| 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 \(\{-11, 9, 5, 15, -10, -15, 12, -25, -22\}\), we follow these steps: 1. **Calculate the mean (average) of the data set.** The sum of the data points is: \[ -11 + 9 + 5 + 15 + (-10) + (-15) + 12 + (-25) + (-22) = -42 \] The number of data points is 9. Therefore, the mean \(\bar{x}\) is: \[ \bar{x} = \frac{-42}{9} = -4.6667 \] 2. **Calculate the squared differences from the mean for each data point.** \[ (-11 - (-4.6667))^2 = (-11 + 4.6667)^2 = (-6.3333)^2 = 40.1111 \] \[ (9 - (-4.6667))^2 = (9 + 4.6667)^2 = 13.6667^2 = 186.7778 \] \[ (5 - (-4.6667))^2 = (5 + 4.6667)^2 = 9.6667^2 = 93.4444 \] \[ (15 - (-4.6667))^2 = (15 + 4.6667)^2 = 19.6667^2 = 386.7778 \] \[ (-10 - (-4.6667))^2 = (-10 + 4.6667)^2 = (-5.3333)^2 = 28.4444 \] \[ (-15 - (-4.6667))^2 = (-15 + 4.6667)^2 = (-10.3333)^2 = 106.7778 \] \[ (12 - (-4.6667))^2 = (12 + 4.6667)^2 = 16.6667^2 = 277.7778 \] \[ (-25 - (-4.6667))^2 = (-25 + 4.6667)^2 = (-20.3333)^2 = 413.4444 \] \[ (-22 - (-4.6667))^2 = (-22 + 4.6667)^2 = (-17.3333)^2 = 300.4444 \] 3. **Sum the squared differences.** \[ 40.1111 + 186.7778 + 93.4444 + 386.7778 + 28.4444 + 106.7778 + 277.7778 + 413.4444 + 300.4444 = 1834.0000 \] 4. **Calculate the variance.** The variance \(s^2\) is the mean of the squared differences: \[ s^2 = \frac{1834.0000}{9 - 1} = \frac{1834.0000}{8} = 229.25 \] 5. **Calculate the sample standard deviation.** The sample standard deviation \(s\) is the square root of the variance: \[ s = \sqrt{229.25} = 15.1410 \] Thus, the sample standard deviation is: \[ \boxed{15.1410} \] |
\frac{\sqrt{917}}{2} |
| deepseek-r1 | To compute the sample standard deviation of the data set \({-11, 9, 5, 15, -10, -15, 12, -25, -22}\), we follow these steps: 1. **Calculate the mean**: \[ \text{Mean} = \frac{-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22}{9} = \frac{-42}{9} = -\frac{14}{3} \approx -4.6667 \] 2. **Compute the squared deviations from the mean**: \[ \begin{aligned} (-11 - (-14/3))^2 &= \left(-\frac{19}{3}\right)^2 = \frac{361}{9} \\ (9 - (-14/3))^2 &= \left(\frac{41}{3}\right)^2 = \frac{1681}{9} \\ (5 - (-14/3))^2 &= \left(\frac{29}{3}\right)^2 = \frac{841}{9} \\ (15 - (-14/3))^2 &= \left(\frac{59}{3}\right)^2 = \frac{3481}{9} \\ (-10 - (-14/3))^2 &= \left(-\frac{16}{3}\right)^2 = \frac{256}{9} \\ (-15 - (-14/3))^2 &= \left(-\frac{31}{3}\right)^2 = \frac{961}{9} \\ (12 - (-14/3))^2 &= \left(\frac{50}{3}\right)^2 = \frac{2500}{9} \\ (-25 - (-14/3))^2 &= \left(-\frac{61}{3}\right)^2 = \frac{3721}{9} \\ (-22 - (-14/3))^2 &= \left(-\frac{52}{3}\right)^2 = \frac{2704}{9} \\ \end{aligned} \] 3. **Sum the squared deviations**: \[ \frac{361 + 1681 + 841 + 3481 + 256 + 961 + 2500 + 3721 + 2704}{9} = \frac{16506}{9} = 1834 \] 4. **Calculate the sample variance**: \[ \text{Variance} = \frac{1834}{9-1} = \frac{1834}{8} = 229.25 \] 5. **Compute the sample standard deviation**: \[ \text{Standard Deviation} = \sqrt{229.25} \approx 15.14 \] Thus, the sample standard deviation of the data set is \(\boxed{15.14}\). |
\frac{\sqrt{917}}{2} |
| gpto1 | To compute the sample standard deviation of the data set \(-11, 9, 5, 15, -10, -15, 12, -25, -22\), we follow these steps: **1. Calculate the mean (average) of the data:** First, sum all the data values: \[ \text{Sum} = (-11) + 9 + 5 + 15 + (-10) + (-15) + 12 + (-25) + (-22) = -42 \] Divide by the number of data points (\(n = 9\)): \[ \bar{x} = \frac{-42}{9} = -\frac{14}{3} \] **2. Calculate each data point's deviation from the mean and square it:** For each \(x_i\), compute \((x_i - \bar{x})^2\): - \((-11) - \left( -\frac{14}{3} \right) = -11 + \frac{14}{3} = -\frac{19}{3},\quad \left( -\frac{19}{3} \right)^2 = \frac{361}{9}\) - \(9 - \left( -\frac{14}{3} \right) = 9 + \frac{14}{3} = \frac{41}{3},\quad \left( \frac{41}{3} \right)^2 = \frac{1681}{9}\) - Repeat this process for all data points. **3. Sum all the squared deviations:** Add up all the squared deviations: \[ \sum (x_i - \bar{x})^2 = \frac{361}{9} + \frac{1681}{9} + \frac{841}{9} + \frac{3481}{9} + \frac{256}{9} + \frac{961}{9} + \frac{2500}{9} + \frac{3721}{9} + \frac{2704}{9} = \frac{16506}{9} \] Simplify: \[ \sum (x_i - \bar{x})^2 = \frac{16506}{9} = 1834 \] **4. Compute the sample variance:** \[ s^2 = \frac{1}{n - 1} \sum (x_i - \bar{x})^2 = \frac{1}{8} \times 1834 = 229.25 \] **5. Compute the sample standard deviation:** \[ s = \sqrt{s^2} = \sqrt{229.25} = \sqrt{\frac{917}{4}} = \frac{\sqrt{917}}{2} \] Therefore, the sample standard deviation is: \[ \boxed{\dfrac{\sqrt{917}}{2}} \] |
\frac{\sqrt{917}}{2} |
| 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: \(-11, 9, 5, 15, -10, -15, 12, -25, -22\). **Step 1: Calculate the mean.** The mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(n\) is the number of data points. Here, \(n = 9\). First, calculate the sum of the data points: \[ -11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22 = -42 \] Now, calculate the mean: \[ \bar{x} = \frac{-42}{9} = -\frac{42}{9} = -4.6667 \] **Step 2: Calculate the variance.** The sample variance \(s^2\) is given by: \[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \] Calculate each squared deviation from the mean: \[ (-11 - (-4.6667))^2 = (-11 + 4.6667)^2 = (-6.3333)^2 = 40.1111 \] \[ (9 - (-4.6667))^2 = (9 + 4.6667)^2 = 13.6667^2 = 186.7778 \] \[ (5 - (-4.6667))^2 = (5 + 4.6667)^2 = 9.6667^2 = 93.4444 \] \[ (15 - (-4.6667))^2 = (15 + 4.6667)^2 = 19.6667^2 = 386.7778 \] \[ (-10 - (-4.6667))^2 = (-10 + 4.6667)^2 = (-5.3333)^2 = 28.4444 \] \[ (-15 - (-4.6667))^2 = (-15 + 4.6667)^2 = (-10.3333)^2 = 106.7778 \] \[ (12 - (-4.6667))^2 = (12 + 4.6667)^2 = 16.6667^2 = 277.7778 \] \[ (-25 - (-4.6667))^2 = (-25 + 4.6667)^2 = (-20.3333)^2 = 413.4444 \] \[ (-22 - (-4.6667))^2 = (-22 + 4.6667)^2 = (-17.3333)^2 = 300.4444 \] Sum of squared deviations: \[ 40.1111 + 186.7778 + 93.4444 + 386.7778 + 28.4444 + 106.7778 + 277.7778 + 413.4444 + 300.4444 = 1834 \] Now, calculate the variance: \[ s^2 = \frac{1834}{9-1} = \frac{1834}{8} = 229.25 \] **Step 3: Calculate the standard deviation.** The sample standard deviation \(s\) is the square root of the variance: \[ s = \sqrt{229.25} \approx 15.1342 \] Thus, the sample standard deviation is: \[ \boxed{15.1342} \] |
\frac{\sqrt{917}}{2} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample standard deviation of the given list \({-11, 9, 5, 15, -10, -15, 12, -25, -22}\), we will follow these steps: 1. **Calculate the mean (average) of the numbers:** \[ \text{Mean} = \frac{-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22}{9} \] Let's add the numbers step by step: \[ -11 + 9 = -2 \] \[ -2 + 5 = 3 \] \[ 3 + 15 = 18 \] \[ 18 - 10 = 8 \] \[ 8 - 15 = -7 \] \[ -7 + 12 = 5 \] \[ 5 - 25 = -20 \] \[ -20 - 22 = -42 \] So, the sum of the numbers is \(-42\). Now, divide by the number of elements (9): \[ \text{Mean} = \frac{-42}{9} = -4.6667 \] 2. **Calculate the variance:** For each number, subtract the mean and square the result: \[ (-11 - (-4.6667))^2 = (-11 + 4.6667)^2 = (-6.3333)^2 = 40.1111 \] \[ (9 - (-4.6667))^2 = (9 + 4.6667)^2 = 13.6667^2 = 186.7778 \] \[ (5 - (-4.6667))^2 = (5 + 4.6667)^2 = 9.6667^2 = 93.4444 \] \[ (15 - (-4.6667))^2 = (15 + 4.6667)^2 = 19.6667^2 = 386.7778 \] \[ (-10 - (-4.6667))^2 = (-10 + 4.6667)^2 = -5.3333^2 = 28.4444 \] \[ (-15 - (-4.6667))^2 = (-15 + 4.6667)^2 = -10.3333^2 = 106.7778 \] \[ (12 - (-4.6667))^2 = (12 + 4.6667)^2 = 16.6667^2 = 277.7778 \] \[ (-25 - (-4.6667))^2 = (-25 + 4.6667)^2 = -20.3333^2 = 413.4444 \] \[ (-22 - (-4.6667))^2 = (-22 + 4.6667)^2 = -17.3333^2 = 300.4444 \] Now, sum these squared differences: \[ 40.1111 + 186.7778 + 93.4444 + 386.7778 + 28.4444 + 106.7778 + 277.7778 + 413.4444 + 300.4444 = 1634.0444 \] 3. **Calculate the sample standard deviation:** The variance is \(1634.0444\). The sample standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{1634.0444} \approx 40.42 \] Thus, the sample standard deviation is: \[ \boxed{40.42} \] |
\frac{\sqrt{917}}{2} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Calculate the mean of the given data set. To find the mean, we sum all the numbers and then divide by the total count of numbers. The data set is ${-11, 9, 5, 15, -10, -15, 12, -25, -22}$. The sum is $-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22 = -42$. There are 9 numbers in the data set, so the mean is $-42 / 9 = -4.67$. ## Step 2: Calculate the variance of the data set. To find the variance, we first need to find the deviation of each number from the mean, then square each deviation, sum these squared deviations, and finally divide by the number of items minus one (for sample variance). The deviations from the mean are $(-11 - (-4.67)), (9 - (-4.67)), (5 - (-4.67)), (15 - (-4.67)), (-10 - (-4.67)), (-15 - (-4.67)), (12 - (-4.67)), (-25 - (-4.67)), (-22 - (-4.67))$ which simplifies to $-6.33, 13.67, 9.67, 19.67, -5.33, -10.33, 16.67, -20.33, -17.33$. Squaring each gives $40.07, 186.89, 93.45, 386.89, 28.41, 106.69, 277.89, 413.69, 300.29$. Summing these gives $1834.27$. Dividing by $9 - 1 = 8$ gives a variance of $229.28$. ## Step 3: Calculate the sample standard deviation. The sample standard deviation is the square root of the variance. Therefore, we calculate $\sqrt{229.28}$. The final answer is: $\boxed{15.13}$ |
\frac{\sqrt{917}}{2} |
| 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 ${-11, 9, 5, 15, -10, -15, 12, -25, -22}$. The sum of these numbers is $-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22 = -42$. There are 9 numbers in the data set. Therefore, the mean is $-42 / 9 = -4.67$. ## Step 2: Subtract the mean from each number in the data set. Next, we subtract the mean from each number in the data set to find the deviations from the mean. The deviations are $(-11 - (-4.67)), (9 - (-4.67)), (5 - (-4.67)), (15 - (-4.67)), (-10 - (-4.67)), (-15 - (-4.67)), (12 - (-4.67)), (-25 - (-4.67)), (-22 - (-4.67))$. This simplifies to $-6.33, 13.67, 9.67, 19.67, -5.33, -10.33, 16.67, -20.33, -17.33$. ## Step 3: Square each deviation. Now, we square each deviation. The squared deviations are $(-6.33)^2, (13.67)^2, (9.67)^2, (19.67)^2, (-5.33)^2, (-10.33)^2, (16.67)^2, (-20.33)^2, (-17.33)^2$. This simplifies to $40.07, 186.89, 93.45, 388.09, 28.41, 106.69, 277.89, 412.41, 299.69$. ## Step 4: Calculate the sum of the squared deviations. We then calculate the sum of the squared deviations. The sum is $40.07 + 186.89 + 93.45 + 388.09 + 28.41 + 106.69 + 277.89 + 412.41 + 299.69 = 1833.09$. ## Step 5: Divide the sum of the squared deviations by the number of items minus one. Since we are calculating the sample standard deviation, we divide the sum of the squared deviations by the number of items minus one. There are 9 items, so we divide by 8. The result is $1833.09 / 8 = 229.14$. ## Step 6: Take the square root of the result. Finally, we take the square root of the result to find the sample standard deviation. The square root of $229.14$ is $15.13$. The final answer is: $\boxed{15.13}$ |
\frac{\sqrt{917}}{2} |
| 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: \(-11, 9, 5, 15, -10, -15, 12, -25, -22\). Step 1: Calculate the mean \[ \text{Mean} = \frac{-11 + 9 + 5 + 15 + (-10) + (-15) + 12 + (-25) + (-22)}{9} \] \[ \text{Mean} = \frac{-47}{9} = -5.2222... \] Step 2: Subtract the mean from each number \[ -11 - (-5.2222...) = -5.7778 \] \[ 9 - (-5.2222...) = 14.2222 \] \[ 5 - (-5.2222...) = 10.2222 \] \[ 15 - (-5.2222...) = 20.2222 \] \[ -10 - (-5.2222...) = -4.7778 \] \[ -15 - (-5.2222...) = -9.7778 \] \[ 12 - (-5.2222...) = 17.2222 \] \[ -25 - (-5.2222...) = -19.7778 \] \[ -22 - (-5.2222...) = -16.7778 \] Step 3: Square each deviation \[ (-5.7778)^2 = 33.4000 \] \[ (14.2222)^2 = 202.2778 \] \[ (10.2222)^2 = 104.4999 \] \[ (20.2222)^2 = 408.9333 \] \[ (-4.7778)^2 = 22.8222 \] \[ (-9.7778)^2 = 95.6000 \] \[ (17.2222)^2 = 296.4999 \] \[ (-19.7778)^2 = 391.1556 \] \[ (-16.7778)^2 = 281.5556 \] Step 4: Find the average of these squared deviations (variance) \[ \text{Variance} = \frac{33.4000 + 202.2778 + 104.4999 + 408.9333 + 22.8222 + 95.6000 + 296.4999 + 391.1556 + 281.5556}{9} \] \[ \text{Variance} = \frac{1836.7443}{9} = 204.0827 \] Step 5: Take the square root of the variance to get the standard deviation \[ \text{Standard Deviation} = \sqrt{204.0827} = 14.2857 \] Therefore, the sample standard deviation of the given set of numbers is approximately \(14.2857\). \(\boxed{14.2857}\) |
\frac{\sqrt{917}}{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, 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: ${-11, 9, 5, 15, -10, -15, 12, -25, -22}$, let's follow these steps: 1. Calculate the mean: Mean = (-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22) / 9 Mean = (-32) / 9 Mean ≈ -3.5556 2. For each number, subtract the mean and square the result: (-11 - (-3.5556))^2 = (-7.4444)^2 ≈ 55.3333 (9 - (-3.5556))^2 = 12.5556^2 ≈ 157.7778 (5 - (-3.5556))^2 = 8.5556^2 ≈ 73.1778 (15 - (-3.5556))^2 = 18.5556^2 ≈ 344.2222 (-10 - (-3.5556))^2 = (-6.4444)^2 ≈ 41.5333 (-15 - (-3.5556))^2 = (-11.4444)^2 ≈ 131.1111 (12 - (-3.5556))^2 = 15.5556^2 ≈ 241.9778 (-25 - (-3.5556))^2 = (-21.4444)^2 ≈ 460.3333 (-22 - (-3.5556))^2 = (-18.4444)^2 ≈ 340.0000 3. Sum up these squared differences: Sum ≈ 55.3333 + 157.7778 + 73.1778 + 344.2222 + 41.5333 + 131.1111 + 241.9778 + 460.3333 + 340.0000 Sum ≈ 1845.4678 4. Divide by the number of observations minus one (since it's a sample standard deviation): Variance ≈ 1845.4678 / (9 - 1) Variance ≈ 1845.4678 / 8 Variance ≈ 230.6835 5. Take the square root of the variance to get the standard deviation: Standard Deviation ≈ √230.6835 Standard Deviation ≈ 15.1896 Therefore, the sample standard deviation of the given numbers is approximately 15.1896. Please note that the calculations were done using approximate values due to the nature of the arithmetic operations involved. The final answer should be rounded to a reasonable number of decimal places for practical purposes. In this case, we can round to two decimal places: $ \boxed{15.19} $ |
\frac{\sqrt{917}}{2} |
| doubao | Detailed exception information is listed below. req_id: 2024101011423800004DAFB5C2A12357FB code_n: 1709701 code: ClientSDKRequestError message: MaaS SDK request error: HTTPSConnectionPool(host='maas-api.ml-platform-cn-beijing.volces.com', port=443): Read timed out. (read timeout=60) |
\frac{\sqrt{917}}{2} |
| gpto1 | To compute the sample standard deviation of the data set \(-11, 9, 5, 15, -10, -15, 12, -25, -22\), we follow these steps: **1. Calculate the mean (average) of the data:** First, sum all the data values: \[ \text{Sum} = (-11) + 9 + 5 + 15 + (-10) + (-15) + 12 + (-25) + (-22) = -42 \] Divide by the number of data points (\(n = 9\)): \[ \bar{x} = \frac{-42}{9} = -\frac{14}{3} \] **2. Calculate each data point's deviation from the mean and square it:** For each \(x_i\), compute \((x_i - \bar{x})^2\): - \((-11) - \left( -\frac{14}{3} \right) = -11 + \frac{14}{3} = -\frac{19}{3},\quad \left( -\frac{19}{3} \right)^2 = \frac{361}{9}\) - \(9 - \left( -\frac{14}{3} \right) = 9 + \frac{14}{3} = \frac{41}{3},\quad \left( \frac{41}{3} \right)^2 = \frac{1681}{9}\) - Repeat this process for all data points. **3. Sum all the squared deviations:** Add up all the squared deviations: \[ \sum (x_i - \bar{x})^2 = \frac{361}{9} + \frac{1681}{9} + \frac{841}{9} + \frac{3481}{9} + \frac{256}{9} + \frac{961}{9} + \frac{2500}{9} + \frac{3721}{9} + \frac{2704}{9} = \frac{16506}{9} \] Simplify: \[ \sum (x_i - \bar{x})^2 = \frac{16506}{9} = 1834 \] **4. Compute the sample variance:** \[ s^2 = \frac{1}{n - 1} \sum (x_i - \bar{x})^2 = \frac{1}{8} \times 1834 = 229.25 \] **5. Compute the sample standard deviation:** \[ s = \sqrt{s^2} = \sqrt{229.25} = \sqrt{\frac{917}{4}} = \frac{\sqrt{917}}{2} \] Therefore, the sample standard deviation is: \[ \boxed{\dfrac{\sqrt{917}}{2}} \] |
\frac{\sqrt{917}}{2} |
| kimi | To compute the sample standard deviation of the given data set \(\{-11, 9, 5, 15, -10, -15, 12, -25, -22\}\), we will follow these steps: 1. **Calculate the mean (\(\bar{x}\)) of the data set.** 2. **Calculate the squared differences from the mean for each data point.** 3. **Compute the average of these squared differences (variance).** 4. **Take the square root of the variance to get the standard deviation.** ### Step 1: Calculate the mean (\(\bar{x}\)) First, sum all the data points: \[ -11 + 9 + 5 + 15 + (-10) + (-15) + 12 + (-25) + (-22) \] Perform the addition step-by-step: \[ -11 + 9 = -2 \] \[ -2 + 5 = 3 \] \[ 3 + 15 = 18 \] \[ 18 - 10 = 8 \] \[ 8 - 15 = -7 \] \[ -7 + 12 = 5 \] \[ 5 - 25 = -20 \] \[ -20 - 22 = -42 \] The sum of the data points is \(-42\). Next, divide the sum by the number of data points (9): \[ \bar{x} = \frac{-42}{9} = -4.6667 \] ### Step 2: Calculate the squared differences from the mean For each data point \(x_i\), compute \((x_i - \bar{x})^2\): \[ (-11 - (-4.6667))^2 = (-11 + 4.6667)^2 = (-6.3333)^2 = 40.1111 \] \[ (9 - (-4.6667))^2 = (9 + 4.6667)^2 = 13.6667^2 = 186.7778 \] \[ (5 - (-4.6667))^2 = (5 + 4.6667)^2 = 9.6667^2 = 93.4444 \] \[ (15 - (-4.6667))^2 = (15 + 4.6667)^2 = 19.6667^2 = 386.7778 \] \[ (-10 - (-4.6667))^2 = (-10 + 4.6667)^2 = (-5.3333)^2 = 28.4444 \] \[ (-15 - (-4.6667))^2 = (-15 + 4.6667)^2 = (-10.3333)^2 = 106.7778 \] \[ (12 - (-4.6667))^2 = (12 + 4.6667)^2 = 16.6667^2 = 277.7778 \] \[ (-25 - (-4.6667))^2 = (-25 + 4.6667)^2 = (-20.3333)^2 = 412.4444 \] \[ (-22 - (-4.6667))^2 = (-22 + 4.6667)^2 = (-17.3333)^2 = 299.4444 \] ### Step 3: Compute the average of these squared differences (variance) Sum the squared differences: \[ 40.1111 + 186.7778 + 93.4444 + 386.7778 + 28.4444 + 106.7778 + 277.7778 + 412.4444 + 299.4444 = 1770.6666 \] Divide by \(n-1\) (where \(n\) is the number of data points, 9): \[ s^2 = \frac{1770.6666}{8} = 221.3333 \] ### Step 4: Take the square root of the variance to get the standard deviation \[ s = \sqrt{221.3333} \approx 14.8750 \] Thus, the sample standard deviation is: \[ \boxed{14.875} \] |
\frac{\sqrt{917}}{2} |
| minimax | To compute the sample standard deviation of the given data set, we will follow these steps: 1. Find the mean (average) of the data set. 2. Subtract the mean from each data point and square the result. 3. Find the average of those squared differences. 4. Take the square root of that average to get the standard deviation. Let's go through these steps: 1. Calculate the mean: $$ \text{Mean} = \frac{\sum{x_i}}{n} = \frac{-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22}{9} = \frac{-52}{9} \approx -5.78 $$ 2. Subtract the mean from each data point and square the result: $$ (-11 - (-5.78))^2 = (-5.22)^2 \approx 27.2484 $$ $$ (9 - (-5.78))^2 = (14.78)^2 \approx 218.4524 $$ $$ (5 - (-5.78))^2 = (10.78)^2 |
\frac{\sqrt{917}}{2} |
| wenxinyiyan | 计算过程如下： 首先，我们需要计算这组数据的平均值。 平均值 $\bar{x}$ 的计算公式是： $$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$$ 其中 $n$ 是数据的数量，$x_i$ 是每一个数据点。 给定的数据是 ${-11, 9, 5, 15, -10, -15, 12, -25, -22}$，共有 $n = 9$ 个数据点。 计算平均值： $$\bar{x} = \frac{-11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22}{9}$$ $$\bar{x} = \frac{-42}{9}$$ $$\bar{x} = -4.666666666666667$$ 接下来，我们计算每个数据点与平均值的差的平方，然后求这些平方差的平均值。 方差 $s^2$ 的计算公式是： $$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$$ 计算方差： $$s^2 = \frac{(-11 + 4.666666666666667)^2 + (9 + 4.666666666666667)^2 + \cdots + (-22 + 4.666666666666667)^2}{9-1}$$ $$s^2 = \frac{384.6666666666667 + 189.3333333333333 + \cdots + 301.3333333333333}{8}$$ $$s^2 = \frac{1977.333333333333}{8}$$ $$s^2 = 247.16666666666666$$ 最后，我们计算方差的平方根，得到样本标准偏差。 样本标准偏差 $s$ 的计算公式是： $$s = \sqrt{s^2}$$ 计算样本标准偏差： $$s = \sqrt{247.16666666666666}$$ $$s \approx 15.72164596631226$$ 所以，样本标准偏差是 $\boxed{15.72164596631226}$。 |
\frac{\sqrt{917}}{2} |
| zhipu | 1. **Calculate the mean of the data set**: The data set is ${-11, 9, 5, 15, -10, -15, 12, -25, -22}$. First, find the sum of these numbers: \[ -11 + 9 + 5 + 15 - 10 - 15 + 12 - 25 - 22 = -62 \] There are 9 numbers in the set, so the mean (average) is: \[ \text{Mean} = \frac{-62}{9} \approx -6.89 \] 2. **Calculate the squared deviations from the mean**: For each number in the set, subtract the mean and square the result: \[ \begin{align*} (-11 + 6.89)^2 & = (-4.11)^2 = 16.8921, \\ (9 + 6.89)^2 & = (15.89)^2 = 252.7921, \\ (5 + 6.89)^2 & = (11.89)^2 = 141.1321, \\ (15 + 6.89)^2 & = (21.89)^2 = 480.7921, \\ (-10 + 6.89)^2 & = (-3.11)^2 = 9.6721, \\ (-15 + 6.89)^2 & = (-8.11)^2 = 65.7321, \\ (12 + 6.89)^2 & = (18.89)^2 = 355.9121, \\ (-25 + 6.89)^2 & = (-18.11)^2 = 328.7921, \\ (-22 + 6.89)^2 & = (-15.11)^2 = 228.7921. \end{align*} \] 3. **Sum the squared deviations**: \[ 16.8921 + 252.7921 + 141.1321 + 480.7921 + 9.6721 + 65.7321 + 355.9121 + 328.7921 + 228.7921 = 1779.6089 \] 4. **Calculate the variance**: The variance is the average of the squared deviations. Since this is a sample, we divide by $n-1$ (where $n$ is the number of observations): \[ \text{Variance} = \frac{1779.6089}{9-1} = \frac{1779.6089}{8} = 222.4501125 \] 5. **Calculate the standard deviation**: The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{222.4501125} \approx 14.91 \] Thus, the sample standard deviation of the data set is $\\boxed{14.91}$. |
\frac{\sqrt{917}}{2} |
| deepseek-v3 | To compute the **sample standard deviation** of the dataset \(\{-11, 9, 5, 15, -10, -15, 12, -25, -22\}\), follow these steps: ### 1. Calculate the Mean (\(\bar{x}\)) First, find the mean of the data set. \[ \bar{x} = \frac{(-11) + 9 + 5 + 15 + (-10) + (-15) + 12 + (-25) + (-22)}{9} \] \[ \bar{x} = \frac{-42}{9} = -4.6667 \] ### 2. Compute Each Deviation from the Mean Subtract the mean from each data point to find the deviations. \[ \begin{align*} -11 - (-4.6667) &= -6.3333 \\ 9 - (-4.6667) &= 13.6667 \\ 5 - (-4.6667) &= 9.6667 \\ 15 - (-4.6667) &= 19.6667 \\ -10 - (-4.6667) &= -5.3333 \\ -15 - (-4.6667) &= -10.3333 \\ 12 - (-4.6667) &= 16.6667 \\ -25 - (-4.6667) &= -20.3333 \\ -22 - (-4.6667) &= -17.3333 \\ \end{align*} \] ### 3. Square Each Deviation Square the deviations to eliminate negative values. \[ \begin{align*} (-6.3333)^2 &= 40.1111 \\ 13.6667^2 &= 186.7778 \\ 9.6667^2 &= 93.4444 \\ 19.6667^2 &= 386.7778 \\ (-5.3333)^2 &= 28.4444 \\ (-10.3333)^2 &= 106.7778 \\ 16.6667^2 &= 277.7778 \\ (-20.3333)^2 &= 413.4444 \\ (-17.3333)^2 &= 300.4444 \\ \end{align*} \] ### 4. Sum the Squared Deviations Add up all the squared deviations. \[ 40.1111 + 186.7778 + 93.4444 + 386.7778 + 28.4444 + 106.7778 + 277.7778 + 413.4444 + 300.4444 = 1833.0000 \] ### 5. Calculate the Variance Divide the sum of squared deviations by \(n - 1\) (since it's a sample). \[ \text{Variance} = \frac{1833.0000}{9 - 1} = \frac{1833.0000}{8} = 229.1250 \] ### 6. Find the Standard Deviation Take the square root of the variance to get the standard deviation. \[ \text{Standard Deviation} = \sqrt{229.1250} \approx 15.1369 \] ### Final Answer \[ \boxed{15.14} \] |
\frac{\sqrt{917}}{2} |