The Normal Distribution

6.1 Z-scores

The normal distribution is the bell curve. Two numbers pin it down. The mean \(\mu\) says where the peak sits, and the standard deviation \(\sigma\) says how wide it spreads. Every normal curve has the same shape, and only the position and the scale differ. That is why one trick works for all of them. The z-score

\[z = \frac{x - \mu}{\sigma}\]

counts how many standard deviations a value sits from its mean. A score of 77 on a test with \(\mu = 65\), \(\sigma = 8\) is \(z = 1.5\). A score of 620 with \(\mu = 500\), \(\sigma = 100\) is \(z = 1.2\). The first student did better, even though 620 is the bigger number. Drag both scores and watch them land on the standard normal curve, which has \(\mu = 0\) and \(\sigma = 1\). Move a slider and the raw score stays, but its z changes.

Reading: OpenStax Introductory Statistics, Section 6.1.

6.2 Normal areas and the empirical rule

As with any density, a normal probability is an area under the curve, written \(P(a \le X \le b) = \Phi\!\left(\frac{b-\mu}{\sigma}\right) - \Phi\!\left(\frac{a-\mu}{\sigma}\right)\), where \(\Phi\) is the standard normal cumulative distribution. There is no simple formula for \(\Phi\). Tables and software compute it, and so does this page.

Three areas are worth memorising. Press the buttons. About 68 % of the area lies within one standard deviation of the mean, 95 % within two, and 99.7 % within three. This is the empirical rule. Change \(\mu\) and \(\sigma\) and press the buttons again. The numbers do not move, because they belong to the shape rather than the scale. Now drag one edge on its own to see how fast the tail area shrinks past two standard deviations.

Reading: OpenStax Introductory Statistics, Section 6.2.