The sample variance is a measure of the spread of data points in a sample, while the population variance is a measure of the spread of data points in an entire population.
Sample variance is calculated by taking the sum of the squared differences between each data point and the sample mean, dividing by the number of data points minus one, and then taking the square root of the result. Population variance is calculated in a similar way, but the sum of squared differences is divided by the total number of data points in the population.
In general, sample variance tends to underestimate the true variance of a population, especially for small sample sizes. This is because sample variance is calculated using only a subset of the data, rather than the entire population.
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Google Gemini AI
Jun 19
Sample Variance
- Calculated from a sample of observations from a population. - Estimates the population variance based on the available data. - Represented by the symbol s². - Uses the (n-1) degrees of freedom, where n is the sample size.
Population Variance
- Calculates the variance of all observations in a population. - Unknown in most practical situations, as it requires data from the entire population. - Represented by the symbol σ². - Uses the (N-1) degrees of freedom, where N is the population size.
Key Differences:
- Source: Sample variance is estimated from a sample, while population variance refers to the variance of the entire population. - Estimation: Sample variance estimates population variance with an unknown degree of accuracy. Population variance provides an exact measure of variance. - Degrees of Freedom: Sample variance uses (n-1) degrees of freedom, while population variance uses (N-1) degrees of freedom. - Accuracy: Sample variance is an unbiased estimator of population variance, meaning that in the long run, the average of sample variances will approach population variance. However, any individual sample variance may differ from the population variance.
Relationship:
The sample variance is an unbiased estimator of the population variance, meaning that the expected value of the sample variance is equal to the population variance. However, the sample variance is not necessarily equal to the population variance due to sampling error.
Usage:
Sample variance is used in statistical inference to:
- Construct confidence intervals for population variance - Test hypotheses about population variance - Compare the variances of different populations
Population variance is rarely used directly, as it is typically unknown. However, it is a theoretical concept that is important for understanding statistical theory and for developing statistical methods.