The Normal Distribution & Z-Scores

Heights, exam marks, measurement errors, the noise on a sensor — astonishingly many things pile up into the same symmetric bell curve. Learn to read that curve and you can turn almost any messy real-world quantity into a clean probability.

The shape of the bell

A normal distribution is a continuous, symmetric curve that is tallest in the middle and tails off smoothly on both sides. Its centre sits at the mean μ, and because the shape is perfectly symmetric the mean, median and mode all land at exactly the same place. The width of the bell is set by the standard deviation σ: a small σ gives a tall, narrow spike; a large σ gives a low, wide hump.

The single most important idea is that area under the curve is probability. The whole curve encloses a total area of exactly 1— a probability of 100% that a value lands somewhere. The chance of a value falling between two points is just the slice of area sitting above that stretch of the axis.

The bell is a budget of exactly one unit of probability, shared out by area.

See it: drag the curve

Try it below. Slide μ to move the whole bell left or right, slide σ to squeeze it thin or spread it wide, then drag the two bounds to shade a region. The shaded area is the probability of a value landing in that range, and the z-scores of the bounds appear alongside.

Standardising with z-scores

There is not one normal distribution but infinitely many — one for every pair of μ and σ. We could never tabulate them all. The fix is to standardise: measure how many standard deviations a value sits away from the mean.

A z-score of +2, for instance, means “two standard deviations above the mean” whatever the original units were — centimetres, marks or grams. That comparability is why z-scores are the workhorse of the topic.

The 68–95–99.7 rule

Because the standard normal is fixed, a few areas are worth memorising. About 68%of values lie within one standard deviation of the mean (−1 < z < 1), about 95% within two, and about 99.7% within three. This empirical rule lets you sanity-check any answer at a glance: a value three σ from the mean is genuinely rare.

Sampling and the Central Limit Theorem

Take a sample of size n and work out its mean x̄. Do it again and you get a slightly different x̄. The collection of all these sample means has its own distribution — the sampling distribution of the mean— and it is remarkably well behaved.

Slide n upward in the widget below and watch what happens: the distribution of the sample mean stays centred on μ but grows tall and narrow, its spread shrinking to the standard error σ/√n.

Common mistakes

Practice

Where next?

Once you can turn values into z-scores, hypothesis testing is the natural sequel: comparing a sample mean against a claimed μ is just another standard-error calculation dressed up as a decision.