Calculate The Variance of The Following Sample 2 5 8

Variance is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A high variance indicates that the data points are spread out over a wide range, while a low variance indicates that the data points are clustered closely around the mean.

What is Variance?

Variance is a fundamental concept in statistics that measures how far each number in a dataset is from the mean (average) of the dataset. It provides insight into the spread of the data points and is widely used in various fields including finance, quality control, and social sciences.

There are two main types of variance:

Population Variance: Measures the spread of all values in an entire population.
Sample Variance: Estimates the spread of values in a sample drawn from a larger population.

For this calculation, we'll focus on sample variance, which is more commonly used when working with a subset of data.

How to Calculate Variance

The formula for sample variance (s²) is:

Sample Variance Formula

s² = Σ (xᵢ - x̄)² / (n - 1)

Where:

xᵢ = each individual data point
x̄ = the sample mean
n = number of data points

To calculate variance:

Find the mean (average) of the data set.
For each data point, subtract the mean and square the result.
Sum all the squared differences.
Divide the sum by (n - 1) where n is the number of data points.

Key Notes

We divide by (n - 1) in sample variance to get an unbiased estimate of the population variance. This adjustment accounts for the fact that we're working with a sample rather than the entire population.

Step-by-Step Example

Let's calculate the variance for the sample data: 2, 5, 8.

Find the mean:
(2 + 5 + 8) / 3 = 15 / 3 = 5
Calculate each squared difference from the mean:
- (2 - 5)² = (-3)² = 9
- (5 - 5)² = 0² = 0
- (8 - 5)² = 3² = 9
Sum the squared differences:
9 + 0 + 9 = 18
Divide by (n - 1):
18 / (3 - 1) = 18 / 2 = 9

The variance of the sample 2, 5, 8 is 9.

Interpreting Variance

A variance of 9 means that, on average, the data points in the sample are 9 units squared away from the mean. To get a more intuitive understanding, you can take the square root of the variance to get the standard deviation:

√9 = 3

This means the data points typically deviate by about 3 units from the mean of 5.

In practical terms, this indicates that the sample data has a moderate amount of variability. Values are somewhat spread out around the mean, but not extremely so.

Frequently Asked Questions

What is the difference between variance and standard deviation?

Variance measures the average of the squared differences from the mean, while standard deviation is the square root of the variance. Standard deviation is often preferred because it's in the same units as the original data, making it more interpretable.

Why do we divide by (n - 1) in sample variance?

Dividing by (n - 1) rather than n provides an unbiased estimate of the population variance. This adjustment accounts for the fact that we're working with a sample rather than the entire population.

How does variance differ from range?

Range is simply the difference between the maximum and minimum values in a dataset, while variance considers all data points and their distances from the mean. Variance provides a more comprehensive measure of data spread.