Table of Content

• What is Variance?
• What is Sample Variance?
• Formula for Sample Variance
• What is Population Variance?
• Formula for Population Variance
• Differences Between Sample and Population Variance
• When to Use Sample Variance vs Population Variance
• Conclusion
• FAQs on Sample Variance and Population Variance

Aspect	Sample Variance ( s2 )	Population Variance ( σ2 )
Definition	Measure of dispersion in a sample	Measure of dispersion in an entire population
Formula	[Tex]s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i \bar{x})^2[/Tex]	[Tex]\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i \mu)^2[/Tex]
Mean Used	Sample mean ([Tex]\bar{x}[/Tex])	Population mean (μ)
Denominator	n − 1 (degrees of freedom)	N (total number of data points)
Purpose	Estimates the population variance from a sample	Measures the true variance of the population
Bias Adjustment	Dividing by n − 1 corrects the bias in estimation	No bias adjustment needed, uses entire population data
Usage	When only a subset of the population is available	When the entire population data is available
Data Set	Sample data (a subset of the population)	Population data (all data points in the population)

When to Use Sample Variance vs Population Variance

Use sample variance when you are working with a subset (sample) of the entire population and you want to estimate the variability within the population. We can use sample variance for following scenarios:

• Survey Research: When you survey a sample of people to infer the behavior or characteristics of the entire population.
• Experimental Studies: When you conduct an experiment on a sample to generalize the results to the larger population.
• Quality Control: When you inspect a sample of products to determine the quality of a large batch.
• Biological Studies: When you study a sample of individuals from a species to understand the traits of the species as a whole.

Use population variance when you have access to data for the entire population, and you want to measure the true variability within that population. We can use population variance for following scenarios:

• Census Data: When analyzing data from a national census that includes every individual in the population.
• Complete Data Sets: When you have a complete set of data points for the population of interest, such as all students in a particular school.
• Controlled Experiments: In certain controlled experiments where every member of the target population is included.

Conclusion

In conclusion, sample variance is used to estimate the variability within a population based on a subset of data, while population variance measures the true variability using all data points. Use sample variance when you have limited data, and population variance when complete data is available. Understanding when to apply each ensures accurate statistical analysis and meaningful insights from your data.

Read More,

• Bernoulli Trials
• Mean
• Probability Mass Function
• Variance and Standard Deviation
• Skewness

FAQs on Sample Variance and Population Variance

Define sample variance.

Sample variance is a measure of how spread out the data points are in a sample. It estimates the variability of a population based on a subset of data.

Define population variance.

Population variance measures the true spread of all data points in an entire population. It calculates the average of the squared differences from the population mean.

When should I use sample variance?

Use sample variance when you have a subset of data from a larger population and you want to estimate the population’s variability. It is commonly used when collecting data from the entire population is impractical.

When should I use population variance?

Use population variance when you have data for the entire population and want to measure the true variability without making estimates.

Why does sample variance use n−1 in the formula?

Sample variance uses n−1 (degrees of freedom) to correct for bias in the estimation of population variance. This adjustment accounts for the fact that a sample mean is used instead of the true population mean.

How do I calculate sample variance?

Calculate the sample mean, find the squared differences from the mean for each data point, sum these squared differences, and divide by n−1n-1n−1.