For z-scores, it always holds (by definition) that a score of 1.5 means “1.5 standard deviations higher than average”. The histogram below illustrates this: if a variable is roughly normally distributed, z-scores will roughly follow a standard normal distribution. For instance, it's well known that some 2.5% of values are larger than two and some 68% of values are between -1 and 1. This is a common procedure in statistics because values that (roughly) follow a standard normal distribution are easily interpretable. If a normally distributed variable is standardized, it will follow a standard normal distribution. Due to the central limit theorem, this holds especially for test statistics. The reason may be that many variables actually do follow normal distributions. So why do people relate z-scores to normal distributions? We saw earlier that standardizing scores doesn't change the shape of their distribution in any way distribution don't become any more or less “normal”. In a similar vein, if we had plotted scores versus squared scores, our line would have been curved in contrast to standardizing, taking squares is a non linear transformation. It contains 100 points but many end up right on top of each other. What we mean by this, is that if we run a scatterplot of scores versus z-scores, all dots will be exactly on a straight line (hence, “linear”). Z-scores are linearly transformed scores. That is, standardizing scores doesn't make their distribution more “normal” in any way. Other than that, however, z-scores follow the exact same distribution as original scores. If you look closely, you'll notice that the z-scores indeed have a mean of zero and a standard deviation of 1. We did so and ran a histogram on our z-scores, which is shown below.
In practice, we obviously have some software compute z-scores for us. In a similar vein, the screenshot below shows the z-scores for all distinct values of our first IQ test added to the data. These two steps are the same as the following formula:Īs shown by the table below, our 100 scores have a mean of 3.45 and a standard deviation of 1.70.īy entering these numbers into the formula, we see why a score of 5 corresponds to a z-score of 0.91: then dividing each remainder by the standard deviation over all scores.first subtracting the mean over all scores from each individual score and.We suggested earlier on that giving scores a common standard of zero mean and unity standard deviation facilitates their interpretation. Note that these scores are clearly not normally distributed. This pattern is known as a uniform distribution and we typically see this when we roll a die a lot of times: numbers 1 through 6 are equally likely to come up. The histogram confirms that scores range from 1 through 6 and each of these scores occurs about equally frequently. However, we'll gain much more insight into these scores by inspecting their histogram as shown below. Scores - HistogramĪ quick peek at some of our 100 scores on our first IQ test shows a minimum of 1 and a maximum of 6. What we see here is that standardizing scores facilitates the interpretation of a single test score. However, if my score of 5 corresponds to a z-score of 0.91, you'll know it was pretty good: it's roughly a standard deviation higher than the average (which is always zero for z-scores). So is that good or bad? At this point, there's no way of telling because we don't know what people typically score on this test. This standard is a mean of zero and a standard deviation of 1.Ĭontrary to what many people believe, z-scores are not necessarily normally distributed.Ī group of 100 people took some IQ test. Z-scores are also known as standardized scores they are scores (or data values) that have been given a common standard. Z-Scores – What and Why? By Ruben Geert van den Berg under Statistics A-Z & T-Tests