Distributions

The shape of your data

A distribution describes how often each value occurs. The shape of that distribution drives which statistical tools are valid and which conclusions are sound.

The normal distribution

The bell curve — symmetric around the mean, with most values clustered near the center and thinning out at the tails. Heights, test scores, and measurement errors often approximate it.

The 68-95-99.7 rule

For a normal distribution, standard deviation partitions the data predictably:

±1σ  → ~68%  of the data
±2σ  → ~95%  of the data
±3σ  → ~99.7% of the data

So if mean test score is 70 with σ = 10, about 95% of students score between 50 and 90.

The 68-95-99.7 rule lets you eyeball whether a value is "unusual": anything beyond two standard deviations from the mean is in the rarest 5%.

Skewness

• Right-skewed (positive) — long tail to the right. Mean > median. e.g. income.
• Left-skewed (negative) — long tail to the left. Mean < median. e.g. retirement age.

The uniform distribution

Every value in a range is equally likely — a fair die roll is uniform across 1–6. Variance is higher than a peaked normal with the same range.

Sampling bias

If your sample isn't representative of the population, every downstream statistic is misleading. A classic example: a 1936 poll predicted Landon would beat Roosevelt, but it sampled mostly wealthy phone owners during the Depression. The sample was large and wrong.