How To Calculate Z-Score (With Examples)

If you have been schooled in a traditional American setting, you have most likely heard of a bell curve. It may even bring on some high school flashbacks — whether those are good memories or bad memories depends on what kind of student you were.

Don’t worry. We won’t tell anyone if you were a bad student. It happens to the best of us.

The bell curve was used in school to calculate the grading scale. Its use is often debated, but it was most certainly a commonly used grading process for a long time. Teachers and professors often employed it when the class’ marks on a specific exam were well below expectations.

It was typically curved off of the highest score. So, if one person scored significantly higher than all others, it could negatively impact the curve.

What we know as the bell curve is also known as normal distribution. Normal distribution is a distribution of data with an equal mean, median, and mode.

The mean sits at the center of the curve, the highest point in the “bell,” with 50% of the values in the data set less than the mean and 50% of the values in the data set greater than the mean.

It creates this beautifully symmetrical data representation that looks very much like a bell, thus the more commonly used nickname.

When we convert the scores in a normal distribution to a standard score, more commonly referred to as a z-score, we get a standard normal distribution (SND). This will also present a very bell-like visual but will always have a mean of zero and a standard deviation (SD) of one.

As the name suggests, z-scores help to standardize distribution. It doesn’t change any of the data. It simply rescales (or standardizes) it.

What is a Z-Score (Standard Score)?

A z-score, or standard score, is a numerical value that measures the standard deviation units between a data point and the population mean. It does this by converting (or standardizing) the raw scores in a normal distribution.

Z-scores will measure the exact amount of standard deviations above or below the population mean for each data point. Standard deviation reflects the amount of variability within a data set.

To find the standard deviation, you would first calculate the distance between each individual data point and the population mean. With standard normal distribution (SDN), this will always be represented the same.

The z-score measures the number of standard deviations for each data point. With a standard normal distribution (SND), the standard deviation will always be one. Typically, z-scores will fall within a +3 to -3 range. If a z-score falls above +3 or below -3, it would be considered unusual or non-standard.

A positive z-score value will lie above the population mean or to the right of the “bell” center, while a negative z-score value will lie below the population mean, or to the left of the “bell” center.

Like a normal distribution, a standard normal distribution (SND) will be visually symmetrical.

Why Are Z-Scores Important?

Z-scores can be extremely useful when comparing data sets. The standardization of results allows:

1. The calculation of probability. How likely is it for a specific score or result to occur within the normal distribution? For example, it could help determine how likely a score of 2200 on the SATs is.

2. The comparison of scores from different samples. The standardized values allow for two data points from different normal distributions, which may have different means or standard deviations, to be compared with ease. For example, you could compare LSAT scores and GPAs or SATs vs. ACTs.

In the simplest of terms, it simplifies.

Calculating Z-Score with the Z-Score Formula

To calculate z-score, you will need to know the value of the following variables:

1. Raw score. This will be represented as “x” in the z-score formula.

2. Population mean. This will be represented as “μ” (the Greek letter Mu) in the z-score formula.

3. Population standard deviation. This will be represented as “σ” (the lowercase Greek letter Sigma) in the z-score formula.

Z-score is calculated by taking the raw score minus the population mean, divided by the population standard deviation. The formula is:

z = (x-μ)/σ

This can also be represented plainly as:

z = (data point – mean)/standard deviation

Or as:

z = (raw score – mean)/standard deviation

If the population mean or standard deviation are unknown, the z-score can be calculated using the sample mean, x̄ (pronounced x bar), and sample standard deviation, s, as estimations of the population values.

z = (x̄-μ)/s

You can calculate z-score by hand (if you’re mathematically inclined), by using Microsoft Excel, by using a Ti-83 calculator, or by using one of the many online z-score calculators available.

Example Answer: Sample Z-Score: MCAT

Let’s use the MCAT exam as an example. You are an aspiring doctor preparing your med school applications. First, you need to take the MCAT examination. You studied for months and managed to score a 510. Is that a good score?

The MCAT is scored on a scale for each section (118-132) and the exam as a whole (472-528).

According to AAMC, the makers of the exam, the mean MCAT score for the 2020-21 testing cycle was 506.4 with a standard deviation of 9.2.

1. Step 1. Input your x-value into the z-score formula. In this example, that is your MCAT score: 510.

   z = (510-μ)/σ

2. Step 2. Input your population mean, μ, into the z-score formula. For our example, this is the mean MCAT score: 506.4.

   z = (510-506.4)/σ

3. Step 3. Input the standard deviation, σ, into the z-score formula. According to the AAMC, this is 9.6.

   z = (510-506.4)/9.6

4. Calculate. Follow the order of operations, use your calculator or Excel, and find the z-score.

   z = (510-506.4)/9.6
   z = (3.6)/9.6
   z = 0.375

   Your z-score, 0.375, indicates that your score was 0.375 standard deviations above the mean. If you were using a z-table, you would be able to see what percentage of test-takers you scored above or below.

Interpreting Z-Score

A z-score will tell you whether or not a score is typical for a data set. It can also tell you how well you did on an exam or in a course.

The value of your z-score will tell you how many standard deviations you are away from the mean. Below SD indicates standard deviation:

The mean in any standard normal distribution (SND) will be represented as zero. The standard deviation (SD) of any standard normal distribution (SND) will always equal one. The standard deviation simply represents the number, or distance, that a data point is from the population mean.

So, one standard deviation of the data point, or raw score, can be converted to one z-score unit. The standard deviation can fall anywhere between +3 and -3, very rarely falling outside of that range.

Z-scores will always be expressed in terms of standard deviations from the means of the data set. In our example, our z-score indicates that you were only 0.375 standard deviations from the mean. While a score of 510 is decent, it is not far above the mean and may limit your opportunities for med school acceptance.

It should be noted that while z-scores standardize data sets and make interpretation simpler, they are only as accurate as the data that is input. Incorrect data can skew Z-scores.

Z-Score Uses in Real Life

If you are learning about a topic in your statistics class, then it is likely that a use for it exists outside an academic setting. Sometimes not too far, though.

One of the common uses of z-scores is to determine standardized scores on exams like the SAT, MCAT, LSAT, GRE, or GMAT.

It can also be used in many research-related fields and situations.

You may see z-scores used to calculate weight-for-height or weight-for-age scores, typically in children.

It is often used in finance to determine things like market volatility. Edward Altman, a professor at New York University, developed his version of the z-score formula in the late 1960s to determine how close a company was to bankruptcy.

There are plenty of uses for z-score outside of the classroom. It is very commonly used in research and sometimes used in business settings.

Z-Score vs. T-Score

Both t-scores and z-scores are used to convert individual raw scores into a more standardized form. It can cause a lot of confusion for students when determining which formula to use.

Typically, you would use a t-score if the sample size or population is below 30 and there is an unknown standard deviation. Otherwise, the z-score is standard.

Author

Samantha Goddiess

Samantha is a lifelong writer who has been writing professionally for the last six years. After graduating with honors from Greensboro College with a degree in English & Communications, she went on to find work as an in-house copywriter for several companies including Costume Supercenter, and Blueprint Education.