zzz....z-scores!

Link to Youtube: z score calculations in Excel


Introduction
The z score of any individual observation tells us how far that individual observation is away from the mean of all the observations. The larger the z score, the further away. If the z score was zero, then the observation would be the same as the mean. It is òn`the mean, so no distance away. We use the z scores to detect outlying observations, and it is fundamentally important in inferential statistics. Before we start, make sure you're familiar with the concept of the standard deviation.

How we calculate  z scores
The unit of measurement is the standard deviation....think about this. A metre or an inch is a unit of measurement. Why not use a standard deviation? The z score is measured in numbers of standard deviations. So the z score for any particular observation is the number of standard deviations that observation is away from the mean. Take a look at the sketch below.

We find the a z score by using this equation:

So the z score for any individual observation (that is the meaning of the little i) is the difference between the observation and the mean, divided by the standard deviation. So for example, if the mean was 10, the observation was 15, and one standard deviation was 5, then the z score would be 1. 15 is one standard deviation above the mean.

A z score can be negative as well as positive. A negative z score tells us that the observation is smaller than the mean. It is to the left of the mean.

z scores and outliers
We define an outlier as being an observation that is more than three standard deviations away from the mean. Because a z score is in units of standard deviations that means we can use the z score directly. This is what you do:

1. If you want to find the z score for any particular observation you can do the calculations by hand. Example: the observation is 600. The mean is 500. The standard deviation is 50. So the z score is:

(600-500)/50 = 2. So 600 has a z score of 2. 600 is two standard deviations away from the mean. It is not an outlier.

2. If you have a number of observations, you can use Excel. Please see the Youtube which uses the Earnings per share dataset, which is available to you in the Datasets folder. Go on, practise!