Z-Scores Explained: How to Standardize Data

How do you compare a student's score on a math test graded out of 50 to their score on an English test graded out of 100? The raw numbers are meaningless side by side because the scales are different. Z-scores solve this problem by converting any data point into a standardized value that tells you exactly where it falls relative to its distribution — regardless of the original units.

This guide explains what z-scores are, how to calculate them, how to use the standard normal (z) table, and how z-scores apply in real-world scenarios from academics to finance to quality control.

What Is a Z-Score?

A z-score (also called a standard score) tells you how many standard deviations a data point is from the mean of its distribution. A positive z-score means the value is above the mean, a negative z-score means it is below the mean, and a z-score of zero means the value equals the mean exactly.

The power of z-scores lies in standardization. Once you convert different measurements to z-scores, they all share the same scale: mean of 0, standard deviation of 1. This makes direct comparisons possible across completely different data sets, units, and distributions.

The Z-Score Formula

The formula has an elegant interpretation: the numerator (x minus the mean) measures how far the data point is from the center, and dividing by the standard deviation expresses that distance in standardized units. A value 10 points above the mean has a z-score of 1.0 if the standard deviation is 10, but a z-score of 2.0 if the standard deviation is only 5.

Step-by-Step Calculation Examples

Understanding the Standard Normal Distribution

When you convert data to z-scores, you transform the original distribution into the standard normal distribution — a bell curve with a mean of 0 and a standard deviation of 1. This universal distribution allows you to determine the percentage of values above, below, or between any z-score values.

The standard normal distribution has several key properties:

• It is perfectly symmetric around z = 0
• The total area under the curve equals 1 (representing 100% of all data)
• 50% of values fall below z = 0 and 50% fall above
• 68.27% of values fall between z = -1 and z = +1
• 95.45% of values fall between z = -2 and z = +2
• 99.73% of values fall between z = -3 and z = +3

Using the Z-Table

A z-table (also called the standard normal table) shows the cumulative probability from the far left of the distribution up to a given z-score. Here is how to read it:

1. Find your z-score's row. The rows represent the ones and tenths digits. For z = 1.96, find the row for 1.9.
2. Find the hundredths column. For z = 1.96, find the column for 0.06.
3. Read the intersection. The value at row 1.9 and column 0.06 is 0.9750, meaning 97.50% of data falls below z = 1.96.
4. Calculate tail probability. To find the area above z = 1.96: 1 - 0.9750 = 0.0250, or 2.50%.

For negative z-scores, the table works the same way due to symmetry. The area below z = -1.96 equals the area above z = +1.96, which is 2.50%.

Practical Applications of Z-Scores

Z-scores appear across many fields. Here are the most common real-world applications.

Z-Score Reference Table

This quick reference table shows commonly used z-scores, their cumulative probabilities, and their two-tailed significance levels (used in hypothesis testing and confidence intervals).

Tips for Working with Z-Scores

Z-scores are a foundational statistical concept. These tips will help you use them correctly and effectively in your analyses.

• Always verify the mean and standard deviation first. A z-score is only as accurate as the mean and standard deviation used to calculate it. If these summary statistics are distorted by outliers or calculated from insufficient data, your z-scores will be misleading. Check for outliers and verify your data before standardizing.
• Use z-scores for cross-variable comparison. Whenever you need to compare measurements on different scales — test scores in different subjects, performance metrics in different units, financial returns from different asset classes — z-scores put everything on a common scale. This is their primary practical value.
• Remember that z-scores preserve the shape of the distribution. Standardizing data changes the scale (centering at 0, scaling by standard deviation) but does not make non-normal data normal. If the original data is skewed, the z-scores will also be skewed. The z-table percentages only apply if the underlying data is approximately normal.
• Apply z-scores to detect outliers systematically. Rather than relying on visual inspection, use z-scores to identify data points beyond ±2 or ±3 standard deviations. This provides a consistent, objective criterion across different data sets and different analysts.
• Combine z-scores cautiously. If you average z-scores from different tests (as in creating a composite score), the average is meaningful only if the tests are reasonably comparable in what they measure. Averaging a z-score from a math test and a z-score from a personality questionnaire produces a number without clear interpretation.

Common Mistakes to Avoid

• Applying z-table probabilities to non-normal data. The standard normal table gives exact probabilities only for normally distributed data. If your data is heavily skewed or has multiple peaks, the percentages derived from z-scores will be inaccurate. Use non-parametric methods or transform the data before standardizing in these cases.
• Confusing z-scores with z-tests. A z-score converts a single data point to a standardized value. A z-test is a hypothesis testing procedure that uses the standard normal distribution to evaluate whether a sample statistic differs significantly from a hypothesized value. The z-test uses z-scores as test statistics, but the two concepts are distinct.
• Forgetting that z-scores are relative measures. A z-score of 2.0 on one test is not the same achievement as a z-score of 2.0 on a different test unless the distributions are comparable. A z-score of 2.0 in a highly competitive group is a greater absolute achievement than a z-score of 2.0 in a less competitive group.
• Standardizing when the raw scale is meaningful. If the original units have practical meaning (dollars, centimeters, minutes), converting to z-scores removes that interpretability. Use z-scores for comparison, but report results in original units when the audience needs practical context.
• Using z-scores with extremely small samples. Z-scores calculated from small samples (under 10) are unreliable because the sample mean and standard deviation are poor estimates of the population parameters. For small samples, use the t-distribution instead of the standard normal distribution for inference.

Last updated: February 23, 2026