How a Normal Distribution Works

Quick Answer

A normal distribution is a bell curve fixed by a mean and a sigma. The z-score is the bound minus the mean, divided by sigma. For example, mean 0 and sigma 1 at bound 1 give z = 1, with area near 0.84. This normal distribution calculator shades the left tail for the bound you set. Drag the bound and the shade follows it across the curve.

What is a normal distribution?

A normal distribution is a symmetric bell curve set by its mean and its standard deviation. Values near the mean are common and values in the tails are rare. For example, class heights and repeated measurements follow this shape. Sigma controls the width and the mean slides the peak.

How does the z-score become the shaded area?

Divide the distance from the mean by sigma, then read the left-tail probability as the shade. The formula for that distance is z = (bound - mean) / sigma. For example, mean 2 and sigma 2 with bound 4 give z = 1. The shaded left area is near 0.84 and the right area is what remains. The two areas add to 1.

How do the 68, 95, and 99.7 bands work?

Each band is the share of area within a whole number of sigma of the mean. About 68 percent lies within one sigma. About 95 percent lies within two sigma, and about 99.7 percent lies within three. A bound of 2 on the standard normal selects band 2. Unlike the shade, the band is a label and not a probability.

Why do the shaded area and the z-score disagree?

They disagree because z counts distance and the shade counts probability. A z-score of 1 is one sigma above the mean, while the left area is near 0.84. The normal distribution vs the empirical rule is the same kind of split. The curve is the full model and the rule is only three middle bands.

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