Normal Distribution Properties

Imagine a histogram of measurements with most values clustered around the middle and fewer values towards either extreme. A Normal curve provides a smooth model of this pattern. It describes the overall distribution rather than tracing every irregularity in a sample.

What the bell shape tells you

A Normal curve has one peak at its centre. Moving away from the centre in either direction, the curve becomes lower: values far from the centre are less common than values close to it.

The two sides are mirror images. This symmetry means that equally sized intervals at equal distances on opposite sides of the centre contain equal proportions of values. Half the values lie below the centre and half lie above it.

For an exact Normal distribution, the mean, median and mode are all equal:

• The mean is the average and lies at the centre of the balanced distribution.
• The median divides the distribution into two equal halves.
• The mode is at the single peak, where values are most concentrated.

A symmetric shape alone is not enough to identify a Normal distribution: it must also have the characteristic single-peaked bell shape.

Centre and spread describe different properties

The mean determines the horizontal position of the centre. Increasing the mean while keeping the spread unchanged shifts the curve to the right without changing its shape.

The standard deviation measures spread around the mean. A small standard deviation produces a narrow, tall curve, because values are closely clustered. A larger standard deviation produces a wider, lower curve, because values are more spread out. If the mean stays unchanged, both curves remain centred at the same value.

For example, suppose two machines fill bottles with a mean volume of 500 ml. One machine has a standard deviation of 2 ml and the other has a standard deviation of 4 ml, with both distributions modelled as Normal. Their curves have the same centre, but the first is narrower: its fill volumes are more consistent. Its taller peak does not mean that its average fill volume is greater.

Reading Normal distribution notation

A Normal distribution can be written as $N(\mu,\sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance.

Variance is the standard deviation squared. The mean and standard deviation use the same units as the measurements; variance uses squared units. The notation therefore identifies both the centre and the spread of the distribution.

Deciding whether the model is suitable

To judge whether a Normal model is reasonable, consider the measurements and their overall pattern. The data should be continuous, with a roughly symmetric, bell-shaped distribution and most values near the middle. The sample mean, median and mode should be approximately equal.

Real samples will not usually form a perfectly smooth bell shape or have exactly equal averages. Look for the overall pattern rather than expecting perfection. A clearly skewed distribution, with a much longer tail on one side, is not well represented by a Normal model.