To calculate the variance of a dataset, we follow these steps:
- Calculate the mean (average) of the data.
- Subtract the mean from each data point and square the result (the squared difference).
- Calculate the average of these squared differences.
Let's start with the data you provided:
217, 235, 212, 253, 233, 171, 184, 301,
256, 297, 247, 226, 237, 241, 195, 204,
197, 214, 166, 309, 281, 303, 309, 299
Step 1: Calculate the Mean
To calculate the mean:
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \]
Sum of the data points: \[ 217 + 235 + 212 + 253 + 233 + 171 + 184 + 301 + 256 + 297 + 247 + 226 + 237 + 241 + 195 + 204 + 197 + 214 + 166 + 309 + 281 + 303 + 309 + 299 = 5728 \]
Number of data points: There are 24 data points.
\[ \text{Mean} = \frac{5728}{24} \approx 238.67 \]
Step 2: Calculate Squared Differences
Now we will calculate the squared difference from the mean for each data point.
\[ \text{Squared Differences} = (x_i - \text{Mean})^2 \]
Calculating for each \(x_i\):
\[ \begin{align*} (217 - 238.67)^2 & \approx 475.11 \ (235 - 238.67)^2 & \approx 13.49 \ (212 - 238.67)^2 & \approx 706.68 \ (253 - 238.67)^2 & \approx 201.11 \ (233 - 238.67)^2 & \approx 32.11 \ (171 - 238.67)^2 & \approx 4471.51 \ (184 - 238.67)^2 & \approx 3033.11 \ (301 - 238.67)^2 & \approx 3845.11 \ (256 - 238.67)^2 & \approx 298.78 \ (297 - 238.67)^2 & \approx 3397.31 \ (247 - 238.67)^2 & \approx 68.78 \ (226 - 238.67)^2 & \approx 160.11 \ (237 - 238.67)^2 & \approx 2.78 \ (241 - 238.67)^2 & \approx 5.56 \ (195 - 238.67)^2 & \approx 1917.78 \ (204 - 238.67)^2 & \approx 1219.11 \ (197 - 238.67)^2 & \approx 1737.78 \ (214 - 238.67)^2 & \approx 612.11 \ (166 - 238.67)^2 & \approx 5152.11 \ (309 - 238.67)^2 & \approx 4940.11 \ (281 - 238.67)^2 & \approx 1768.11 \ (303 - 238.67)^2 & \approx 4143.11 \ (309 - 238.67)^2 & \approx 4940.11 \ (299 - 238.67)^2 & \approx 3640.11 \ \end{align*} \]
Step 3: Calculate the Variance
Sum of squared differences: \[ 475.11 + 13.49 + 706.68 + 201.11 + 32.11 + 4471.51 + 3033.11 + 3845.11 + 298.78 + 3397.31 + 68.78 + 160.11 + 2.78 + 5.56 + 1917.78 + 1219.11 + 1737.78 + 612.11 + 5152.11 + 4940.11 + 1768.11 + 4143.11 + 4940.11 + 3640.11 \approx 23272.9 \]
Variance (Population): \[ \text{Variance} = \frac{\text{Sum of Squared Differences}}{N} = \frac{23272.9}{24} \approx 969.70 \]
Variance (Sample): \[ \text{Sample Variance} = \frac{\text{Sum of Squared Differences}}{N-1} = \frac{23272.9}{23} \approx 1011.46 \]
Final Result
The population variance of the data is approximately 969.70, and the sample variance is approximately 1011.46. Depending on your requirement (whether you need sample or population variance), you can consider the appropriate value.