4.1 Normal distribution

If a normal distribution has a mean of \(\mu\) and a standard deviation of \(\sigma\) then we can write the distribution as \(N(\mu, \sigma)\).

The mean and standard deviation are called the parameters of the normal distribution. A normal distribution with a mean of 0 and standard deviation of 1 is called the standard normal distribution. The above example uses a standardization technique called the Z-score. The Z-score is defined as the number of standard deviations that a case falls above or below the mean. If the observation is 1 standard deviation above the mean, then it has a Z-score of 1. If it is 1.5 standard deviations below the mean, it has a Z-score of -1.5. It is defined mathematically as \(Z=(x-\mu)\div\sigma\).

The percentile of an observation is the fraction of observations with lower values. Using the above example, it is the fraction of people who scored less than Ann. If we graphed a curve of all test scores, it would be equal to the area under the curve, up to Ann’s score on the x-axis. This is generally calculated on a computer or graphing calculator.