| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | First, let's calculate the mean of the data set: \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{1 - 21 - 21 - 18 + 8}{5} = \frac{-51}{5} = -10.2 \] Next, we calculate the variance: \[ \text{Variance} = \frac{1}{5-1} \left[ (1 + 10.2)^2 + (-21 + 10.2)^2 + (-21 + 10.2)^2 + (-18 + 10.2)^2 + (8 + 10.2)^2 \right] \] \[ = \frac{1}{4} \left[ (11.2)^2 + (-10.8)^2 + (-10.8)^2 + (-7.8)^2 + (18.2)^2 \right] \] \[ = \frac{1}{4} \left[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 \right] \] \[ = \frac{1}{4} \left[ 750.8 \right] = 187.7 \] Finally, we calculate the sample standard deviation: \[ \text{Sample Standard Deviation} = \sqrt{187.7} \approx 13.7 \] Thus, the final answer is: \[ \boxed{13.7} \] |
\sqrt{\frac{1877}{10}} |
| deepseek-r1 | To compute the sample standard deviation of the data set \(\{1, -21, -21, -18, 8\}\): 1. **Calculate the mean**: \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Find the squared differences from the mean**: - For \(1\): \((1 - (-10.2))^2 = (11.2)^2 = 125.44\) - For \(-21\): \((-21 - (-10.2))^2 = (-10.8)^2 = 116.64\) - For \(-21\): Same as above, \(116.64\) - For \(-18\): \((-18 - (-10.2))^2 = (-7.8)^2 = 60.84\) - For \(8\): \((8 - (-10.2))^2 = (18.2)^2 = 331.24\) 3. **Sum the squared differences**: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] 4. **Calculate the sample variance**: \[ \text{Variance} = \frac{750.8}{5 - 1} = \frac{750.8}{4} = 187.7 \] 5. **Compute the sample standard deviation**: \[ \text{Standard Deviation} = \sqrt{187.7} \approx 13.7 \] Thus, the sample standard deviation is \(\boxed{13.7}\). |
\sqrt{\frac{1877}{10}} |
| gpto1 | To compute the sample standard deviation of the data set \({1, -21, -21, -18, 8}\), we follow these steps: **Step 1: Find the sample mean (\(\bar{x}\))** \[ \bar{x} = \frac{\sum x_i}{n} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] **Step 2: Compute each 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** \[ \text{Sum of squared deviations} = 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] **Step 4: Compute the sample variance** \[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{750.8}{5 - 1} = \frac{750.8}{4} = 187.7 \] **Step 5: Compute the sample standard deviation** \[ s = \sqrt{s^2} = \sqrt{187.7} \] So, the sample standard deviation is \(\sqrt{187.7}\). \(\boxed{\sqrt{187.7}}\) |
\sqrt{\frac{1877}{10}} |
| gpt-4o-2024-08-06 | To compute the sample standard deviation, we first need to find the mean of the data set, then calculate the variance, and finally take the square root of the variance. Given the data set: \( \{1, -21, -21, -18, 8\} \). **Step 1: Calculate the mean.** The mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] Calculating the sum of the data: \[ 1 - 21 - 21 - 18 + 8 = -51 \] Thus, the mean is: \[ \bar{x} = \frac{-51}{5} = -10.2 \] **Step 2: Calculate the variance.** The variance \(s^2\) is calculated using the formula: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] 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: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] Calculate the variance: \[ s^2 = \frac{750.8}{5-1} = \frac{750.8}{4} = 187.7 \] **Step 3: Calculate the standard deviation.** The sample standard deviation \(s\) is the square root of the variance: \[ s = \sqrt{187.7} \approx 13.7 \] Thus, the sample standard deviation is: \[ \boxed{13.7} \] |
\sqrt{\frac{1877}{10}} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample standard deviation of the set \(\{1, -21, -21, -18, 8\}\), we follow these steps: 1. **Calculate the mean (average) of the numbers:** \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] \[ \text{Mean} = \frac{1 - 21 - 21 - 18 + 8}{5} \] \[ \text{Mean} = \frac{1 - 21 - 21 - 18 + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Calculate the variance:** The variance is the average of the squared differences from the mean. \[ \text{Variance} = \frac{(1 - (-10.2))^2 + (-21 - (-10.2))^2 + (-21 - (-10.2))^2 + (-18 - (-10.2))^2 + (8 - (-10.2))^2}{5} \] \[ \text{Variance} = \frac{(1 + 10.2)^2 + (-21 + 10.2)^2 + (-21 + 10.2)^2 + (-18 + 10.2)^2 + (8 + 10.2)^2}{5} \] \[ \text{Variance} = \frac{11.2^2 + (-10.8)^2 + (-10.8)^2 + (-7.8)^2 + 18.2^2}{5} \] \[ \text{Variance} = \frac{125.44 + 116.64 + 116.64 + 60.84 + 331.24}{5} \] \[ \text{Variance} = \frac{751.8}{5} = 150.36 \] 3. **Calculate the standard deviation:** The standard deviation is the square root of the variance. \[ \text{Standard Deviation} = \sqrt{150.36} \approx 12.26 \] Therefore, the sample standard deviation of \(\{1, -21, -21, -18, 8\}\) is: \[ \boxed{12.26} \] |
\sqrt{\frac{1877}{10}} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Calculate the mean of the given sample To find the sample standard deviation, we first need to calculate the mean of the given sample. The mean is calculated by summing all the values and then dividing by the number of values. The given sample is ${1, -21, -21, -18, 8}$. The sum of these values is $1 - 21 - 21 - 18 + 8 = -51$. There are $5$ values in the sample. Therefore, the mean is $\frac{-51}{5} = -10.2$. ## Step 2: Calculate the deviations from the mean Next, we calculate the deviations of each value from the mean. This is done by subtracting the mean from each value. 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 variance The variance is the average of the squared deviations. Since this is a sample, we divide the sum of the squared deviations by the number of items minus one (Bessel's correction), which is $5 - 1 = 4$. Therefore, the variance is $\frac{750.8}{4} = 187.7$. ## Step 6: Calculate the sample standard deviation Finally, the sample standard deviation is the square root of the variance. Therefore, we calculate $\sqrt{187.7}$ to find the sample standard deviation. The final answer is: $\boxed{13.69}$ |
\sqrt{\frac{1877}{10}} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Calculate the mean of the given data set To find the sample standard deviation, we first need to calculate the mean of the given data set. The mean is calculated by summing up all the numbers and then dividing by the total count of numbers. The given data set is ${1, -21, -21, -18, 8}$. The sum of these numbers is $1 + (-21) + (-21) + (-18) + 8 = -51$. There are $5$ numbers in the data set. Therefore, the mean is $\frac{-51}{5} = -10.2$. ## Step 2: Subtract the mean from each data point to find the deviations Next, we subtract the mean from each data point to find the deviations from the mean. The deviations are calculated as follows: - For $1$: $1 - (-10.2) = 11.2$ - For $-21$: $-21 - (-10.2) = -10.8$ - For $-21$: $-21 - (-10.2) = -10.8$ - For $-18$: $-18 - (-10.2) = -7.8$ - For $8$: $8 - (-10.2) = 18.2$ ## Step 3: Square each deviation Now, we square each deviation: - $(11.2)^2 = 125.44$ - $(-10.8)^2 = 116.64$ - $(-10.8)^2 = 116.64$ - $(-7.8)^2 = 60.84$ - $(18.2)^2 = 331.24$ ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of these squared deviations: $125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 751.8$ ## Step 5: Divide the sum of the squared deviations by the number of data points minus one To calculate the sample variance, we divide the sum of the squared deviations by the number of data points minus one. In this case, we divide by $5 - 1 = 4$: $\frac{751.8}{4} = 187.95$ ## Step 6: Take the square root of the result to find the sample standard deviation Finally, we take the square root of the result to find the sample standard deviation: $\sqrt{187.95} \approx 13.7$ The final answer is: $\boxed{13.7}$ |
\sqrt{\frac{1877}{10}} |
| 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 its deviation from the mean. 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. Let's apply these steps to the given set of numbers: ${1, -21, -21, -18, 8}$. Step 1: Calculate the mean. \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] Step 2: Subtract the mean from each number. \[ 1 - (-10.2) = 11.2 \\ -21 - (-10.2) = -10.8 \\ -21 - (-10.2) = -10.8 \\ -18 - (-10.2) = -7.8 \\ 8 - (-10.2) = 18.2 \] Step 3: Square each deviation. \[ (11.2)^2 = 125.44 \\ (-10.8)^2 = 116.64 \\ (-10.8)^2 = 116.64 \\ (-7.8)^2 = 60.84 \\ (18.2)^2 = 331.24 \] Step 4: Find the average of these squared deviations (variance). \[ \text{Variance} = \frac{125.44 + 116.64 + 116.64 + 60.84 + 331.24}{5} = \frac{750.8}{5} = 150.16 \] Step 5: Take the square root of the variance to get the standard deviation. \[ \text{Standard Deviation} = \sqrt{150.16} \approx 12.25 \] Therefore, the sample standard deviation of the given set of numbers is approximately $\boxed{12.25}$. |
\sqrt{\frac{1877}{10}} |
| baichuan4 | To compute the sample standard deviation of a set of numbers, we first need to calculate the mean (average) of the numbers. Then, for each number, we subtract the mean and square the result. We sum these squared differences, divide by the number of observations minus one (since it's a sample standard deviation), and finally take the square root of that result. Given the numbers: ${1, -21, -21, -18, 8}$ 1. Calculate the mean: Mean = (1 + (-21) + (-21) + (-18) + 8) / 5 Mean = (-51) / 5 Mean = -10.2 2. For each number, subtract the mean and square the result: (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 these squared differences: 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 4. Divide by the number of observations minus one: 750.8 / (5 - 1) = 750.8 / 4 = 187.7 5. Take the square root of the result: Sqrt(187.7) ≈ 13.7 Therefore, the sample standard deviation is approximately 13.7. Final answer: $\boxed{13.7}$ |
\sqrt{\frac{1877}{10}} |
| doubao | First find the mean: $\bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5}$ $= \frac{1 - 21 - 21 - 18 + 8}{5}$ $= \frac{-41}{5} = -8.2$. Then find the deviations from the mean: $1 - (-8.2) = 9.2$; $-21 - (-8.2) = -12.8$; $-21 - (-8.2) = -12.8$; $-18 - (-8.2) = -9.8$; $8 - (-8.2) = 16.2$. Square the deviations: $9.2^2 = 84.64$; $(-12.8)^2 = 163.84$; $(-12.8)^2 = 163.84$; $(-9.8)^2 = 96.04$; $16.2^2 = 262.44$. Find the sum of the squared deviations: $84.64 + 163.84 + 163.84 + 96.04 + 262.44 = 770.8$. Divide by the number of data points minus 1 (5 - 1 = 4): $770.8 \div 4 = 192.7$. The sample standard deviation is the square root of this value: $\sqrt{192.7} \approx 13.9$. So the answer is $\boxed{13.9}$. |
\sqrt{\frac{1877}{10}} |
| gpto1 | To compute the sample standard deviation of the data set \({1, -21, -21, -18, 8}\), we follow these steps: **Step 1: Find the sample mean (\(\bar{x}\))** \[ \bar{x} = \frac{\sum x_i}{n} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] **Step 2: Compute each 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** \[ \text{Sum of squared deviations} = 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] **Step 4: Compute the sample variance** \[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{750.8}{5 - 1} = \frac{750.8}{4} = 187.7 \] **Step 5: Compute the sample standard deviation** \[ s = \sqrt{s^2} = \sqrt{187.7} \] So, the sample standard deviation is \(\sqrt{187.7}\). \(\boxed{\sqrt{187.7}}\) |
\sqrt{\frac{1877}{10}} |
| kimi | To compute the sample standard deviation 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. \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} \] Simplify the numerator: \[ 1 - 21 - 21 - 18 + 8 = -51 \] 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. **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. \[ s^2 = \frac{746.8}{5 - 1} = \frac{746.8}{4} = 186.7 \] 5. **Calculate the sample standard deviation (\(s\)):** The sample standard deviation is the square root of the sample variance. \[ s = \sqrt{186.7} \approx 13.66 \] Therefore, the sample standard deviation of the data set is \(\boxed{13.66}\). |
\sqrt{\frac{1877}{10}} |
| minimax | To compute the sample standard deviation 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. Find the mean of the squared deviations. 5. Take the square root of the mean of the squared deviations to get the standard deviation. 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: $$ 1 - (-10.2) = 1 + 10.2 = 11.2 $$ $$ -21 - (-10.2) = -21 + 10.2 = -10.8 $$ $$ -21 - (-10.2) = -21 + 10.2 = -10.8 $$ $$ -1 |
\sqrt{\frac{1877}{10}} |
| wenxinyiyan | 计算平均值： $\bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2$ 计算方差 $s^2$： $s^2 = \frac{1}{5-1} \left( (1 - (-10.2))^2 + (-21 - (-10.2))^2 + (-21 - (-10.2))^2 + (-18 - (-10.2))^2 + (8 - (-10.2))^2 \right)$ $= \frac{1}{4} \left( (11.2)^2 + (-10.8)^2 + (-10.8)^2 + (-7.8)^2 + (18.2)^2 \right)$ $= \frac{1}{4} \left( 125.44 + 116.64 + 116.64 + 60.84 + 331.24 \right)$ $= \frac{1}{4} \times 750.8 = 187.7$ 计算标准差 $s$： $s = \sqrt{187.7} \approx 13.700$ 所以，样本标准差为 $\boxed{13.700}$。 |
\sqrt{\frac{1877}{10}} |
| zhipu | 1. **Calculate the mean of the data set**: \[ \text{Mean} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Calculate the squared deviations from the mean for each data point**: - For $1$: $(1 + 10.2)^2 = 11.2^2 = 125.44$ - For $-21$: $(-21 + 10.2)^2 = (-10.8)^2 = 116.64$ - For $-21$: $(-21 + 10.2)^2 = (-10.8)^2 = 116.64$ - For $-18$: $(-18 + 10.2)^2 = (-7.8)^2 = 60.84$ - For $8$: $(8 + 10.2)^2 = 18.2^2 = 331.24$ 3. **Sum the squared deviations**: \[ 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] 4. **Calculate the variance** (using $n-1$ as this is a sample): \[ \text{Variance} = \frac{750.8}{5-1} = \frac{750.8}{4} = 187.7 \] 5. **Calculate the standard deviation**: \[ \text{Standard Deviation} = \sqrt{187.7} \approx 13.7 \] Thus, the sample standard deviation of the data set is $\\boxed{13.7}$. |
\sqrt{\frac{1877}{10}} |
| deepseek-v3 | To compute the **sample standard deviation** of the dataset \(\{1, -21, -21, -18, 8\}\), follow these steps: 1. **Calculate the Mean (\(\bar{x}\))**: \[ \bar{x} = \frac{1 + (-21) + (-21) + (-18) + 8}{5} = \frac{-51}{5} = -10.2 \] 2. **Find the Squared Deviations from the Mean**: \[ \begin{align*} (1 - (-10.2))^2 &= (11.2)^2 = 125.44 \\ (-21 - (-10.2))^2 &= (-10.8)^2 = 116.64 \\ (-21 - (-10.2))^2 &= (-10.8)^2 = 116.64 \\ (-18 - (-10.2))^2 &= (-7.8)^2 = 60.84 \\ (8 - (-10.2))^2 &= (18.2)^2 = 331.24 \\ \end{align*} \] \[ \text{Sum of squared deviations} = 125.44 + 116.64 + 116.64 + 60.84 + 331.24 = 750.8 \] 3. **Compute the Variance (\(s^2\))**: \[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{750.8}{4} = 187.7 \] 4. **Determine the Standard Deviation (\(s\))**: \[ s = \sqrt{187.7} \approx 13.7 \] Therefore, the sample standard deviation is: \[ \boxed{13.7} \] |
\sqrt{\frac{1877}{10}} |