Normal Distribution and Z Score

Interpreting a z score

A z score measures the distance from the mean in standard deviations. Subtract the mean and divide by the positive standard deviation. For a mean of 10, standard deviation of 2 and observed value of 12, the z score is 1.

Probabilities and quantiles

The lower tail is the probability of a value at or below the chosen threshold. The upper tail covers larger values. The inverse calculation accepts a proportion such as 0.975 and finds its quantile. These probabilities assume a normal distribution. Check whether that model describes your observations before interpreting the results.

A tail versus a single value

Mean, standard deviation and observed value must describe the same quantity in the same units. Do not enter variance as standard deviation. Variance has squared units and is not used directly in standardization. A positive z score places the observation above the mean, and a negative score below it.

For example, let the mean be 50, deviation 5 and observation 50. The score is (50−50)/5 = 0. Normal symmetry gives a lower tail of 0.5 and upper tail of 0.5. In inverse mode, a lower tail probability of 0.5 with the same parameters returns 50.

This does not mean that the probability of observing exactly 50 is one half. In a continuous model, probability describes interval area. For an interval, subtract the lower boundary's cumulative probability from the upper boundary's cumulative probability. Enter a proportion rather than a percentage number in inverse mode. Assess the distribution assumption separately. Bounded, skewed or mixed group data may behave differently from a normal model.

Frequently asked questions

How do you calculate a z score from the mean and standard deviation?

With an example mean 100, standard deviation 15 and observation 130, subtracting the mean and dividing by standard deviation gives a z score of 2. Under a normal model the lower tail probability is 0.9772498681, and this probability interpretation requires the data to fit that model, which must be checked separately.

Calculate with these values