Collect a huge amount of naturally varying data — like height, test scores, weight — and plot it on a graph, and remarkably, a similar shape appears again and again. Values pile up heavily near the mean, and get rarer and rarer the further they get from it, tracing a smooth bell shape. This curve is called the normal distribution.
The shape of a normal distribution is determined by exactly two numbers. The mean μ decides where the center of the curve sits, and the standard deviation σ decides how widely the curve is spread. If σ is small, values cluster tightly near the mean, so the curve gets sharp and narrow; if σ is large, values scatter widely, so the curve gets flat and wide. Change the mean μ, and the curve's shape stays the same — it just moves left or right as a whole.
The normal distribution has one extremely useful rule: about 68% of everything falls within one standard deviation of the mean (±1σ), about 95% within ±2σ, and about 99.7% within ±3σ. This 68-95-99.7 rule is a remarkable property that always holds true, no matter what the mean and standard deviation are — that is, for any normal distribution at all.
But comparing two different values is hard using raw scores alone when the distributions have different means and standard deviations. For example, which is relatively better: an 85 on a math test with mean 70 and standard deviation 10, or a 78 on a Korean test with mean 60 and standard deviation 15? This is exactly where the standard score (z-score) comes in. It's calculated as z = (x − μ) / σ, and this value represents "how many standard deviations my score sits from the mean."
Calculate it for the example above, and math gives z = (85−70)/10 = 1.5, while Korean gives z = (78−60)/15 = 1.2. The math score has the larger z-value, meaning it sits relatively farther from the mean — so you can say the math score is relatively the better performance of the two tests.
The normal distribution is also deeply connected to the binomial distribution. As the number of trials n grows very large, the jagged bars of a binomial distribution get closer and closer to a smooth bell shape. Grow n larger in the "Binomial Distribution Lab" and you can watch that change happen directly. On the activity page, you can change the mean and standard deviation with sliders and watch the shape of the curve change, and on the standard score tab, you can enter a score yourself and check the z-value and what it means.