Normal Distribution

Some datasets have histograms that look bell-shaped. Heights are a common example: most measurements are near the middle, with fewer at either end. We say these data approximately follow the normal curve. Other datasets have histograms that look different. Incomes and house prices, for example, are often skewed rather than bell-shaped.

The normal curve is a smooth model. The histogram of real measurements does not have to trace it exactly. The area under the curve represents probability, and the total area is 1 (NationalInstituteofStandardsandTechnology).

The empirical rule

If data approximately follow the normal curve, the empirical rule is a useful rule of thumb (NationalInstituteofStandardsandTechnology):

• About 68% of the data fall within one Standard Deviation of the mean. That is roughly two-thirds.
• About 95% fall within two standard deviations.
• About 99.7% fall within three standard deviations.

The curve is symmetric, so the area left over is split between the two sides. After one standard deviation, about 16% is in each tail. After two standard deviations, about 2.5% is in each tail.

For example, suppose fathers' heights have a sample mean of $\bar{x}=68.3$ inches and a sample standard deviation of $s=1.8$ inches. The empirical rule suggests that about 95% of their heights fall between $68.3-2(1.8)=64.7$ inches and $68.3+2(1.8)=71.9$ inches.

Standardising data

To standardise a height, subtract the sample mean and divide by the sample standard deviation:

$z=\frac{\text{height}-\bar{x}}{s}.$

The result is a z-score. It has no units: in this example, inches in the numerator and denominator cancel. A z-score of $1.5$ means a height is 1.5 standard deviations above the average. A z-score of $-0.52$ means it is 0.52 standard deviations below the average.

When we standardise every value using the same sample mean and standard deviation, the resulting values have a mean of 0 and a standard deviation of 1. If the original values follow a normal distribution, the standardised values follow the standard normal distribution. Its curve is given by (NationalInstituteofStandardsandTechnology):

$\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}.$

The height of that curve is a density. To find a probability, we look at the area under it.

Computing a percentile

What is the 30th percentile of the fathers' heights in the example? It is the height below which about 30% of the values fall. A standard normal table gives $z\approx-0.52$ for an area of 30% to the left.

We can undo the standardisation to turn the z-score back into a height:

$\text{height}=\bar{x}+zs=68.3+(-0.52)(1.8)\approx67.4\text{ inches}.$

So the normal model estimates the 30th percentile at about 67.4 inches.

References

National Institute of Standards and Technology. What do we mean by Normal data? NIST/SEMATECH e-Handbook of Statistical Methods. Accessed 5 October 2026. URL: https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm (visited on 2026-10-05). ↩ ^{1 2 3}