A sample variance is a measure of the spread of data points around the sample mean, and can be illustrated with a specific example.
Calculating Sample Variance
Here's how the sample variance is calculated and an example using provided data:
Example Data Set:
Let's use the data set: 3, 21, 98, 17, and 9 from the reference.
Step 1: Calculate the Sample Mean
First, find the mean (average) of the data points.
- Sum of data points: 3 + 21 + 98 + 17 + 9 = 148
- Number of data points (n): 5
- Sample mean (x̄) = 148 / 5 = 29.6
Step 2: Calculate Deviations from the Mean
Next, subtract the sample mean from each data point.
- 3 - 29.6 = -26.6
- 21 - 29.6 = -8.6
- 98 - 29.6 = 68.4
- 17 - 29.6 = -12.6
- 9 - 29.6 = -20.6
Step 3: Square the Deviations
Now, square each of these deviations.
- (-26.6)^2 = 707.56
- (-8.6)^2 = 73.96
- (68.4)^2 = 4678.56
- (-12.6)^2 = 158.76
- (-20.6)^2 = 424.36
Step 4: Sum of the Squared Deviations
Add all the squared deviations together:
- 707.56 + 73.96 + 4678.56 + 158.76 + 424.36 = 6043.2
Step 5: Divide by (n-1)
Finally, divide the sum of the squared deviations by (n-1), where n is the number of data points. In this case, n=5 so (n-1) = 4
- Sample variance (s^2) = 6043.2 / (5 - 1) = 6043.2 / 4 = 1510.8
Result
The calculated sample variance for this dataset is 1510.8
Understanding Sample Variance
The sample variance is an essential concept in statistics. It helps to determine how spread out data is, using the sample mean as a reference. A higher sample variance means that the data points are more scattered around the mean, while a lower variance indicates that the data points are closer to the mean. Sample variance is used for sample data, whereas population variance is used for the population.
