Why for sample variance is divided by n-1?? ??

??Understanding Variance, Standard Deviation, Population, Sample, and the Importance of Dividing by (n-1) in sample variance

What is Variance?

Variance measures how far a set of numbers are spread out from their mean. It quantifies data dispersion but is expressed in squared units, making it less interpretable.

What is Standard Deviation?

The standard deviation is the square root of the variance. It provides a measure of data dispersion while being in the same units as the original dataset, making it easier to interpret.

??Population vs. Sample in the Context of Variance and Standard Deviation:

?Population:

The entire set of data or all possible observations that could be studied. For example, if studying the heights of all adult men in a country, the population includes every adult male’s height.

Sample:

?A subset of the population selected for analysis. Since studying an entire population is often impractical, researchers analyze a sample to make inferences about the population.

Why is Sample Variance Divided by (n-1)?

When calculating the variance of a sample, we divide by (n-1) instead of n. This adjustment, known as Bessel’s correction, ensures an unbiased estimate of the population variance.

??Understanding the Need for (n-1):

Dataset Well distributed:

Sampling from a Well-Distributed Population?:

When randomly selecting a sample from a population, the sample mean (x?) is usually close to the population mean (μ), making it a good estimate. Sampling from a Skewed Population:

If the sample is not representative and comes from a specific cluster within the population, the sample mean (x?) may be significantly different from (μ), leading to an underestimated variance.

Correction for Bias:

Since a sample tends to underestimate the true population variance, dividing by (n-1) instead of n inflates the variance slightly, compensating for this bias. This adjustment ensures that the sample variance provides a better estimate of the true population variance.

Conclusion

The use of (n-1) in sample variance calculations corrects for the natural bias that occurs when estimating population variance from a sample. By making this adjustment, we ensure that our statistical estimates are more accurate and reliable, bringing the sample variance closer to the true population variance.