Why do most results cluster around the average? Commuting times, meal-prep times, daily phone usage, test scores — a huge range of real-world data roughly follows the same bell-shaped pattern, built entirely out of the mean and the standard deviation.

The Shape A smooth bell curve centred on the mean

Most results sit close to the mean. The farther a value is — left or right — the fewer observations there are out there. How quickly the curve tapers off is set by the standard deviation.

The Rule 68–95–99.7

The standard deviation tells you how widely the data spreads around the mean . Moving out from the mean by whole numbers of standard deviations always covers the same fixed share of the data:

1 standard deviation each side of the mean — about 68% of the data
2 standard deviations each side — about 95% of the data
3 standard deviations each side — about 99.7% of the data

Example Test scores: mean 70, standard deviation 10

Assuming the scores follow a normal distribution:

1 standard deviation:

between

and

— about

of students

2 standard deviations:

between

and

— about

of students

With just the mean and the standard deviation, a teacher can estimate who may need extra support, spot high performers, and read the overall pass rate — without checking every single score.

Summary

The normal distribution is a smooth, symmetric bell curve centred on the mean.
The standard deviation controls how spread out the curve is.
68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
Mean , standard deviation : about 68% of scores fall between and , and about 95% fall between and .

The Normal Distribution