Chegg Calculate The Variance of A Sample for Which N

Calculating the variance of a sample is a fundamental statistical operation used to measure the spread of data points around the mean. This guide explains how to calculate sample variance when you have N observations, provides a step-by-step calculator, and offers practical interpretation of the results.

What is Sample Variance?

Sample variance is a measure of how far each number in a sample is from the mean of the sample. It quantifies the dispersion of data points around the sample mean. A higher variance indicates that the data points are more spread out, while a lower variance indicates that the data points are closer to the mean.

Variance is particularly useful in statistics for understanding the consistency of data. In research and quality control, it helps identify whether processes are stable or if there's significant variation that needs investigation.

How to Calculate Sample Variance

To calculate the sample variance, follow these steps:

Collect your sample data points
Calculate the sample mean (average)
For each data point, subtract the mean and square the result
Sum all these squared differences
Divide the sum by (n-1) where n is the number of observations

The division by (n-1) rather than n is called Bessel's correction and provides an unbiased estimate of the population variance.

The Formula

Sample Variance Formula

The formula for sample variance (s²) is:

s² = Σ(xᵢ - x̄)² / (n - 1)

Where:

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

This formula calculates the average of the squared differences from the mean, providing a measure of data spread.

Worked Example

Let's calculate the sample variance for the following data set: 4, 7, 13, 16.

Calculate the mean: (4 + 7 + 13 + 16) / 4 = 40 / 4 = 10
Calculate each squared difference:
- (4 - 10)² = 36
- (7 - 10)² = 9
- (13 - 10)² = 9
- (16 - 10)² = 36
Sum the squared differences: 36 + 9 + 9 + 36 = 90
Divide by (n-1): 90 / (4-1) = 30

The sample variance is 30. This means the data points are, on average, 30 units squared from the mean.

Interpreting the Result

The sample variance provides several important insights:

It measures the spread of your data points around the mean
A higher variance indicates more variability in your data
A lower variance indicates more consistent data points
It's particularly useful for comparing different data sets

When interpreting variance, it's important to consider the units of your data. Since variance is squared, it's often more intuitive to look at the standard deviation (the square root of variance) which is in the same units as your original data.

FAQ

Why do we use (n-1) instead of n in the sample variance formula?

Using (n-1) provides an unbiased estimate of the population variance. This adjustment accounts for the fact that we're estimating the population variance from a sample, not calculating it from the entire population.

What's the difference between variance and standard deviation?

Variance measures the spread of data points in squared units, while standard deviation is the square root of variance and is in the same units as the original data. Standard deviation is often more interpretable for many applications.

When would I use sample variance instead of population variance?

Use sample variance when you're analyzing a subset of a larger population. Population variance is used when you have data for the entire population, which is less common in practice.