How To Calculate Z-Score (With Examples)

Statistical concepts play a crucial role in both the business and academic landscapes, even if you didn’t enjoy your statistics class. One such essential concept is the z-score, also known as the standard score.

In this article, you’ll discover what a z-score is, how to calculate and interpret it, and its practical applications in the real world.
Key Takeaways

• A z-score quantifies the distance between a data point and the mean of a dataset.

• Various professionals, including scientists, healthcare providers, and educators, utilize z-scores to compare specific data points against their datasets.

• One common application of z-scores is in evaluating how an individual’s standardized test score compares to those of other test-takers.

What is a Z-Score (Standard Score)?

A z-score, or standard score, is a numerical representation indicating how many standard deviations a data point is from the mean of a population. It standardizes raw scores within a normal distribution.

Z-scores indicate the precise number of standard deviations a data point is above or below the population mean, where standard deviation measures the variability within a dataset.

To calculate the standard deviation, first determine the distance between each data point and the population mean. In a standard normal distribution (SND), this value is consistently represented.

The z-score reflects the number of standard deviations for each data point within a standard normal distribution, where the standard deviation is always one. Typically, z-scores range from +3 to -3; values outside this range are considered atypical.

A positive z-score indicates a value above the population mean, while a negative z-score signifies a value below it.

Similar to a normal distribution, a standard normal distribution is visually symmetrical.

Calculating Z-Score with the Z-Score Formula

To compute the z-score, you need to know the following variables:

1. Raw score. Denoted as “x” in the z-score formula.

2. Population mean. Represented as “μ” (the Greek letter Mu) in the z-score formula.

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

The z-score is calculated by subtracting the population mean from the raw score and then dividing by the population standard deviation:

z = (x-μ)/σ

This can also be expressed as:

z = (data point – mean)/standard deviation

Or as:

z = (raw score – mean)/standard deviation

If the population mean or standard deviation is unknown, you can estimate the z-score using the sample mean (x̄) and sample standard deviation (s):

z = (x̄-μ)/s

You can calculate z-scores manually, use software like Microsoft Excel, employ a Ti-83 calculator, or utilize any of the numerous online z-score calculators available.

Example Answer: Sample Z-Score: MCAT

Let’s consider the MCAT exam as an example. As an aspiring medical student, your first step is to take the MCAT. After months of preparation, you score a 510. How does that score measure up?

The MCAT is scored on a scale of 118-132 for each section, with an overall score range of 472-528.

According to AAMC, the average MCAT score for the 2022-2023 testing cycle was approximately 511, with a standard deviation of 9.2.

1. Step 1. Insert your x-value into the z-score formula. Here, that’s your MCAT score of 510.

   z = (510-μ)

2. Step 2. Substitute the population mean, μ, into the formula. In this case, the mean MCAT score is 511.

   z = (510-511)/σ

3. Step 3. Input the standard deviation, σ, into the formula. For our example, it is 9.2.

   z = (510-511)/9.2

4. Calculate. Follow the order of operations to find the z-score:

   z = (510-511)/9.2
   z = (-1)/9.2
   z = -0.109

   This z-score of -0.109 indicates that your score was slightly below the mean. By consulting a z-table, you can determine the percentage of test-takers you performed better than.

Interpreting Z-Score

A z-score provides insight into whether a score is typical within a dataset and indicates your performance on exams or in academic settings.

The z-score value reveals how many standard deviations you are from the mean. Below is a breakdown of standard deviations:

In any standard normal distribution (SND), the mean equals zero, while the standard deviation (SD) is always one. The standard deviation quantifies the distance of a data point from the mean.

Thus, one standard deviation of a data point translates to one z-score unit, with the standard deviation typically falling within the range of +3 to -3.

Z-scores indicate how far a data point is from the data set’s mean. In this example, a z-score of -0.109 suggests that a score of 510 is slightly below the mean, which may affect your chances of medical school acceptance.

It’s crucial to remember that while z-scores standardize datasets for easier interpretation, their accuracy relies on the quality of the input data; incorrect data can lead to misleading z-scores.

Why Are Z-Scores Important?

Z-scores are invaluable when it comes to comparing datasets. The standardization of results allows:

1. The calculation of probability. You can assess the likelihood of a specific score occurring within a normal distribution, like determining the probability of achieving a score of 2200 on the SAT.

2. The comparison of scores from diverse samples. Standardized values facilitate the comparison of data points from different normal distributions, even when their means or standard deviations differ. For instance, this allows for comparisons between LSAT scores and GPAs or SATs compared to ACTs.

In essence, z-scores simplify analysis.

Z-Score Uses in Real Life

Understanding z-scores is not limited to academia; they have practical applications in various fields:

• Comparing test scores. Z-scores are frequently used to evaluate and compare standardized test scores on exams like the SAT, MCAT, LSAT, GRE, or GMAT.

• Benchmarking children’s growth. Z-scores are useful for calculating growth metrics, such as weight-for-height or weight-for-age scores in children, aiding doctors and parents in monitoring development.

• Predicting financial trends. Financial analysts use z-scores to evaluate market volatility. Edward Altman developed a version of the z-score formula in the late 1960s to assess the bankruptcy risk of companies.

• Conducting biological or zoological research. Scientific researchers utilize z-scores to analyze measurements of plants and animals. For instance, z-scores are used to compare the size of a newborn elephant to other newborns, helping identify healthy growth patterns.

• In healthcare. Medical professionals use z-scores to compare vital signs, such as blood pressure, to population norms and individual medical history, assisting in identifying potential health issues.

Z-Score vs. T-Score

Both t-scores and z-scores are utilized to standardize raw scores, which can lead to confusion regarding their appropriate applications.

Generally, a t-score is preferred when the sample size is below 30 and the standard deviation is unknown; otherwise, the z-score is commonly used.

Final Thoughts

If you’ve experienced a traditional American education, you’re likely familiar with the concept of the bell curve. While your memories of it may vary, this statistical principle remains relevant.

Historically, the bell curve was employed to calculate grading scales, often sparking debate about its fairness. Teachers might curve grades when class performance fell below expectations, often basing curves off the highest scores.

The bell curve represents normal distribution, where data is symmetrically distributed with an equal mean, median, and mode. The mean sits at the curve’s peak, with half the data points below and half above it.

This symmetrical representation of data resembles a bell, hence its name. When converting scores in a normal distribution to z-scores, we achieve a standard normal distribution (SND), characterized by a mean of zero and a standard deviation of one.

As the title suggests, z-scores are instrumental in standardizing distributions. They don’t alter the data; rather, they rescale it for clearer interpretation.

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.