| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the sample variance of the set \(\{9, 14, 5, 4, -20, -13, -5, 13\}\), we follow these steps: 1. **Calculate the mean (average) of the data set:** \[ \text{Mean} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} \] \[ \text{Mean} = \frac{3 \times 13 + 14 + 5 + 4 - 20 - 13 - 5}{8} \] \[ \text{Mean} = \frac{39 + 14 + 5 + 4 - 20 - 13 - 5}{8} \] \[ \text{Mean} = \frac{39 + 14 + 4 - 20 - 13}{8} \] \[ \text{Mean} = \frac{39 + 14 - 20 - 13}{8} \] \[ \text{Mean} = \frac{39 - 20}{8} \] \[ \text{Mean} = \frac{19}{8} \] \[ \text{Mean} = 2.375 \] 2. **Calculate the squared deviations from the mean for each data point:** \[ (9 - 2.375)^2 = 6.625^2 = 43.890625 \] \[ (14 - 2.375)^2 = 11.625^2 = 135.140625 \] \[ (5 - 2.375)^2 = 2.625^2 = 6.890625 \] \[ (4 - 2.375)^2 = 1.625^2 = 2.640625 \] \[ (-20 - 2.375)^2 = -22.375^2 = 500.640625 \] \[ (-13 - 2.375)^2 = -15.375^2 = 236.390625 \] \[ (-5 - 2.375)^2 = -7.375^2 = 54.390625 \] \[ (13 - 2.375)^2 = 10.625^2 = 112.890625 \] 3. **Sum the squared deviations:** \[ \sum (x_i - \bar{x})^2 = 43.890625 + 135.140625 + 6.890625 + 2.640625 + 500.640625 + 236.390625 + 54.390625 + 112.890625 \] \[ \sum (x_i - \bar{x})^2 = 1092.875 \] 4. **Divide the sum of squared deviations by \(n-1\) (where \(n\) is the number of data points):** \[ \text{Sample Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{1092.875}{7} \] \[ \text{Sample Variance} = 156.125 \] Thus, the sample variance is: \[ \boxed{156.125} \] |
\frac{8599}{56} |
| deepseek-r1 | To compute the sample variance of the dataset \(\{9, 14, 5, 4, -20, -13, -5, 13\}\), we follow these steps: 1. **Calculate the sample mean**: \[ \text{Mean} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} = \frac{7}{8} = 0.875 \] 2. **Find the squared differences from the mean**: \[ \begin{aligned} (9 - 0.875)^2 &= 66.015625 \\ (14 - 0.875)^2 &= 172.265625 \\ (5 - 0.875)^2 &= 17.015625 \\ (4 - 0.875)^2 &= 9.765625 \\ (-20 - 0.875)^2 &= 435.765625 \\ (-13 - 0.875)^2 &= 192.515625 \\ (-5 - 0.875)^2 &= 34.515625 \\ (13 - 0.875)^2 &= 147.015625 \\ \end{aligned} \] 3. **Sum the squared differences**: \[ 66.015625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.515625 + 34.515625 + 147.015625 = 1074.875 \] 4. **Divide by \(n - 1\) (where \(n = 8\))**: \[ \text{Sample Variance} = \frac{1074.875}{7} = \frac{8599}{56} \] Thus, the sample variance of the dataset is \(\boxed{\dfrac{8599}{56}}\). |
\frac{8599}{56} |
| gpto1 | To compute the sample variance of the dataset \({9,\ 14,\ 5,\ 4,\ -20,\ -13,\ -5,\ 13}\), we follow these steps: **1. Calculate the Sample Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{\text{sum of all }x_i}{n} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} = \frac{7}{8} = 0.875 \] **2. Compute Each Squared Deviation (\((x_i - \bar{x})^2\)):** For each data point: - Calculate the deviation from the mean (\(x_i - \bar{x}\)). - Square this deviation. | \(x_i\) | \(x_i - \bar{x}\) | \((x_i - \bar{x})^2\) | |---------|--------------------|----------------------------| | 9 | \(9 - 0.875 = 8.125\) | \(8.125^2 = 66.015625\) | | 14 | \(14 - 0.875 = 13.125\) | \(13.125^2 = 172.265625\) | | 5 | \(5 - 0.875 = 4.125\) | \(4.125^2 = 17.015625\) | | 4 | \(4 - 0.875 = 3.125\) | \(3.125^2 = 9.765625\) | | -20 | \(-20 - 0.875 = -20.875\) | \((-20.875)^2 = 435.765625\) | | -13 | \(-13 - 0.875 = -13.875\) | \((-13.875)^2 = 192.515625\) | | -5 | \(-5 - 0.875 = -5.875\) | \((-5.875)^2 = 34.515625\) | | 13 | \(13 - 0.875 = 12.125\) | \(12.125^2 = 147.015625\) | **3. Sum All Squared Deviations:** \[ \sum(x_i - \bar{x})^2 = 66.015625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.515625 + 34.515625 + 147.015625 = 1074.875 \] **4. Calculate the Sample Variance (\(s^2\)):** \[ s^2 = \frac{\sum(x_i - \bar{x})^2}{n - 1} = \frac{1074.875}{7} \] To express the variance as a fraction: - Recognize that \(1074.875 = 1074 + \frac{7}{8}\). - Convert \(1074 + \frac{7}{8}\) to an improper fraction: \[ 1074 + \frac{7}{8} = \frac{(1074 \times 8) + 7}{8} = \frac{8599}{8} \] - Divide by \(n - 1 = 7\): \[ s^2 = \frac{\frac{8599}{8}}{7} = \frac{8599}{8} \times \frac{1}{7} = \frac{8599}{56} \] **Final Answer:** \[ \boxed{\dfrac{8599}{56}} \] |
\frac{8599}{56} |
| gpt-4o-2024-08-06 | To compute the sample variance, we first need to find the sample mean and then use it to calculate the variance. Given the data set: \( \{9, 14, 5, 4, -20, -13, -5, 13\} \). **Step 1: Calculate the sample mean.** The sample mean \(\bar{x}\) is given by: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \( n \) is the number of data points. Here, \( n = 8 \). First, calculate the sum of the data points: \[ 9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13 = 7 \] Now, calculate the mean: \[ \bar{x} = \frac{7}{8} = 0.875 \] **Step 2: Calculate the sample variance.** The sample variance \( s^2 \) is given by: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] Calculate each squared deviation from the mean: \[ (9 - 0.875)^2 = 66.015625 \] \[ (14 - 0.875)^2 = 173.140625 \] \[ (5 - 0.875)^2 = 17.015625 \] \[ (4 - 0.875)^2 = 9.765625 \] \[ (-20 - 0.875)^2 = 428.640625 \] \[ (-13 - 0.875)^2 = 194.640625 \] \[ (-5 - 0.875)^2 = 34.765625 \] \[ (13 - 0.875)^2 = 147.015625 \] Sum of squared deviations: \[ 66.015625 + 173.140625 + 17.015625 + 9.765625 + 428.640625 + 194.640625 + 34.765625 + 147.015625 = 1070 \] Now, calculate the sample variance: \[ s^2 = \frac{1070}{8-1} = \frac{1070}{7} = 152.85714285714286 \] Thus, the sample variance is: \[ \boxed{152.85714285714286} \] |
\frac{8599}{56} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the sample variance of the given data set \({9, 14, 5, 4, -20, -13, -5, 13}\), we will follow these steps: 1. **Calculate the mean (average)** of the data set. 2. **Calculate the squared differences** from the mean for each data point. 3. **Sum the squared differences**. 4. **Divide by the number of data points minus one** (n-1) to get the sample variance. Let's go through these steps in detail: ### Step 1: Calculate the mean The mean \(\mu\) is calculated by summing all the data points and dividing by the number of data points. \[ \mu = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} \] First, sum the data points: \[ 9 + 14 + 5 + 4 - 20 - 13 - 5 + 13 = 10 \] Now, divide by the number of data points (8): \[ \mu = \frac{10}{8} = 1.25 \] ### Step 2: Calculate the squared differences from the mean For each data point, calculate the squared difference from the mean: \[ (9 - 1.25)^2 = 7.75^2 = 60.0625 \] \[ (14 - 1.25)^2 = 12.75^2 = 162.5625 \] \[ (5 - 1.25)^2 = 3.75^2 = 14.0625 \] \[ (4 - 1.25)^2 = 2.75^2 = 7.5625 \] \[ (-20 - 1.25)^2 = -21.25^2 = 451.5625 \] \[ (-13 - 1.25)^2 = -14.25^2 = 203.0625 \] \[ (-5 - 1.25)^2 = -6.25^2 = 39.0625 \] \[ (13 - 1.25)^2 = 11.75^2 = 138.0625 \] ### Step 3: Sum the squared differences Sum all the squared differences: \[ 60.0625 + 162.5625 + 14.0625 + 7.5625 + 451.5625 + 203.0625 + 39.0625 + 138.0625 \] Calculate the sum: \[ 60.0625 + 162.5625 + 14.0625 + 7.5625 + 451.5625 + 203.0625 + 39.0625 + 138.0625 = 1016 \] ### Step 4: Divide by the number of data points minus one The number of data points is 8, so we divide by \(8 - 1 = 7\): \[ \text{Sample Variance} = \frac{1016}{7} = 145.142857 \] Rounding to two decimal places, the sample variance is: \[ \boxed{145.14} \] |
\frac{8599}{56} |
| 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. The sample mean is the average of all the numbers in the dataset. We add up all the numbers and then divide by the total count of numbers. ## Step 2: Add up all the numbers in the dataset The dataset is ${9, 14, 5, 4, -20, -13, -5, 13}$. Let's add them up: $9 + 14 + 5 + 4 - 20 - 13 - 5 + 13 = 7$. ## Step 3: Calculate the sample mean There are 8 numbers in the dataset. The sample mean is the sum of all the numbers divided by the count of numbers. So, the sample mean is $7 / 8 = 0.875$. ## Step 4: Calculate the deviations from the mean Next, we need to find the deviation of each number from the sample mean. This is done by subtracting the sample mean from each number. ## Step 5: Calculate the squared deviations After finding the deviations, we need to square each deviation. This is because the variance is based on squared differences. ## Step 6: Calculate the squared deviations for each data point - For 9: $(9 - 0.875)^2 = (8.125)^2 = 66.015625$ - For 14: $(14 - 0.875)^2 = (13.125)^2 = 172.265625$ - For 5: $(5 - 0.875)^2 = (4.125)^2 = 17.015625$ - For 4: $(4 - 0.875)^2 = (3.125)^2 = 9.765625$ - For -20: $(-20 - 0.875)^2 = (-20.875)^2 = 435.765625$ - For -13: $(-13 - 0.875)^2 = (-13.875)^2 = 192.515625$ - For -5: $(-5 - 0.875)^2 = (-5.875)^2 = 34.515625$ - For 13: $(13 - 0.875)^2 = (12.125)^2 = 147.015625$ ## Step 7: Sum up the squared deviations Now, we add up all the squared deviations: $66.015625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.515625 + 34.515625 + 147.015625 = 1075.078125$ ## Step 8: Calculate the sample variance The sample variance is the sum of the squared deviations divided by the count of numbers minus one (n-1). In this case, it's $1075.078125 / (8 - 1)$. ## Step 9: Calculate the final sample variance $1075.078125 / 7 = 153.582591$ (rounded to six decimal places for the intermediate step, but will be rounded as required for the final answer). The final answer is: $\boxed{153.58}$ |
\frac{8599}{56} |
| 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. The values given are ${9, 14, 5, 4, -20, -13, -5, 13}$. The sum of these values is $9 + 14 + 5 + 4 - 20 - 13 - 5 + 13 = 7$. There are 8 values in total. Therefore, the sample mean is $7 / 8 = 0.875$. ## Step 2: Subtract the sample mean from each value Next, we subtract the sample mean from each value to find the deviations from the mean. The deviations are $(9 - 0.875), (14 - 0.875), (5 - 0.875), (4 - 0.875), (-20 - 0.875), (-13 - 0.875), (-5 - 0.875), (13 - 0.875)$. This simplifies to $8.125, 13.125, 4.125, 3.125, -20.875, -13.875, -5.875, 12.125$. ## Step 3: Square each deviation Now, we square each deviation. The squared deviations are $8.125^2, 13.125^2, 4.125^2, 3.125^2, -20.875^2, -13.875^2, -5.875^2, 12.125^2$. This simplifies to $65.765625, 172.265625, 17.015625, 9.765625, 435.765625, 192.765625, 34.515625, 147.265625$. ## Step 4: Calculate the sum of the squared deviations We then calculate the sum of these squared deviations. The sum is $65.765625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.765625 + 34.515625 + 147.265625 = 1075.5$. ## Step 5: Calculate the sample variance Finally, we calculate the sample variance by dividing the sum of the squared deviations by the number of values minus one (since this is sample variance). The number of values is 8, so we divide by 7. The sample variance is $1075.5 / 7 = 153.643$ (rounded to three decimal places for the intermediate step, but we'll provide the exact fraction for the final answer). ## Step 6: Convert the sample variance to a fraction for the final answer To convert the sample variance to a fraction, we keep it as is because the format requires a boxed number, and the exact fraction would be quite complex. However, following the format strictly means providing the final number in the box without further explanation of its fractional form. The final answer is: $\boxed{153.643}$ |
\frac{8599}{56} |
| Qwen2-72B-Instruct | To compute the sample variance of a set of numbers, we first need to calculate the mean (average) of the numbers, then find the squared differences from the mean for each number, and finally average those squared differences. Given the set of numbers: ${9, 14, 5, 4, -20, -13, -5, 13}$ First, let's calculate the mean: \[ \text{Mean} = \frac{\sum x_i}{n} = \frac{9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13}{8} \] \[ = \frac{2}{8} = 0.25 \] Next, we calculate the squared differences from the mean for each number: - For $x_1 = 9$: $(9 - 0.25)^2 = 72.25$ - For $x_2 = 14$: $(14 - 0.25)^2 = 176.0625$ - For $x_3 = 5$: $(5 - 0.25)^2 = 18.0625$ - For $x_4 = 4$: $(4 - 0.25)^2 = 11.0625$ - For $x_5 = -20$: $(-20 - 0.25)^2 = 441.0625$ - For $x_6 = -13$: $(-13 - 0.25)^2 = 182.0625$ - For $x_7 = -5$: $(-5 - 0.25)^2 = 25.0625$ - For $x_8 = 13$: $(13 - 0.25)^2 = 156.0625$ Now, we sum these squared differences and divide by $n-1$ (since this is a sample variance): \[ \text{Sample Variance} = s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1} \] \[ = \frac{72.25 + 176.0625 + 18.0625 + 11.0625 + 441.0625 + 182.0625 + 25.0625 + 156.0625}{8-1} \] \[ = \frac{1082.625}{7} = 154.66071428571428 \] Rounded to two decimal places, the sample variance is approximately $154.66$. Therefore, the final answer is $\boxed{154.66}$. |
\frac{8599}{56} |
| baichuan4 | To compute the sample variance of a set of numbers, we first need to find the mean (average) of the numbers. Then, for each number, we subtract the mean and square the result. Finally, we sum these squared differences and divide by the number of items minus one (since it's a sample variance) Given Given the set of numbers: ${9, 14, 5, 4, -20, -13, -5, 13}$, let's calculate the sample variance step by step. 1. Calculate the mean: Mean = (9 + 14 + 5 + 4 - 20 - 13 - 5 + 13) / 8 Mean = 0 / 8 Mean = 0 2. For each number, subtract the mean and square the result: (9 - 0)^2 = 81 (14 - 0)^2 = 196 (5 - 0)^2 = 25 (4 - 0)^2 = 16 (-20 - 0)^2 = 400 (-13 - 0)^2 = 169 (-5 - 0)^2 = 25 (13 - 0)^2 = 169 3. Sum these squared differences: 81 + 196 + 25 + 16 + 400 + 169 + 25 + 169 = 1081 4. Divide by the number of items minus one (since it's a sample variance): Sample Variance = 1081 / (8 - 1) Sample Variance = 1081 / 7 Sample Variance = 154.42857142857142 Therefore, the sample variance of the given set of numbers is approximately 154.43. We can put this answer in a box as requested: $ \boxed{154.43} $ |
\frac{8599}{56} |
| doubao | First find the mean: \[ \begin{align*} &\frac{9+14+5+4-20-13-5+13}{8}\\ =&\frac{23+5+4-20-13-5+13}{8}\\ =&\frac{28+4-20-13-5+13}{8}\\ =&\frac{32-20-13-5+13}{8}\\ =&\frac{12-13-5+13}{8}\\ =&\frac{-1-5+13}{8}\\ =&\frac{-6+13}{8}\\ =&\frac{7}{8} \end{align*} \] Then calculate the sum of the squared differences from the mean: \[ \begin{align*} &[(9-\frac{7}{8})^2+(14-\frac{7}{8})^2+(5-\frac{7}{8})^2+(4-\frac{7}{8})^2+(-20-\frac{7}{8})^2+(-13-\frac{7}{8})^2+(-5-\frac{7}{8})^2+(13-\frac{7}{8})^2]\\ =&(\frac{65}{8})^2+(\frac{105}{8})^2+(\frac{33}{8})^2+(\frac{25}{8})^2+(\frac{-167}{8})^2+(\frac{-111}{8})^2+(\frac{-47}{8})^2+(\frac{97}{8})^2\\ =&\frac{4225}{64}+\frac{11025}{64}+\frac{1089}{64}+\frac{625}{64}+\frac{27889}{64}+\frac{12321}{64}+\frac{2209}{64}+\frac{9409}{64}\\ =&\frac{4225+11025+1089+625+27889+12321+2209+9409}{64}\\ =&\frac{66053}{64} \end{align*} \] Finally, divide the sum of the squared differences by the number of items to get the sample variance: \[ \frac{66053}{64}\div8=\frac{66053}{64\times8}=\frac{66053}{512}=\boxed{\frac{66053}{512}} \] |
\frac{8599}{56} |
| gpto1 | To compute the sample variance of the dataset \({9,\ 14,\ 5,\ 4,\ -20,\ -13,\ -5,\ 13}\), we follow these steps: **1. Calculate the Sample Mean (\(\bar{x}\)):** \[ \bar{x} = \frac{\text{sum of all }x_i}{n} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} = \frac{7}{8} = 0.875 \] **2. Compute Each Squared Deviation (\((x_i - \bar{x})^2\)):** For each data point: - Calculate the deviation from the mean (\(x_i - \bar{x}\)). - Square this deviation. | \(x_i\) | \(x_i - \bar{x}\) | \((x_i - \bar{x})^2\) | |---------|--------------------|----------------------------| | 9 | \(9 - 0.875 = 8.125\) | \(8.125^2 = 66.015625\) | | 14 | \(14 - 0.875 = 13.125\) | \(13.125^2 = 172.265625\) | | 5 | \(5 - 0.875 = 4.125\) | \(4.125^2 = 17.015625\) | | 4 | \(4 - 0.875 = 3.125\) | \(3.125^2 = 9.765625\) | | -20 | \(-20 - 0.875 = -20.875\) | \((-20.875)^2 = 435.765625\) | | -13 | \(-13 - 0.875 = -13.875\) | \((-13.875)^2 = 192.515625\) | | -5 | \(-5 - 0.875 = -5.875\) | \((-5.875)^2 = 34.515625\) | | 13 | \(13 - 0.875 = 12.125\) | \(12.125^2 = 147.015625\) | **3. Sum All Squared Deviations:** \[ \sum(x_i - \bar{x})^2 = 66.015625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.515625 + 34.515625 + 147.015625 = 1074.875 \] **4. Calculate the Sample Variance (\(s^2\)):** \[ s^2 = \frac{\sum(x_i - \bar{x})^2}{n - 1} = \frac{1074.875}{7} \] To express the variance as a fraction: - Recognize that \(1074.875 = 1074 + \frac{7}{8}\). - Convert \(1074 + \frac{7}{8}\) to an improper fraction: \[ 1074 + \frac{7}{8} = \frac{(1074 \times 8) + 7}{8} = \frac{8599}{8} \] - Divide by \(n - 1 = 7\): \[ s^2 = \frac{\frac{8599}{8}}{7} = \frac{8599}{8} \times \frac{1}{7} = \frac{8599}{56} \] **Final Answer:** \[ \boxed{\dfrac{8599}{56}} \] |
\frac{8599}{56} |
| kimi | To compute the sample variance of the given data set \(\{9, 14, 5, 4, -20, -13, -5, 13\}\), we will follow these steps: 1. **Calculate the mean (\(\bar{x}\)) of the data set.** 2. **Find the squared differences from the mean for each data point.** 3. **Calculate the average of these squared differences.** ### Step 1: Calculate the mean (\(\bar{x}\)) First, sum all the data points: \[ 9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13 \] Perform the addition: \[ 9 + 14 = 23 \] \[ 23 + 5 = 28 \] \[ 28 + 4 = 32 \] \[ 32 - 20 = 12 \] \[ 12 - 13 = -1 \] \[ -1 - 5 = -6 \] \[ -6 + 13 = 7 \] The sum of the data points is \(7\). Next, divide the sum by the number of data points (which is 8): \[ \bar{x} = \frac{7}{8} \] ### Step 2: Find the squared differences from the mean For each data point \(x_i\), calculate \((x_i - \bar{x})^2\): \[ (9 - \frac{7}{8})^2 = \left(\frac{72}{8} - \frac{7}{8}\right)^2 = \left(\frac{65}{8}\right)^2 = \frac{4225}{64} \] \[ (14 - \frac{7}{8})^2 = \left(\frac{112}{8} - \frac{7}{8}\right)^2 = \left(\frac{105}{8}\right)^2 = \frac{11025}{64} \] \[ (5 - \frac{7}{8})^2 = \left(\frac{40}{8} - \frac{7}{8}\right)^2 = \left(\frac{33}{8}\right)^2 = \frac{1089}{64} \] \[ (4 - \frac{7}{8})^2 = \left(\frac{32}{8} - \frac{7}{8}\right)^2 = \left(\frac{25}{8}\right)^2 = \frac{625}{64} \] \[ (-20 - \frac{7}{8})^2 = \left(-\frac{160}{8} - \frac{7}{8}\right)^2 = \left(-\frac{167}{8}\right)^2 = \frac{27889}{64} \] \[ (-13 - \frac{7}{8})^2 = \left(-\frac{104}{8} - \frac{7}{8}\right)^2 = \left(-\frac{111}{8}\right)^2 = \frac{12321}{64} \] \[ (-5 - \frac{7}{8})^2 = \left(-\frac{40}{8} - \frac{7}{8}\right)^2 = \left(-\frac{47}{8}\right)^2 = \frac{2209}{64} \] \[ (13 - \frac{7}{8})^2 = \left(\frac{104}{8} - \frac{7}{8}\right)^2 = \left(\frac{97}{8}\right)^2 = \frac{9409}{64} \] ### Step 3: Calculate the average of these squared differences Sum all the squared differences: \[ \frac{4225}{64} + \frac{11025}{64} + \frac{1089}{64} + \frac{625}{64} + \frac{27889}{64} + \frac{12321}{64} + \frac{2209}{64} + \frac{9409}{64} \] Combine the fractions: \[ \frac{4225 + 11025 + 1089 + 625 + 27889 + 12321 + 2209 + 9409}{64} \] Sum the numerators: \[ 4225 + 11025 = 15250 \] \[ 15250 + 1089 = 16339 \] \[ 16339 + |
\frac{8599}{56} |
| minimax | To compute the sample variance of the given data set ${9, 14, 5, 4, -20, -13, -5, 13}$, we will follow these steps: 1. Calculate 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 all the squared deviations. 5. Divide the sum by $n - 1$, where $n$ is the number of data points in the set. Let's go through these steps: 1. Calculate the mean: $$ \text{mean} = \frac{9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13}{8} = \frac{5}{8} = 0.625 $$ 2. Calculate the deviations from the mean and square them: $$ (9 - 0.625)^2 = 76.515625 $$ $$ (14 - 0.625)^2 = 185.515625 $$ $$ |
\frac{8599}{56} |
| 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 $\bar{x}$ 是所有数据之和除以数据的个数。 $\bar{x} = \frac{9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13}{8}$ $\bar{x} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8}$ $\bar{x} = \frac{2}{8}$ $\bar{x} = \frac{1}{4}$ 接下来，我们计算每个数据与平均值的差的平方。 $(9 - \frac{1}{4})^2 = (\frac{36}{4} - \frac{1}{4})^2 = (\frac{35}{4})^2$ $(14 - \frac{1}{4})^2 = (\frac{56}{4} - \frac{1}{4})^2 = (\frac{55}{4})^2$ $(5 - \frac{1}{4})^2 = (\frac{20}{4} - \frac{1}{4})^2 = (\frac{19}{4})^2$ $(4 - \frac{1}{4})^2 = (\frac{16}{4} - \frac{1}{4})^2 = (\frac{15}{4})^2$ $(-20 - \frac{1}{4})^2 = (-\frac{80}{4} - \frac{1}{4})^2 = (-\frac{81}{4})^2$ $(-13 - \frac{1}{4})^2 = (-\frac{52}{4} - \frac{1}{4})^2 = (-\frac{53}{4})^2$ $(-5 - \frac{1}{4})^2 = (-\frac{20}{4} - \frac{1}{4})^2 = (-\frac{21}{4})^2$ $(13 - \frac{1}{4})^2 = (\frac{52}{4} - \frac{1}{4})^2 = (\frac{51}{4})^2$ 现在，我们计算这些平方差的平均值，即样本方差 $s^2$。 $s^2 = \frac{1}{8} \left[ (\frac{35}{4})^2 + (\frac{55}{4})^2 + (\frac{19}{4})^2 + (\frac{15}{4})^2 + (-\frac{81}{4})^2 + (-\frac{53}{4})^2 + (-\frac{21}{4})^2 + (\frac{51}{4})^2 \right]$ $s^2 = \frac{1}{8} \left[ \frac{1225}{16} + \frac{3025}{16} + \frac{361}{16} + \frac{225}{16} + \frac{6561}{16} + \frac{2809}{16} + \frac{441}{16} + \frac{2601}{16} \right]$ $s^2 = \frac{1}{8} \times \frac{17248}{16}$ $s^2 = \frac{17248}{128}$ $s^2 = \frac{134.75}{8}$ $s^2 = 16.84375$ 所以，样本方差是 $\boxed{16.84375}$。 |
\frac{8599}{56} |
| zhipu | To compute the sample variance of the data set \(\{9, 14, 5, 4, -20, -13, -5, 13\}\), follow these steps: 1. **Calculate the sample mean (\(\bar{x}\))**: \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(n\) is the number of data points. Here, \(n = 8\): \[ \bar{x} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} = \frac{9 + 14 + 5 + 4 - 20 - 13 - 5 + 13}{8} = \frac{9}{8} = 1.125 \] 2. **Calculate the squared deviations from the mean**: For each data point \(x_i\), compute \((x_i - \bar{x})^2\): \[ (9 - 1.125)^2 = (7.875)^2 = 61.640625 \] \[ (14 - 1.125)^2 = (12.875)^2 = 165.890625 \] \[ (5 - 1.125)^2 = (3.875)^2 = 15.015625 \] \[ (4 - 1.125)^2 = (2.875)^2 = 8.265625 \] \[ (-20 - 1.125)^2 = (-21.125)^2 = 445.765625 \] \[ (-13 - 1.125)^2 = (-14.125)^2 = 199.765625 \] \[ (-5 - 1.125)^2 = (-6.125)^2 = 37.515625 \] \[ (13 - 1.125)^2 = (11.875)^2 = 141.609375 \] 3. **Sum the squared deviations**: \[ \sum_{i=1}^{n} (x_i - \bar{x})^2 = 61.640625 + 165.890625 + 15.015625 + 8.265625 + 445.765625 + 199.765625 + 37.515625 + 141.609375 = 1075.60625 \] 4. **Calculate the sample variance (\(s^2\))**: \[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} \] Here, \(n - 1 = 8 - 1 = 7\): \[ s^2 = \frac{1075.60625}{7} \approx 153.658036 \] Thus, the sample variance of the data set is approximately \(153.658\). \[ \boxed{153.658} \] |
\frac{8599}{56} |
| deepseek-v3 | To compute the **sample variance** of the dataset \(\{9, 14, 5, 4, -20, -13, -5, 13\}\), follow these steps: ### 1. **Calculate the Mean (\(\bar{x}\))** First, find the average of the data points. \[ \bar{x} = \frac{9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13}{8} \] \[ \bar{x} = \frac{7}{8} = 0.875 \] ### 2. **Compute Each Deviation from the Mean** Subtract the mean from each data point. \[ \begin{align*} 9 - 0.875 &= 8.125 \\ 14 - 0.875 &= 13.125 \\ 5 - 0.875 &= 4.125 \\ 4 - 0.875 &= 3.125 \\ -20 - 0.875 &= -20.875 \\ -13 - 0.875 &= -13.875 \\ -5 - 0.875 &= -5.875 \\ 13 - 0.875 &= 12.125 \\ \end{align*} \] ### 3. **Square Each Deviation** Square the result of each deviation. \[ \begin{align*} 8.125^2 &= 66.015625 \\ 13.125^2 &= 172.265625 \\ 4.125^2 &= 17.015625 \\ 3.125^2 &= 9.765625 \\ (-20.875)^2 &= 435.765625 \\ (-13.875)^2 &= 192.515625 \\ (-5.875)^2 &= 34.515625 \\ 12.125^2 &= 147.015625 \\ \end{align*} \] ### 4. **Sum the Squared Deviations** Add up all the squared deviations. \[ 66.015625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.515625 + 34.515625 + 147.015625 = 1074.875 \] ### 5. **Calculate the Sample Variance** Since this is a sample (not the entire population), we divide by \(n - 1\), where \(n\) is the number of data points. \[ s^2 = \frac{1074.875}{8 - 1} = \frac{1074.875}{7} \approx 153.55357142857 \] Rounding to two decimal places: \[ s^2 \approx 153.55 \] However, for precise calculations, we'll keep the fraction: \[ s^2 = \frac{1074.875}{7} = \frac{8599}{56} \approx 153.5536 \] But to present it neatly: \[ s^2 = \frac{8599}{56} \] However, considering the initial sum: \[ \text{Sum of squared deviations} = 1074.875 = \frac{8599}{8} \] Thus, \[ s^2 = \frac{8599}{8 \times 7} = \frac{8599}{56} \] But to simplify: \[ s^2 = \frac{1074.875}{7} = 153.55357142857 \] For the final answer, we'll use the decimal form: \[ \boxed{153.55} \] However, if we prefer the fractional form: \[ \boxed{\frac{8599}{56}} \] But typically, sample variance is presented in decimal form rounded to two decimal places. Therefore: \[ \boxed{153.55} \] **Final Answer:** \[ \boxed{153.55} \] |
\frac{8599}{56} |