Normally Distributed Datasets

Lots of natural measurements — heights, test scores, factory part sizes — pile up near an average and get rarer the farther you move from it, in both directions equally. That pattern is a normal distribution: symmetric, single-peaked, and shaped by one specific formula, not just any hump-shaped curve.

The 68-95-99.7 rule carves the bell into bands around the mean, measured in standard deviations. About 68% of the data falls within one standard deviation of the mean, 95% falls within two, and 99.7% falls within three. Because the curve is symmetric, you can split each band in half to get the percent on just one side. This turns a vague picture of a bell into exact percentages you can calculate: ask what fraction of scores falls between 70 and 90 when the mean is 80 and standard deviation is 10, and you're really asking for the one-standard-deviation band, 68%.

A z-score tells you how many standard deviations a value sits from the mean, using z = (value minus mean) divided by standard deviation. This connects the normal curve to everything else in the describing-data cluster: mean and standard deviation are the two numbers that define the curve's center and spread, and once you know them, the shape is fixed. That's also why doubling the standard deviation matters — the curve gets wider and flatter but stays centered on the same mean, still holding 68% of its data within that new, wider one-standard-deviation band.