Calculate The Variance of The Following Sample 1 5 9

Variance is a measure of how spread out the numbers in a data set are. In this guide, we'll show you how to calculate the variance of the sample 1, 5, 9 using our free online calculator.

What is variance?

Variance is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low variance indicates that the data points tend to be close to the mean (average) of the set, while a high variance indicates that the data points are spread out over a wider range.

Variance is calculated as the average of the squared differences from the mean. The square root of the variance is known as the standard deviation, which is often more intuitive to interpret.

How to calculate variance

To calculate the variance of a sample, follow these steps:

Find the mean (average) of the numbers.
For each number, subtract the mean and square the result.
Find the average of these squared differences.

Variance formula

For a sample of size n:

s² = Σ(xᵢ - x̄)² / (n - 1)

Where:

s² = sample variance
xᵢ = each individual value in the sample
x̄ = sample mean
n = number of values in the sample

Note: For population variance, the denominator is n instead of n-1. This calculator uses the sample variance formula.

Example calculation

Let's calculate the variance of the sample 1, 5, 9 step by step.

Step 1: Find the mean

Mean = (1 + 5 + 9) / 3 = 15 / 3 = 5

Step 2: Calculate squared differences from the mean

Value (xᵢ)	Difference from mean (xᵢ - x̄)	Squared difference (xᵢ - x̄)²
1	1 - 5 = -4	(-4)² = 16
5	5 - 5 = 0	0² = 0
9	9 - 5 = 4	4² = 16

Step 3: Calculate the variance

Sum of squared differences = 16 + 0 + 16 = 32

Variance = 32 / (3 - 1) = 32 / 2 = 16

The variance of the sample 1, 5, 9 is 16. The standard deviation would be √16 = 4.

Interpretation

A variance of 16 means that, on average, the numbers in the sample are 16 units squared away from the mean of 5. This indicates that the data points are somewhat spread out from the mean.

To put this in perspective:

The smallest value (1) is 4 units below the mean
The largest value (9) is 4 units above the mean
The middle value (5) is exactly at the mean

This shows that the data is not tightly clustered around the mean, which is reflected in the relatively high variance.

FAQ

What is the difference between variance and standard deviation?: Variance measures the spread of data in squared units, while standard deviation is the square root of variance and measures spread in the same units as the original data.
When would you use variance instead of standard deviation?: Variance is often used in mathematical calculations and statistical models because it preserves the units of measurement when combined with other quantities.
Is sample variance different from population variance?: Yes, sample variance uses n-1 in the denominator to correct for bias when estimating population variance from a sample.
What does a high variance mean?: A high variance indicates that the data points are spread out over a wider range, while a low variance indicates that the data points are clustered more closely around the mean.
Can variance be negative?: No, variance is always a non-negative value because it's based on squared differences.