February 29, 2020
Chapters
How to Calculate ZScores
In other sections of this guide, you learned about the basics of standard scores, including how to calculate them and how they relate to normal distributions. Here, we’ll show you how zscores are applied in the real world and how you can interpret them in various scenarios.
What is a ZScore?
As a reminder, standard scores, also known as zscores, is a score that indicates how usual or typical a data point is within the data set. The zscore uses the mean and standard deviation in order to calculate a score for a data point, letting us know how far away that data point is from the mean in standard deviations.
If this sounds complicated, let’s remind ourselves of what the formula looks like.
Sample Zscore  Population Zscore 


= sample z score  = population z score 
= observation  = raw score, or observation 
= sample mean  = population mean 
= sample SD  = population SD 
From the sample zscore formula, it’s easy to see that a z score simply measures the distance from a data point to the mean and then divides it by the standard deviation. While the zscore itself can tell you a bit about where the data point lies, it becomes more interpretable after finding that zscore in a ztable.
Finding your zscore on a ztable means finding the corresponding percentage that indicates the proportion of the standard normal distribution that lies below your chosen data point.
Uses of Zscores
The best thing about standard scores is their simplicity. Because they have the power to tell us how “normal” or typical a value is for a given data set, the applications of zscores are numerous. From psychology to sports, zscores are perfect for finding the probability of that data point occurring within a standard normal distribution.
One example of how zscores are used in the real world comes from the financial world and is called Altman Zscore. While not strictly a zscore in this sense, it does use the concept of the zscore in order to calculate the likelihood of bankruptcy of a company.
The reason why zscores are so applicable to a diverse range of sectors and subjects is because it:
 Enables people to understand how likely a data point is within a standard normal distribution
 Allows people to compare scores from different samples, which can contain different means and standard deviations
Interpretation of Zscores
The interpretation of the zscore without a ztable is possible but will limit your interpretation. As a reminder of what the zscore means, recall a normal and standard normal distribution. The only difference between a normal distribution and a standard normal distribution is that, while a normal distribution shows the distribution of a data set following a normal curve, a standard normal distribution expresses the data in terms of standard deviation.
Looking at the images above, we can see the normal histogram on the left side and the standard normal distribution on the right. In the histogram, we can see that the distribution follows a normal distribution curve, with the mean in the centre. What the standard normal curve does is that it standardizes all of these data points.
In other words, all the data points from a normal distribution are taken and standardized. Take a look at the standardization formula and see if you notice where it’s from.
That’s right, standardizing any observation is the same as taking the zscore, which means the interpretation is the same as a standard normal distribution. Recall the 689599 rule, which states that:
 68% of a standard normal distribution have a zscore of 1 to 1
 95% of a standard normal distribution have a zscore between 2 and 2
 99% of a standard normal distribution have a zscore between 3 and 3
Interpreting the zscore, then, will follow some general rules expressed in the table below.
Zscore  Meaning 
<1  1 SD less than the mean 
<0  Less than the mean 
Equal to 0  Equal to the mean 
>0  Greater than the mean 
>1  1 SD greater than the mean 
This can be extended to zscores greater or less than 2, 3 and 4.
ZScore Versus Standard Deviation
While there are many similarities to the zscore and standard deviation, there are major differences between what the two measures are and how they are applied in statistics. The standard deviation, as a brief reminder, is a measure of variability. This means that it strives to measure the dispersion around the mean.
Okay, admittedly this sounds very similar to what the zscore does. However, the main difference stems from the fact that the zscore is a measure of how likely a value is to appear for any given mean and standard deviation.
So, while a standard deviation can tell us information about variability within a data set, the zscore expands this by comparing our data set to a standard normal distribution. This is why the zscore utilizes the standard deviation and mean in order to find out how typical a value is given a standard normal distribution.
Problem 1: Calculating the Zscore
Let’s say that you want to calculate the zscore of a given data set but don’t have information about the mean or standard deviation. In the table below, you have information about the data points within two data sets.
Observation  Data Set 1  Data Set 2 
1  45  11 
2  60  90 
3  74  60 
4  25  45 
5  36  90 
Find the zscore of two new data points corresponding to each data set, given below.
Data Set 1  Data Set 2  
New Observation  29  150 
Solution to Problem 1
In this problem, our task was to calculate the zscore of a new observation. First, we need to calculate the mean and standard deviation for each data set. Compare your answers to the table below.
Data Set 1  Data Set 2  

 


Now that we have the mean and the SD, we can calculate the zscores for each new value using the formula for standardization. We attain the following result.
Data Set 1  Data Set 2  
New Observation 


Interpretation  About 1 SD less than the mean  Almost 3 SD above the mean 
Problem 2: ZScore in the Real World
Zscores are particularly relevant in situations involving scores. Let’s say you want to calculate how typical your IQ is. You are given the following information about your population’s mean and standard deviation.
Population  
100  
15 
Pretend that your IQ is 180. How typical is your score compared to the population?
Solution to Problem 2
In this problem, we were asked to:
 Comment on how typical your IQ score is compared to the population
To do this, we must find and interpret the zscore. This is done in the table below.
Your IQ  
zscore 

Interpretation  Your IQ is roughly 5 SD above the population, which happens at a frequency of less than 0.1%. Meaning, this is not typical compared to the population. 