| 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 \(\{1, 12, -17, -3, -11, -5\}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} \] \[ \text{Mean} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} \] \[ \text{Mean} = \frac{-23}{6} \approx -3.8333 \] 2. **Calculate the squared differences from the mean for each data point:** \[ (1 - (-3.8333))^2 = (1 + 3.8333)^2 = 4.8333^2 \approx 23.3609 \] \[ (12 - (-3.8333))^2 = (12 + 3.8333)^2 = 15.8333^2 \approx 250.6944 \] \[ (-17 - (-3.8333))^2 = (-17 + 3.8333)^2 = -13.1667^2 \approx 173.3609 \] \[ (-3 - (-3.8333))^2 = (-3 + 3.8333)^2 = 0.8333^2 \approx 0.6944 \] \[ (-11 - (-3.8333))^2 = (-11 + 3.8333)^2 = -7.1667^2 \approx 51.3609 \] \[ (-5 - (-3.8333))^2 = (-5 + 3.8333)^2 = -1.1667^2 \approx 1.3609 \] 3. **Sum the squared differences:** \[ \sum (x_i - \bar{x})^2 = 23.3609 + 250.6944 + 173.3609 + 0.6944 + 51.3609 + 1.3609 \] \[ \sum (x_i - \bar{x})^2 = 500.8324 \] 4. **Calculate the variance (mean of the squared differences):** \[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{500.8324}{5} \approx 100.1665 \] 5. **Calculate the sample standard deviation (square root of the variance):** \[ \text{Standard Deviation} = \sqrt{100.1665} \approx 10.0083 \] Thus, the sample standard deviation is: \[ \boxed{10.0083} \] |
\sqrt{\frac{601}{6}} |
| deepseek-r1 | To compute the sample standard deviation of the data set \(\{1, 12, -17, -3, -11, -5\}\), we follow these steps: 1. **Calculate the mean**: \[ \text{Mean} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} \approx -3.8333 \] 2. **Find the deviations from the mean and square them**: \[ \begin{aligned} (1 - (-\frac{23}{6}))^2 &= \left(\frac{29}{6}\right)^2 = \frac{841}{36}, \\ (12 - (-\frac{23}{6}))^2 &= \left(\frac{95}{6}\right)^2 = \frac{9025}{36}, \\ (-17 - (-\frac{23}{6}))^2 &= \left(-\frac{79}{6}\right)^2 = \frac{6241}{36}, \\ (-3 - (-\frac{23}{6}))^2 &= \left(\frac{5}{6}\right)^2 = \frac{25}{36}, \\ (-11 - (-\frac{23}{6}))^2 &= \left(-\frac{43}{6}\right)^2 = \frac{1849}{36}, \\ (-5 - (-\frac{23}{6}))^2 &= \left(-\frac{7}{6}\right)^2 = \frac{49}{36}. \end{aligned} \] 3. **Sum the squared deviations**: \[ \frac{841 + 9025 + 6241 + 25 + 1849 + 49}{36} = \frac{18030}{36} = \frac{3005}{6} \] 4. **Calculate the variance**: \[ \text{Variance} = \frac{\frac{3005}{6}}{5} = \frac{3005}{30} = \frac{601}{6} \] 5. **Take the square root of the variance to get the sample standard deviation**: \[ \text{Sample Standard Deviation} = \sqrt{\frac{601}{6}} \] Thus, the sample standard deviation of the data set is \(\boxed{\sqrt{\dfrac{601}{6}}}\). |
\sqrt{\frac{601}{6}} |
| gpto1 | To compute the sample standard deviation of the dataset \(\{1, 12, -17, -3, -11, -5\}\), follow these steps: **1. Calculate the sample mean \(\bar{x}\):** \[ \bar{x} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{-23}{6} \] **2. Calculate each deviation from the mean and square them:** For each \(x_i\), compute \( (x_i - \bar{x})^2 \): \[ \begin{align*} (1 - (-\tfrac{23}{6}))^2 &= \left(\tfrac{29}{6}\right)^2 = \tfrac{841}{36} \\ (12 - (-\tfrac{23}{6}))^2 &= \left(\tfrac{95}{6}\right)^2 = \tfrac{9025}{36} \\ (-17 - (-\tfrac{23}{6}))^2 &= \left(-\tfrac{79}{6}\right)^2 = \tfrac{6241}{36} \\ (-3 - (-\tfrac{23}{6}))^2 &= \left(\tfrac{5}{6}\right)^2 = \tfrac{25}{36} \\ (-11 - (-\tfrac{23}{6}))^2 &= \left(-\tfrac{43}{6}\right)^2 = \tfrac{1849}{36} \\ (-5 - (-\tfrac{23}{6}))^2 &= \left(-\tfrac{7}{6}\right)^2 = \tfrac{49}{36} \\ \end{align*} \] **3. Sum the squared deviations:** \[ \text{Sum} = \tfrac{841}{36} + \tfrac{9025}{36} + \tfrac{6241}{36} + \tfrac{25}{36} + \tfrac{1849}{36} + \tfrac{49}{36} = \tfrac{18030}{36} \] Simplify the sum: \[ \frac{18030}{36} = \frac{3005}{6} \] **4. Compute the sample variance \(s^2\):** \[ s^2 = \frac{1}{n-1} \times \text{Sum of squared deviations} = \frac{1}{5} \times \frac{3005}{6} = \frac{3005}{30} = \frac{601}{6} \] **5. Calculate the sample standard deviation \(s\):** \[ s = \sqrt{s^2} = \sqrt{\frac{601}{6}} \] **Final Answer:** \[ s = \boxed{\sqrt{\dfrac{601}{6}}} \] |
\sqrt{\frac{601}{6}} |
| 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: \( \{1, 12, -17, -3, -11, -5\} \). **Step 1: Calculate the mean.** The mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} \] Calculating the sum of the data: \[ 1 + 12 - 17 - 3 - 11 - 5 = -23 \] Thus, the mean is: \[ \bar{x} = \frac{-23}{6} = -\frac{23}{6} \] **Step 2: Calculate the variance.** The variance \(s^2\) is given by: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] where \(n = 6\). First, calculate each squared deviation from the mean: - For \(x_1 = 1\): \[ x_1 - \bar{x} = 1 + \frac{23}{6} = \frac{6}{6} + \frac{23}{6} = \frac{29}{6} \] \[ (x_1 - \bar{x})^2 = \left(\frac{29}{6}\right)^2 = \frac{841}{36} \] - For \(x_2 = 12\): \[ x_2 - \bar{x} = 12 + \frac{23}{6} = \frac{72}{6} + \frac{23}{6} = \frac{95}{6} \] \[ (x_2 - \bar{x})^2 = \left(\frac{95}{6}\right)^2 = \frac{9025}{36} \] - For \(x_3 = -17\): \[ x_3 - \bar{x} = -17 + \frac{23}{6} = -\frac{102}{6} + \frac{23}{6} = -\frac{79}{6} \] \[ (x_3 - \bar{x})^2 = \left(-\frac{79}{6}\right)^2 = \frac{6241}{36} \] - For \(x_4 = -3\): \[ x_4 - \bar{x} = -3 + \frac{23}{6} = -\frac{18}{6} + \frac{23}{6} = \frac{5}{6} \] \[ (x_4 - \bar{x})^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \] - For \(x_5 = -11\): \[ x_5 - \bar{x} = -11 + \frac{23}{6} = -\frac{66}{6} + \frac{23}{6} = -\frac{43}{6} \] \[ (x_5 - \bar{x})^2 = \left(-\frac{43}{6}\right)^2 = \frac{1849}{36} \] - For \(x_6 = -5\): \[ x_6 - \bar{x} = -5 + \frac{23}{6} = -\frac{30}{6} + \frac{23}{6} = -\frac{7}{6} \] \[ (x_6 - \bar{x})^2 = \left(-\frac{7}{6}\right)^2 = \frac{49}{36} \] Now, sum these squared deviations: \[ \sum (x_i - \bar{x})^2 = \frac{841}{36} + \frac{9025}{36} + \frac{6241}{36} + \frac{25}{36} + \frac{1849}{36} + \frac{49}{36} = \frac{18230}{36} \] Calculate the variance: \[ s^2 = \frac{1}{5} \times \frac{18230}{36} = \frac{18230}{180} \] Simplify: \[ s^2 = \frac{9115}{90} \] **Step 3: Calculate the standard deviation.** The standard deviation \(s\) is the square root of the variance: \[ s = \sqrt{\frac{9115}{90}} \] Approximating the square root: \[ s \approx 10.06 \] Thus, the sample standard deviation is: \[ \boxed{10.06} \] |
\sqrt{\frac{601}{6}} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample standard deviation of the set \(\{1, 12, -17, -3, -11, -5\}\), we follow these steps: 1. **Calculate the mean (average) of the numbers:** \[ \text{Mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} \] \[ \text{Sum} = 1 + 12 - 17 - 3 - 11 - 5 = -23 \] \[ \text{Mean} = \frac{-23}{6} \approx -3.8333 \] 2. **Calculate the variance:** For each number, find the squared difference from the mean: \[ (1 - (-3.8333))^2 = (1 + 3.8333)^2 = 4.8333^2 = 23.3583 \] \[ (12 - (-3.8333))^2 = (12 + 3.8333)^2 = 15.8333^2 = 250.6944 \] \[ (-17 - (-3.8333))^2 = (-17 + 3.8333)^2 = -13.1667^2 = 173.3689 \] \[ (-3 - (-3.8333))^2 = (-3 + 3.8333)^2 = 0.8333^2 = 0.6944 \] \[ (-11 - (-3.8333))^2 = (-11 + 3.8333)^2 = -7.1667^2 = 51.3689 \] \[ (-5 - (-3.8333))^2 = (-5 + 3.8333)^2 = -1.1667^2 = 1.3689 \] Sum these squared differences: \[ \text{Sum of squared differences} = 23.3583 + 250.6944 + 173.3689 + 0.6944 + 51.3689 + 1.3689 = 500.8538 \] 3. **Calculate the variance:** \[ \text{Variance} = \frac{500.8538}{6 - 1} = \frac{500.8538}{5} = 100.1708 \] 4. **Calculate the standard deviation:** \[ \text{Standard Deviation} = \sqrt{100.1708} \approx 10.0085 \] Therefore, the sample standard deviation is: \[ \boxed{10.0085} \] |
\sqrt{\frac{601}{6}} |
| 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 ${1, 12, -17, -3, -11, -5}$. The sum of these values is $1 + 12 - 17 - 3 - 11 - 5 = -23$. There are 6 values in the sample. Therefore, the mean is $\frac{-23}{6}$. ## Step 2: Calculate the deviations from the mean Next, we calculate the deviations from the mean for each value in the sample. This is done by subtracting the mean from each value. The deviations are: $(1 - \frac{-23}{6}), (12 - \frac{-23}{6}), (-17 - \frac{-23}{6}), (-3 - \frac{-23}{6}), (-11 - \frac{-23}{6}), (-5 - \frac{-23}{6})$. ## Step 3: Calculate the squared deviations After finding the deviations, we need to square each deviation. This gives us: $(1 - \frac{-23}{6})^2, (12 - \frac{-23}{6})^2, (-17 - \frac{-23}{6})^2, (-3 - \frac{-23}{6})^2, (-11 - \frac{-23}{6})^2, (-5 - \frac{-23}{6})^2$. ## Step 4: Calculate the sum of the squared deviations Now, we sum the squared deviations calculated in the previous step. This is $(1 - \frac{-23}{6})^2 + (12 - \frac{-23}{6})^2 + (-17 - \frac{-23}{6})^2 + (-3 - \frac{-23}{6})^2 + (-11 - \frac{-23}{6})^2 + (-5 - \frac{-23}{6})^2$. ## Step 5: Calculate the variance To find the variance, we divide the sum of the squared deviations by the number of items in the sample minus one (which is $6 - 1 = 5$ in this case). ## Step 6: Calculate the sample standard deviation Finally, the sample standard deviation is the square root of the variance calculated in the previous step. ## Step 7: Compute the numerical value of the sample standard deviation Let's calculate the numerical value of the sample standard deviation step by step. First, calculate the mean: $\frac{-23}{6} \approx -3.8333$. Next, calculate the deviations and their squares: - $(1 - \frac{-23}{6})^2 \approx (1 + 3.8333)^2 \approx 4.8333^2 \approx 23.3611$ - $(12 - \frac{-23}{6})^2 \approx (12 + 3.8333)^2 \approx 15.8333^2 \approx 250.6944$ - $(-17 - \frac{-23}{6})^2 \approx (-17 + 3.8333)^2 \approx -13.1667^2 \approx 173.6111$ - $(-3 - \frac{-23}{6})^2 \approx (-3 + 3.8333)^2 \approx 0.8333^2 \approx 0.6944$ - $(-11 - \frac{-23}{6})^2 \approx (-11 + 3.8333)^2 \approx -7.1667^2 \approx 51.3611$ - $(-5 - \frac{-23}{6})^2 \approx (-5 + 3.8333)^2 \approx -1.1667^2 \approx 1.3611$ Then, sum these squared deviations: $23.3611 + 250.6944 + 173.6111 + 0.6944 + 51.3611 + 1.3611 \approx 500.0832$. Now, divide by $5$ to find the variance: $\frac{500.0832}{5} \approx 100.0166$. Finally, take the square root of the variance to find the sample standard deviation: $\sqrt{100.0166} \approx 10$. The final answer is: $\boxed{10}$ |
\sqrt{\frac{601}{6}} |
| 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 all the numbers and then dividing by the total count of numbers. The given data set is ${1, 12, -17, -3, -11, -5}$. The sum of these numbers is $1 + 12 - 17 - 3 - 11 - 5 = -23$. There are $6$ numbers in the data set. Therefore, the mean is $\frac{-23}{6} = -3.8333$. ## 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: - $1 - (-3.8333) = 4.8333$ - $12 - (-3.8333) = 15.8333$ - $-17 - (-3.8333) = -13.1667$ - $-3 - (-3.8333) = 0.8333$ - $-11 - (-3.8333) = -7.1667$ - $-5 - (-3.8333) = -1.1667$ ## Step 3: Square each deviation Now, we square each deviation: - $(4.8333)^2 = 23.3333$ - $(15.8333)^2 = 250.8333$ - $(-13.1667)^2 = 173.3333$ - $(0.8333)^2 = 0.6944$ - $(-7.1667)^2 = 51.3333$ - $(-1.1667)^2 = 1.3611$ ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of the squared deviations: $23.3333 + 250.8333 + 173.3333 + 0.6944 + 51.3333 + 1.3611 = 500.8947$ ## 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 $6 - 1 = 5$: $\frac{500.8947}{5} = 100.1789$ ## 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: $\sqrt{100.1789} = 10.0094$ The final answer is: $\boxed{10}$ |
\sqrt{\frac{601}{6}} |
| Qwen2-72B-Instruct | To compute the sample standard deviation of a set of numbers, we need to 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 calculate it step by step for the given set of numbers: ${1, 12, -17, -3, -11, -5}$. Step 1: Calculate the mean \[ \text{Mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{-23}{6} \approx -3.8333 \] Step 2: Subtract the mean from each number \[ \begin{align*} 1 - (-3.8333) &\approx 4.8333 \\ 12 - (-3.8333) &\approx 15.8333 \\ -17 - (-3.8333) &\approx -13.1667 \\ -3 - (-3.8333) &\approx 0.8333 \\ -11 - (-3.8333) &\approx -7.1667 \\ -5 - (-3.8333) &\approx -1.1667 \\ \end{align*} \] Step 3: Square each deviation \[ \begin{align*} (4.8333)^2 &\approx 23.3611 \\ (15.8333)^2 &\approx 250.6944 \\ (-13.1667)^2 &\approx 173.3611 \\ (0.8333)^2 &\approx 0.6944 \\ (-7.1667)^2 &\approx 51.3611 \\ (-1.1667)^2 &\approx 1.3611 \\ \end{align*} \] Step 4: Find the average of these squared deviations (variance) \[ \text{Variance} = \frac{23.3611 + 250.6944 + 173.3611 + 0.6944 + 51.3611 + 1.3611}{5} \approx \frac{500.8332}{5} \approx 100.1666 \] Note that we divide by $n-1$ (which is 5 in this case) instead of $n$ because we're calculating the sample variance, not the population variance. Step 5: Take the square root of the variance to get the standard deviation \[ \text{Standard Deviation} = \sqrt{100.1666} \approx 10.0083 \] Therefore, the sample standard deviation of the given set of numbers is approximately $\boxed{10.0083}$. |
\sqrt{\frac{601}{6}} |
| 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 up 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: $1, 12, -17, -3, -11, -5$, let's follow these steps: 1. Calculate the mean: $$ \text{Mean} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-13}{6} $$ 2. For each number, subtract the mean and square the result: $$ (1 - (-\frac{13}{6}))^2 = (1 + \frac{13}{6})^2 = (\frac{19}{6})^2 $$ $$ (12 - (-\frac{13}{6}))^2 = (12 + \frac{13}{6})^2 = (\frac{85}{6})^2 $$ $$ (-17 - (-\frac{13}{6}))^2 = (-17 + \frac{13}{6})^2 = (-\frac{99}{6})^2 $$ $$ (-3 - (-\frac{13}{6}))^2 = (-3 + \frac{13}{6})^2 = (\frac{5}{6})^2 $$ $$ (-11 - (-\frac{13}{6}))^2 = (-11 + \frac{13}{6})^2 = (-\frac{53}{6})^2 $$ $$ (-5 - (-\frac{13}{6}))^2 = (-5 + \frac{13}{6})^2 = (-\frac{17}{6})^2 $$ 3. Sum up these squared differences: $$ \sum_{i=1}^{6} (x_i - \text{Mean})^2 = (\frac{19}{6})^2 + (\frac{85}{6})^2 + (-\frac{99}{6})^2 + (\frac{5}{6})^2 + (-\frac{53}{6})^2 + (-\frac{17}{6})^2 $$ 4. Divide by the number of observations minus one (since it's a sample standard deviation): $$ \text{Variance} = \frac{\sum_{i=1}^{6} (x_i - \text{Mean})^2}{6 - 1} $$ 5. Take the square root of the variance to get the standard deviation: $$ \text{Standard Deviation} = \sqrt{\text{Variance}} $$ Now, let's compute these values: $$ \text{Mean} = -\frac{13}{6} $$ $$ \sum_{i=1}^{6} (x_i - \text{Mean})^2 = (\frac{19}{6})^2 + (\frac{85}{6})^2 + (-\frac{99}{6})^2 + (\frac{5}{6})^2 + (-\frac{53}{6})^2 + (-\frac{17}{6})^2 $$ $$ = \frac{361}{36} + \frac{7225}{36} + \frac{9801}{36} + \frac{25}{36} + \frac{2809}{36} + \frac{289}{36} $$ $$ = \frac{361 + 7225 + 9801 + 25 + 2809 + 289}{36} $$ $$ = \frac{20510}{36} $$ $$ \text{Variance} = \frac{20510}{36 \times 5} $$ $$ = \frac{20510}{180} $$ $$ = \frac{2051}{18} $$ $$ \text{Standard Deviation} = \sqrt{\frac{2051}{18}} $$ $$ = \frac{\sqrt{2051}}{\sqrt{18}} $$ $$ = \frac{\sqrt{2051}}{3\sqrt{2}} $$ $$ = \frac{\sqrt{2051}}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} $$ $$ = \frac{\sqrt{4102}}{6} $$ The sample standard deviation is approximately: $$ \boxed{\frac{\sqrt{4102}}{6}} $$ This is the final answer for the sample standard deviation of the given set of numbers. |
\sqrt{\frac{601}{6}} |
| doubao | First find the mean: \[ \begin{align*} &(1+12+(-17)+(-3)+(-11)+(-5))\div6\\ &=(13-17-3-11-5)\div6\\ &=(-4-3-11-5)\div6\\ &=(-7-11-5)\div6\\ &=(-18-5)\div6\\ &=-23\div6\\ &=-\frac{23}{6} \end{align*} \] Then calculate the deviations from the mean for each number: \[ \begin{align*} &1-(-\frac{23}{6})=\frac{29}{6}\\ &12-(-\frac{23}{6})=\frac{95}{6}\\ &-17-(-\frac{23}{6})=-\frac{89}{6}\\ &-3-(-\frac{23}{6})=\frac{5}{6}\\ &-11-(-\frac{23}{6})=-\frac{43}{6}\\ &-5-(-\frac{23}{6})=\frac{13}{6} \end{align*} \] Square each deviation: \[ \begin{align*} &(\frac{29}{6})^2=\frac{841}{36}\\ &(\frac{95}{6})^2=\frac{9025}{36}\\ &(-\frac{89}{6})^2=\frac{7921}{36}\\ &(\frac{5}{6})^2=\frac{25}{36}\\ &(-\frac{43}{6})^2=\frac{1849}{36}\\ &(\frac{13}{6})^2=\frac{169}{36} \end{align*} \] Find the sum of the squared deviations: \[ \begin{align*} &\frac{841}{36}+\frac{9025}{36}+\frac{7921}{36}+\frac{25}{36}+\frac{1849}{36}+\frac{169}{36}\\ &=\frac{841+9025+7921+25+1849+169}{36}\\ &=\frac{18860}{36} \end{align*} \] Divide the sum of squared deviations by the number of data points (6) to get the variance: \[ \frac{18860}{36}\div6=\frac{18860}{36\times6}=\frac{18860}{216}=\frac{4715}{54} \] Finally, take the square root of the variance to get the standard deviation: \[ \sqrt{\frac{4715}{54}}=\boxed{\frac{\sqrt{253140}}{54}} \] |
\sqrt{\frac{601}{6}} |
| gpto1 | To compute the sample standard deviation of the dataset \(\{1, 12, -17, -3, -11, -5\}\), follow these steps: **1. Calculate the sample mean \(\bar{x}\):** \[ \bar{x} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{-23}{6} \] **2. Calculate each deviation from the mean and square them:** For each \(x_i\), compute \( (x_i - \bar{x})^2 \): \[ \begin{align*} (1 - (-\tfrac{23}{6}))^2 &= \left(\tfrac{29}{6}\right)^2 = \tfrac{841}{36} \\ (12 - (-\tfrac{23}{6}))^2 &= \left(\tfrac{95}{6}\right)^2 = \tfrac{9025}{36} \\ (-17 - (-\tfrac{23}{6}))^2 &= \left(-\tfrac{79}{6}\right)^2 = \tfrac{6241}{36} \\ (-3 - (-\tfrac{23}{6}))^2 &= \left(\tfrac{5}{6}\right)^2 = \tfrac{25}{36} \\ (-11 - (-\tfrac{23}{6}))^2 &= \left(-\tfrac{43}{6}\right)^2 = \tfrac{1849}{36} \\ (-5 - (-\tfrac{23}{6}))^2 &= \left(-\tfrac{7}{6}\right)^2 = \tfrac{49}{36} \\ \end{align*} \] **3. Sum the squared deviations:** \[ \text{Sum} = \tfrac{841}{36} + \tfrac{9025}{36} + \tfrac{6241}{36} + \tfrac{25}{36} + \tfrac{1849}{36} + \tfrac{49}{36} = \tfrac{18030}{36} \] Simplify the sum: \[ \frac{18030}{36} = \frac{3005}{6} \] **4. Compute the sample variance \(s^2\):** \[ s^2 = \frac{1}{n-1} \times \text{Sum of squared deviations} = \frac{1}{5} \times \frac{3005}{6} = \frac{3005}{30} = \frac{601}{6} \] **5. Calculate the sample standard deviation \(s\):** \[ s = \sqrt{s^2} = \sqrt{\frac{601}{6}} \] **Final Answer:** \[ s = \boxed{\sqrt{\dfrac{601}{6}}} \] |
\sqrt{\frac{601}{6}} |
| kimi | To compute the sample standard deviation of the given data set \(\{1, 12, -17, -3, -11, -5\}\), 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. \[ \bar{x} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} \] First, compute the sum: \[ 1 + 12 - 17 - 3 - 11 - 5 = -23 \] Now, divide by the number of values (6): \[ \bar{x} = \frac{-23}{6} \approx -3.8333 \] 2. **Calculate the squared differences from the mean for each data point:** For each \(x_i\) in the data set, compute \((x_i - \bar{x})^2\). \[ (1 - (-3.8333))^2 = (1 + 3.8333)^2 = 4.8333^2 \approx 23.4259 \] \[ (12 - (-3.8333))^2 = (12 + 3.8333)^2 = 15.8333^2 \approx 250.67 \] \[ (-17 - (-3.8333))^2 = (-17 + 3.8333)^2 = -13.1667^2 \approx 172.71 \] \[ (-3 - (-3.8333))^2 = (-3 + 3.8333)^2 = 0.8333^2 \approx 0.6944 \] \[ (-11 - (-3.8333))^2 = (-11 + 3.8333)^2 = -7.1667^2 \approx 51.31 \] \[ (-5 - (-3.8333))^2 = (-5 + 3.8333)^2 = -1.1667^2 \approx 1.36 \] 3. **Calculate the sum of the squared differences:** Add up all the squared differences: \[ 23.4259 + 250.67 + 172.71 + 0.6944 + 51.31 + 1.36 \approx 500.17 \] 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 data points. \[ s^2 = \frac{500.17}{6-1} = \frac{500.17}{5} \approx 100.034 \] 5. **Calculate the sample standard deviation (\(s\)):** The sample standard deviation is the square root of the sample variance. \[ s = \sqrt{100.034} \approx 10.002 \] Therefore, the sample standard deviation of the given data set is: \[ \boxed{10.002} \] |
\sqrt{\frac{601}{6}} |
| minimax | To compute the sample standard deviation of the given data set ${1, 12, -17, -3, -11, -5}$, 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 these squared deviations. 5. Take the square root of the mean of squared deviations to get the standard deviation. Let's go through these steps: 1. Calculate the mean: $$ \text{Mean} = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{-13}{6} = -2.1667 $$ 2. Calculate the deviations from the mean: $$ 1 - (-2.1667) = 3.1667 $$ $$ 12 - (-2.1667) = 14.1667 $$ $$ -17 - (-2.1667) = -14.8333 $$ $$ -3 - (-2.16 |
\sqrt{\frac{601}{6}} |
| wenxinyiyan | 计算样本标准差首先需要计算样本的均值。 样本均值 $\overline{x}$ 的计算公式为： $\overline{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$ 其中 $n$ 是样本数量，$x_i$ 是每一个样本点。 给定的样本是 ${1, 12, -17, -3, -11, -5}$，样本数量 $n = 6$。 计算均值： $\overline{x} = \frac{1}{6}(1 + 12 - 17 - 3 - 11 - 5) = \frac{1}{6} \times (-23) = -\frac{23}{6}$ 接下来计算样本方差 $s^2$，其公式为： $s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_i - \overline{x})^2$ 将给定的样本和计算出的均值代入方差公式： $s^2 = \frac{1}{5}\left[(1 + \frac{23}{6})^2 + (12 + \frac{23}{6})^2 + (-17 + \frac{23}{6})^2 + (-3 + \frac{23}{6})^2 + (-11 + \frac{23}{6})^2 + (-5 + \frac{23}{6})^2\right]$ $= \frac{1}{5}\left[(\frac{29}{6})^2 + (\frac{95}{6})^2 + (\frac{79}{6})^2 + (\frac{11}{6})^2 + (\frac{49}{6})^2 + (\frac{13}{6})^2\right]$ $= \frac{1}{5}\left[\frac{841}{36} + \frac{9025}{36} + \frac{6241}{36} + \frac{121}{36} + \frac{2401}{36} + \frac{169}{36}\right]$ $= \frac{1}{5} \times \frac{18808}{36}$ $= \frac{18808}{180}$ $= \frac{4702}{45}$ 最后，计算样本标准差 $s$，即方差 $s^2$ 的平方根： $s = \sqrt{\frac{4702}{45}} = \frac{\sqrt{4702}}{\sqrt{45}} = \frac{\sqrt{4702}}{3\sqrt{5}} \approx 10.22$ 所以，样本标准差约为 10.22。 |
\sqrt{\frac{601}{6}} |
| zhipu | 1. **Calculate the mean of the data set**: \[ \text{Mean} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} = -\frac{23}{6} \] 2. **Calculate the squared deviations from the mean**: - For $1$: \[ \left(1 + \frac{23}{6}\right)^2 = \left(\frac{29}{6}\right)^2 = \frac{841}{36} \] - For $12$: \[ \left(12 + \frac{23}{6}\right)^2 = \left(\frac{89}{6}\right)^2 = \frac{7921}{36} \] - For $-17$: \[ \left(-17 + \frac{23}{6}\right)^2 = \left(-\frac{89}{6}\right)^2 = \frac{7921}{36} \] - For $-3$: \[ \left(-3 + \frac{23}{6}\right)^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \] - For $-11$: \[ \left(-11 + \frac{23}{6}\right)^2 = \left(-\frac{43}{6}\right)^2 = \frac{1849}{36} \] - For $-5$: \[ \left(-5 + \frac{23}{6}\right)^2 = \left(-\frac{7}{6}\right)^2 = \frac{49}{36} \] 3. **Sum the squared deviations**: \[ \text{Sum of squared deviations} = \frac{841}{36} + \frac{7921}{36} + \frac{7921}{36} + \frac{25}{36} + \frac{1849}{36} + \frac{49}{36} = \frac{19406}{36} \] 4. **Calculate the variance**: \[ \text{Variance} = \frac{\frac{19406}{36}}{6} = \frac{19406}{216} = \frac{3234.33}{36} \approx 90.39 \] 5. **Calculate the standard deviation**: \[ \text{Standard Deviation} = \sqrt{90.39} \approx 9.51 \] However, to keep exact values: \[ \text{Standard Deviation} = \sqrt{\frac{19406}{216}} = \frac{\sqrt{19406}}{\sqrt{216}} = \frac{139.4}{14.7} \approx 9.45 \] For an exact expression: \[ \text{Standard Deviation} = \frac{\sqrt{19406}}{\sqrt{216}} \] Approximating further with a calculator for a precise decimal: \[ \text{Standard Deviation} \approx 9.7 \] Thus, the sample standard deviation is $\\boxed{9.7}$. |
\sqrt{\frac{601}{6}} |
| deepseek-v3 | To compute the **sample standard deviation** of the dataset \(\{1, 12, -17, -3, -11, -5\}\), follow these steps: 1. **Calculate the Mean (\(\mu\)):** \[ \mu = \frac{1 + 12 + (-17) + (-3) + (-11) + (-5)}{6} = \frac{1 + 12 - 17 - 3 - 11 - 5}{6} = \frac{-23}{6} \approx -3.8333 \] 2. **Compute the Squared Deviations from the Mean:** \[ \begin{align*} (1 - (-3.8333))^2 &= (4.8333)^2 \approx 23.3611 \\ (12 - (-3.8333))^2 &= (15.8333)^2 \approx 250.6944 \\ (-17 - (-3.8333))^2 &= (-13.1667)^2 \approx 173.3611 \\ (-3 - (-3.8333))^2 &= (0.8333)^2 \approx 0.6944 \\ (-11 - (-3.8333))^2 &= (-7.1667)^2 \approx 51.3611 \\ (-5 - (-3.8333))^2 &= (-1.1667)^2 \approx 1.3611 \\ \end{align*} \] Sum of squared deviations: \[ 23.3611 + 250.6944 + 173.3611 + 0.6944 + 51.3611 + 1.3611 = 500.8332 \] 3. **Calculate the Variance (\(s^2\)):** Since this is a sample standard deviation, we divide by \(n - 1\) (where \(n = 6\)): \[ s^2 = \frac{500.8332}{5} = 100.1666 \] 4. **Compute the Sample Standard Deviation (\(s\)):** \[ s = \sqrt{100.1666} \approx 10.0083 \] Rounding to four decimal places: \[ s \approx 10.0083 \] Therefore, the sample standard deviation is: \[ \boxed{10.0083} \] |
\sqrt{\frac{601}{6}} |