Statistics Calculator — Mean, Median, Mode, Standard Deviation

This is the step textbooks state and rarely justify, so here is the reasoning from the ground up.

Variance measures how far values sit from the true mean. When you have the entire population, you know the true mean, and dividing the summed squared deviations by n is exactly right.

With a sample, you do not know the true mean — you estimate it from the same data you are measuring the spread of. And the sample mean is, by construction, the value that makes the sum of squared deviations as small as it can possibly be. Any other number, including the true population mean, would produce a larger sum.

So measuring spread around your own sample mean systematically understates the real spread. Dividing by n − 1 rather than n inflates the result by exactly enough to correct that bias on average. This is Bessel's correction.

The intuition for why it is n − 1 specifically: once you know the mean and any n − 1 of the values, the final value is fixed — it carries no independent information. You have n − 1 degrees of freedom, not n.

The practical consequence is that the correction matters enormously for small samples and barely at all for large ones. At n = 4 it changes variance by 33%; at n = 1,000, by 0.1%.

The mean uses every value, which is its strength and its weakness. It is efficient and mathematically tractable, and a single extreme value can drag it somewhere no observation actually sits.

The classic demonstration: incomes of £25k, £28k, £30k, £32k and £1.5m have a mean of about £323k. That figure describes nobody in the room. The median of £30k describes the group honestly. This is precisely why income and house price statistics are almost always reported as medians.

The mode — the most frequent value — is the only measure that works on categorical data, where "average colour" is meaningless but "most common colour" is not. It can also be absent or multiple, which the other two never are.

A useful diagnostic: when mean and median diverge substantially, the distribution is skewed, and the direction tells you which way. Mean above median means a long right tail, usually a few large values.

Standard deviation expresses typical distance from the mean in the original units. For roughly bell-shaped data it carries a useful rule of thumb: about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

That rule only holds for approximately normal distributions. Applied to skewed data — incomes, waiting times, city populations — it will mislead, and the mean it is built around is already unreliable there.

The interquartile range takes a different approach: sort the data, find the value a quarter of the way through (Q1) and three quarters through (Q3), and take the difference. That range holds the middle half of the data and is completely unaffected by how extreme the extremes are.

This robustness is why box plots use quartiles rather than standard deviations, and why the IQR is the better spread measure whenever you have chosen the median as your measure of centre. The two pair naturally: mean with standard deviation, median with IQR.

Summary numbers discard information, and occasionally the discarded part is the whole story. Anscombe's quartet is the standard demonstration: four datasets with near-identical means, variances and correlations that look completely different when plotted — one linear, one curved, one with a single dominant outlier.

The practical lesson is to plot the data before trusting a summary of it. A histogram takes seconds and reveals skew, bimodality and outliers that no combination of summary statistics will surface.

These measures also say nothing about cause. Two variables can move together for reasons that have nothing to do with either affecting the other, and no amount of descriptive statistics distinguishes those cases.