Calculating Sample Variance and Standard Deviation
To calculate the sample variance and standard deviation for the given data set: 18, 21, 32, 41, 1, follow these steps:
Step 1: Find the Mean
The mean (μ) is calculated by adding all the numbers and dividing by the total count of numbers.
Mean (μ) = (18 + 21 + 32 + 41 + 1) / 5
= 113 / 5
= 22.6
Step 2: Calculate the Variance
The sample variance (s²) is calculated using the following formula:
s² = Σ(xi - μ)² / (n - 1)
Where:
- xi is each value in the data set
- μ is the mean
- n is the number of values
Calculating Each (xi – μ)²
- (18 – 22.6)² = (-4.6)² = 21.16
- (21 – 22.6)² = (-1.6)² = 2.56
- (32 – 22.6)² = (9.4)² = 88.36
- (41 – 22.6)² = (18.4)² = 338.56
- (1 – 22.6)² = (-21.6)² = 466.56
Summing Them Up
Σ(xi - μ)² = 21.16 + 2.56 + 88.36 + 338.56 + 466.56 = 917.20
Calculating Variance
s² = 917.20 / (5 - 1) = 917.20 / 4 = 229.30
Step 3: Calculate the Standard Deviation
The sample standard deviation (s) is simply the square root of the sample variance:
s = √s² = √229.30 = 15.13
Final Results
The sample variance of the data set is 229.30, and the sample standard deviation is 15.13.