How a Z-Score Can Help You Compare Data—Here's How to Calculate It

"You can’t compare apples to oranges." Although the saying is cliché, the message behind it is clear. Comparing two things that are completely different is hard to do. You need a way to make the objects equal so it can give you a fair comparison.

Similarly, when analyzing numbers, experts have to find a way to make the numbers comparable. This allows them to draw parallels between the information. Finding the Z-score can help make sense of different types of data.

“Z-scores are a handy way to determine how a specific value compares to the average,” explains Nathan Brunner, CEO of Salarship. “Researchers find the Z-score very useful because it is standardized, which allows them to perform standardized statistical tests and compute prediction intervals,” he notes.

We’ll look at what the Z-score is, when having a Z-score is helpful, and how you can calculate the score.

What Does a Z-Score Tell You?

A Z-score gives you a composite number for a specific set of data. It helps provide standardized numbers for data that may otherwise be hard to compare. The American Psychological Association notes that it is “the standardized score that results from applying a Z-score transformation to raw data.” By using the formula to get a standardized number, you can compare that finding to other sets of data.

In a recent study, researchers wanted to find out what activity helped improve cognitive function in schizophrenia patients more—watching TV or cognitive remediation on a computer. Because different scores and insights were used to measure responses to the input, researchers had to find a common way to measure both sets of data. Using the raw information to find the Z-score gave them the comparison tool that they needed.

Z-Score vs. Standard Deviation

You may be thinking that since the Z-score finds the average of several sets of data as a part of its formula, it is like finding the standard deviation. That’s not exactly the case.

“The Z-score and standard deviation are related but different concepts. The standard deviation measures the spread or variability of the whole dataset. A larger standard deviation indicates greater variability, while a smaller standard deviation indicates less variability,” Brunner notes.

The Z-score gives a way to compare information. The standard deviation tells you the difference between the two sets of information.

How to Calculate the Z-Score

Calculating the Z-score starts with you having the raw data that you want a composite number for. With that data, you will need to find the mean or average number, as well as the standard deviation value for all the data.

You can find out the mean by adding all the numbers together, then dividing them by the total amount of values that you have. For example, if you have 4 + 5 + 6, you add those numbers together, giving you 15. You then divide 15 by the number 3, because you have 3 separate values. That gives you the number 5, which is the average of those numbers.

The standard deviation is a bit more detailed.

“To get the standard deviation … you need to take the square root of the sum of squared differences from the mean divided by the size of the data set,” Brunner tells. “Once you have both values, you can easily compute the z-score.”

You would use the following formula: Z-score = (the initial data point – mean)/standard deviation.

Brunner uses an example of finding out how a student’s test score of 90 compared with the scores his peers received, which are 75, 80, 85, 90, and 95. First, we find out the mean of this data set, which is 85. Then you calculate the standard deviation, which is approximately 7.9. Plugging the numbers into the formula, the Z-score is (90-85)/7.9, which equals 0.63.

If following the math is confusing, a Z-score calculator can help you find your answer.

Interesting information, but where is it applicable?

Z-Scores and IQ Testing

IQ test scores are highly prized for admission to certain clubs and organizations. Researchers note that Z-scores help you to understand the actual value of the IQ test, as well as where the test score falls in comparison to other IQ scores.

How to Convert a Z-Score to IQ

To compute the IQ score from a z-score, you multiply the Z-score by 15 for most tests, then multiply that number by 100.

For some tests, you multiply the Z-score by 16 initially. With a basic example, if you have a Z-score of 0.10, and you multiply that number by 15, you’ll get 1.5. You then multiply 1.5 by 100, for an IQ score of 150.

Which Z-Score Indicates a Genius-Level IQ?

While this number can fluctuate, Brunner provides insight on a high-level z-score.

“Some psychologists consider an IQ of 145 or above to be genius-level. The z-score is (145 - 100) / 15 = 3 for this value. This is extremely restrictive as only 0.135% of the population has an IQ of 145 or above,” he states.

Z-Score and Other Psychological Tests

Z-scores and the information they provide can also be beneficial in the mental health arena.

Studies have shown that when comparing the impact of different levels of stress on subjects’ mood, hormones, and cognition, Z-scores were extremely useful.

The scores helped to create a uniform way to compare the information.

Another study found that using Z-scores to provide neurofeedback and biofeedback in older adults with memory issues helped to target methods of treatment.

Experts say Z-scores can also help with tracking changes in patient's depression and anxiety
levels.