Introduction

Sample variance is a crucial statistical concept used to quantify the dispersion or variation within a data set. It helps in determining how much the individual data points differ from the sample mean, thus highlighting the consistency and reliability of the data. This article will guide you on how to calculate sample variance using a step-by-step process.

Step 1: Understanding the Sample Variance Formula

The formula for calculating sample variance is as follows:

Sample Variance (s^2) = Σ (x_i – x̄)^2 / (n – 1)

Where:

– s^2 represents the sample variance

– Σ denotes the sum of a series of values

– x_i represents each value in the dataset

– x̄ indicates the mean of the dataset

– n is the number of data points in the dataset, and

– (n – 1) is known as degrees of freedom

Step 2: Gather Your Data

To begin, collect your dataset and make sure it comprises at least two data points. If you have only one data point, it is impossible to calculate variance because there won’t be any dispersion.

Step 3: Calculate the Sample Mean

Begin by adding up all of your collected values. Next, divide this sum by the total number of values in your dataset. This gives you the sample mean (x̄).

For example, let’s say we have six numbers in our dataset (4, 7, 5, 9, 6, and 8):

Sample Mean

x̄ = (4 + 7 + 5 + 9 + 6 + 8) / 6

x̄ = (39) / 6

x̄ = 6.5

Step 4: Find the Deviations from the Mean

Now, we need to calculate the deviations of individual data points from the sample mean. These differences are obtained by subtracting the mean from each data point:

Deviations from Mean = (x_i – x̄)

Deviations = (4-6.5, 7-6.5, 5-6.5, 9-6.5, 6-6.5, 8-6.5)

Step 5: Square the Deviations

To ensure that negative values do not cancel out positive ones, square each deviation value:

Squared Deviations = ((4-6.5)^2, (7-6.5)^2, (5-6.5)^2, (9-6.5)^2, (6-6.5)^2, (8-6.5)^2)

Step 6: Sum the Squared Deviations

Next, find the sum of all squared deviations:

Sum of Squared Deviations = Σ(x_i – x̄)^2

In our example,

Sum of Squared Deviations = 17 + 0.25 + 2.25 + 6.25 + 0.25 + 2.25 = 28

Step 7: Divide the Sum by Degrees of Freedom

Finally, divide the result from Step 6 by the degrees of freedom—(n – 1). This yields the sample variance:

Sample Variance(s^2) = Σ(x_i – x̄)^2 / (n – 1)

In our example,

Sample Variance(s^2) = Sum of Squared Deviations/(n -1)

s^2 = (28)/(6 -1)

s^2 = 28/5

s^2 =