Z-scores: What and Why

Why

When we want to standardise scores and analysing them using the standard normal distribution, we need to convert our scores into standard ones. Z-scores allows us to compare two distributions by placing them on the same scale. We do this by transforming our scores into z-scores.

 

What

The z-score of a score is obtained by doing the following: (score – mean)/ standard deviation. A z-score is expressed in terms of standard deviations and tells us how many standard deviations below/above the mean our score is. So a z-score of 2.5 means that our score is 2.5 standard deviations above the mean, while a z-score of -1.5 means that our score is 1.5 standard deviations below the mean.

 

Z-scores and the standard normal distribution

A SND is a probability distribution because every score of the distribution has a probability. This refers to the probability of randomly selecting that score. Just like every score has a probability associated with it so do score ranges. So there is a 68% probability of getting a z-score between -1 and 1.

Probabilities in the SND

Probabilities in the SND

 

Similarly, there is a 96% probability of getting a z-score between -2 and 2.

 

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