Short Answer 
To calculate the variance of a data set, start by finding the mean by summing all numbers and dividing by the count of data points, yielding a mean of 8. Then, calculate the squared differences from the mean, which sum to 194, and finally divide this by the number of data points minus one to obtain a variance of approximately 27.71.
Step 1: Calculate the Mean
Begin by calculating the mean of the data set. To do this, add all the numbers together and then divide by the total number of data points. For the data set 5, 12, 3, 18, 6, 8, 2, 10:
- Add all the numbers: 5 + 12 + 3 + 18 + 6 + 8 + 2 + 10 = 64
- Divide the sum by the number of data points (8): 64 / 8 = 8
This gives you a mean (average), x, of 8.
Step 2: Calculate Squared Differences from the Mean
Next, you need to find the squared differences between each data point and the mean. Subtract the mean from each data point and then square the result:
- (5 – 8)2 = 9
- (12 – 8)2 = 16
- (3 – 8)2 = 25
- (18 – 8)2 = 100
- (6 – 8)2 = 4
- (8 – 8)2 = 0
- (2 – 8)2 = 36
- (10 – 8)2 = 4
Now sum all the squared differences: 9 + 16 + 25 + 100 + 4 + 0 + 36 + 4 = 194.
Step 3: Calculate the Variance
Finally, to find the variance, divide the total squared differences by the number of data points minus one (n-1) since this is treated as a sample variance:
- Sum of squared differences = 194
- n = 8, so n – 1 = 7
- Calculate variance: 194 / 7 ≈ 27.71
Thus, the variance of this ungrouped data set is approximately 27.71.