To calculate the variance and standard deviation of the data set you've provided, we will follow these steps:
- Calculate the mean (average) of the data.
- Calculate the variance.
- Calculate the standard deviation.
Step 1: Calculate the Mean
First, we need to add all the numbers together. You've already provided the total sum of the data:
\[ \text{Total Sum} = 218 + 234 + 210 + 261 + 265 + 175 + 191 + 308 + 253 + 295 + 253 + 220 + 234 + 232 + 189 + 208 + 190 + 202 + 159 + 304 + 275 + 305 + 308 + 295 = 5784 \]
The number of data points (n) is 24.
The mean (μ) is calculated as follows:
\[ \text{Mean} = \frac{\text{Total Sum}}{n} = \frac{5784}{24} \approx 241 \]
Step 2: Calculate the Variance
Variance is calculated by taking the average of the squared differences from the Mean. Here’s the formula:
\[ \text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n} \]
Where \(x_i\) is each value in the data set.
Let's calculate the squared differences:
-
Calculate each squared difference from the mean:
- \((218 - 241)^2 = 529\)
- \((234 - 241)^2 = 49\)
- \((210 - 241)^2 = 961\)
- \((261 - 241)^2 = 400\)
- \((265 - 241)^2 = 576\)
- \((175 - 241)^2 = 4356\)
- \((191 - 241)^2 = 2500\)
- \((308 - 241)^2 = 4489\)
- \((253 - 241)^2 = 144\)
- \((295 - 241)^2 = 2916\)
- \((253 - 241)^2 = 144\)
- \((220 - 241)^2 = 441\)
- \((234 - 241)^2 = 49\)
- \((232 - 241)^2 = 81\)
- \((189 - 241)^2 = 2704\)
- \((208 - 241)^2 = 1089\)
- \((190 - 241)^2 = 2601\)
- \((202 - 241)^2 = 1521\)
- \((159 - 241)^2 = 6724\)
- \((304 - 241)^2 = 3969\)
- \((275 - 241)^2 = 1156\)
- \((305 - 241)^2 = 4096\)
- \((308 - 241)^2 = 4489\)
- \((295 - 241)^2 = 2916\)
-
Now sum all the squared differences:
\[ 529 + 49 + 961 + 400 + 576 + 4356 + 2500 + 4489 + 144 + 2916 + 144 + 441 + 49 + 81 + 2704 + 1089 + 2601 + 1521 + 6724 + 3969 + 1156 + 4096 + 4489 + 2916 = 25290 \]
- Now calculate the variance:
\[ \text{Variance} = \frac{25290}{24} = 1053.75 \]
Step 3: Calculate Standard Deviation
Standard deviation is the square root of the variance:
\[ \text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}} = \sqrt{1053.75} \approx 32.5 \]
Final Results
- Variance: \(1053.75\)
- Standard Deviation: Approximately \(32.5\)