取平方根得到标准偏差。 标准偏差的计算公式为： $s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}$ 其中，$s$是样本标准偏差，$n$是样本数量，$x_i$是样本中的每一个数值，$\bar{x}$是样本均值。 代入数值进行计算： $s = \sqrt{\frac{1}{2-1}[(22-7.5)^2 + (-7-7.5)^2]}$ $= \sqrt{\frac{1}{1}[(14.5)^2 + (-14.5)^2]}$ $= \sqrt{\frac{1}{1}[210.25 + 210.25]}$ $= \sqrt{420.5}$ $= \sqrt{420.5}$ $\approx 20.506$ 故答案为：$\boxed{20.506}$。 |
\frac{29}{\sqrt{2}} |
| zhipu | 1. **Calculate the mean of the sample**: \[ \text{Mean} = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] 2. **Compute 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. **Calculate the variance** (using $n-1$ as denominator because it's a sample): \[ \text{Variance} = \frac{210.25 + 210.25}{2 - 1} = \frac{420.5}{1} = 420.5 \] 4. **Find the standard deviation**: \[ \text{Standard Deviation} = \sqrt{420.5} \approx 20.507 \] However, for exact value: \[ \text{Standard Deviation} = \sqrt{420.5} \] Thus, the sample standard deviation is $\sqrt{420.5}$, which can be simplified further if needed. \[ \text{Standard Deviation} = \frac{\sqrt{1682}}{2} \] Final answer: \[ \boxed{\frac{\sqrt{1682}}{2}} \] |
\frac{29}{\sqrt{2}} |
| deepseek-v3 | To compute the **sample standard deviation** of the data set \(\{22, -7\}\), follow these steps: ### Step 1: Calculate the Mean (\(\mu\)) \[ \mu = \frac{22 + (-7)}{2} = \frac{15}{2} = 7.5 \] ### Step 2: Compute the Deviations from the Mean and Square Them \[ \begin{align*} (22 - 7.5)^2 &= (14.5)^2 = 210.25 \\ (-7 - 7.5)^2 &= (-14.5)^2 = 210.25 \\ \end{align*} \] ### Step 3: Calculate the Sample Variance (\(s^2\)) \[ s^2 = \frac{210.25 + 210.25}{2 - 1} = \frac{420.5}{1} = 420.5 \] ### Step 4: Determine the Sample Standard Deviation (\(s\)) \[ s = \sqrt{420.5} \approx 20.506 \] ### Final Answer \[ \boxed{20.506} \] |
\frac{29}{\sqrt{2}} |