Mean and Variance

Intuition

The mean tells you where the data is centred; the variance tells you how spread out it is around that centre. Variance is the average squared distance from the mean.

Precise explanation

For data $x_1,\dots,x_n$ with mean $\bar x = \frac1n\sum x_i$:

• Population variance (the data is the whole population): $\sigma^2 = \frac{1}{n}\sum (x_i-\bar x)^2$.
• Sample variance (the data is a sample used to estimate a larger population): $s^2 = \frac{1}{n-1}\sum (x_i-\bar x)^2$. Dividing by $n-1$ (Bessel's correction) makes the estimate unbiased, because deviations from the sample mean are slightly too small on average.
• Standard deviation = square root of variance, in the original units.

Python's statistics.pvariance uses $n$, statistics.variance uses $n-1$; NumPy's np.var defaults to $n$ (ddof=0) — a frequent source of mismatched results.

The two-pass method (compute the mean, then sum squared deviations) is numerically safer than the textbook shortcut $\overline{x^2} - \bar x^2$, which can lose precision or even go negative.

Worked example

Data 2, 4, 4, 4, 5, 5, 7, 9: mean 5; squared deviations 9,1,1,1,0,0,4,16 sum 32. Population variance 32/8 = 4 (sd 2). Sample variance 32/7 ≈ 4.571.

Common mistakes

• Mixing $n$ and $n-1$ between libraries.
• Computing sample variance of a single value (division by zero — it is undefined).
• Forgetting to square deviations (they sum to zero).

Why this matters

Variance drives feature standardisation (ML-08), batch normalisation (DL-23), confidence intervals and the bias–variance trade-off.

Library implementation

The statistics module provides fmean, pvariance (divide by $n$) and variance (divide by $n-1$), plus pstdev/stdev. They use exact arithmetic internally where possible, so tiny differences from your loop are rounding only.

References

• statistics — Mathematical statistics functions — Python Software Foundation (S003)
• Think Stats (3rd edition) — Allen B. Downey (S010)