Collect naturally occurring data on a large scale — like height, test scores, or product weight — and plot it as a graph, and you'll often see a symmetric bell shape: data piled up heavily near the mean, thinning out the further it gets from the mean. A distribution like this is called the normal distribution.
The shape of a normal distribution is completely determined by just two values: the mean (μ) and standard deviation (σ). The mean marks the center position of the curve, and the standard deviation marks how widely the curve is spread. If σ is small, the data clusters sharply near the mean; if σ is large, the curve becomes flat and widely spread.
The normal distribution has one extremely useful rule. No matter what values μ and σ take, within one standard deviation of the mean, about 68% of all the data is always clustered; within two, about 95%; within three, about 99.7%. This is called the empirical rule (the 68-95-99.7 rule). Thanks to this rule, just knowing "how far this value sits from the mean" instantly tells you whether that value is relatively common or quite rare, without needing any complicated calculation.
The topic that flows naturally from here is estimation. Surveying an entire population (say, the height of every single tenth-grade student nationwide) is realistically difficult, so instead we survey only a sample — a portion of that population — and use the result to make a judgment about the whole. The mean calculated from a sample is called the sample mean (x̄), and it changes slightly every time you draw a new sample. That's why, instead of stating flatly "the population mean is exactly this value," it's more honest to present a confidence interval — something like "it's highly likely to fall within this range."
When you know the population standard deviation σ, the confidence interval for the population mean is found in the form x̄ ± z×(σ/√n). Here, n is the sample size, and z is a value fixed by the confidence level (z=1.96 for a 95% confidence level, z=2.58 for 99%). σ/√n is called the standard error, and true to its name, it represents the degree of error in how far the sample mean might sit from the population mean.
The most important insight in this formula is that as sample size n grows, √n grows too, so the standard error (σ/√n) gets smaller and smaller. In other words, the more samples you draw, the narrower the confidence interval gets, and the more accurate the estimate becomes. Conversely, if you want a higher confidence level (say, going from 95% to 99%), the z-value gets larger and the interval widens along with it — wanting more certainty means you have to offer a wider range in return.
On our activity page, you can change the mean and standard deviation with sliders to check the empirical rule directly, and change the sample size and confidence level to watch the confidence interval widen and narrow, in numbers and on a graph at the same time.