Question 1

What is the difference between population variance and sample variance?

Accepted Answer

Population variance (σ²) divides by n and is used when you have data for the entire population. Sample variance (s²) divides by n−1 and is used when your data is a sample drawn from a larger population. The n−1 denominator is Bessel's correction — it makes the sample variance an unbiased estimator of the population variance. If you measured every student in one class and want only to describe that class, use population variance. If you want to estimate variance for all classes in a school from a sample, use sample variance.

Question 2

What is Bessel's correction and why does it matter?

Accepted Answer

When computing variance from a sample, using n (instead of n−1) consistently underestimates the population variance. This happens because the sample mean is calculated from the same data, introducing a slight downward bias. Dividing by n−1 instead of n corrects for this bias. The correction is named after the mathematician Friedrich Bessel. For large samples the difference is negligible; for small samples (n < 30) it matters significantly.

Question 3

What is the relationship between variance and standard deviation?

Accepted Answer

Standard deviation is simply the square root of variance: σ = √σ². They measure the same thing — spread around the mean — but standard deviation is expressed in the original units of the data, making it easier to interpret. For example, if data is in metres, variance is in metres² (squared), while standard deviation is in metres. Most reporting uses standard deviation; variance is used in mathematical derivations, ANOVA, and regression.

Question 4

What does a high variance mean?

Accepted Answer

High variance means values are widely dispersed around the mean — the dataset is heterogeneous. Low variance means values cluster tightly around the mean — the dataset is homogeneous. However, "high" and "low" are relative: always interpret variance in context. A variance of 25 for test scores (scale 0–100) suggests moderate spread; a variance of 25 for ages in a children's school (scale 5–12) would suggest very high spread.

Question 5

Can variance be negative?

Accepted Answer

No. Variance is the average of squared deviations — squaring eliminates negative values. Variance is always ≥ 0. Variance equals exactly 0 only when all values in the dataset are identical (no spread at all). If you get a negative variance result from a calculation, there is an arithmetic error.

Question 6

When should I report variance vs standard deviation?

Accepted Answer

Report standard deviation when communicating results to a non-technical audience — it is in the same units as the data and easier to interpret. Report variance when: (1) performing ANOVA or regression where variance components are partitioned; (2) comparing variance between groups (F-tests); (3) working with error propagation in physics or engineering. In academic papers, both are often reported: "mean ± standard deviation".

Metric	Units	Use when
Variance (s²)	Squared units (e.g. cm²)	Mathematical calculations, ANOVA, regression analysis
Std Dev (s)	Same as data (e.g. cm)	Reporting results to non-statisticians, normal distribution work

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