Tutorial
Understanding Z-Scores and Normal Distribution
Published 15 March 2026 · 9 min read
The normal distribution, often called the bell curve, is the most important probability distribution in statistics. It describes how data naturally clusters around a central value, with fewer observations appearing as you move further from the centre. Z-scores provide a way to measure exactly how far any data point sits from the mean, expressed in units of standard deviation.
This guide explains what z-scores are, how the normal distribution works, and how to use both in practical situations. You will learn the z-score formula, work through examples, and understand the empirical rule that makes the bell curve so useful.
What Is the Normal Distribution?
A normal distribution is a continuous probability distribution that is symmetric about its mean. Its shape is completely determined by two parameters: the mean (the centre) and the standard deviation (the spread).
The Normal Distribution PDF
While you rarely need to compute this formula by hand, it defines the bell-shaped curve that underpins much of statistical analysis.
Key Properties
- The curve is perfectly symmetric about the mean.
- The mean, median, and mode are all equal.
- The total area under the curve equals 1 (representing 100% probability).
- Approximately 68% of data falls within 1 standard deviation of the mean.
- The tails extend infinitely in both directions but never touch the x-axis.
The Empirical Rule (68-95-99.7)
The empirical rule gives a quick summary of how data is distributed in a normal distribution:
68%
of data falls within of the mean
95%
of data falls within of the mean
99.7%
of data falls within of the mean
This means that in any normally distributed dataset, it is extremely rare (only 0.3% of the time) for a data point to be more than 3 standard deviations from the mean.
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. It standardises values from any normal distribution to the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
The Z-Score Formula
- = the individual data point
- = the population mean
- = the population standard deviation
Interpreting Z-Scores
Positive z-score
The data point is above the mean. A z-score of +2 means the value is 2 standard deviations above the mean.
Negative z-score
The data point is below the mean. A z-score of -1.5 means the value is 1.5 standard deviations below the mean.
Worked Examples
Example 1: Exam Scores
A class has a mean exam score of 72 with a standard deviation of 8. A student scored 88. What is their z-score?
The student scored 2 standard deviations above the mean. Using the empirical rule, only about 2.5% of students scored higher.
Example 2: Heights
The heights of adult men in a population are normally distributed with a mean of 175 cm and a standard deviation of 7 cm. What is the z-score for a man who is 165 cm tall?
A z-score of -1.43 means this person is 1.43 standard deviations below average height. Using a z-table, approximately 7.6% of men are shorter than 165 cm.
Example 3: Comparing Scores Across Different Tests
A student scored 78 on a biology test (mean 70, SD 6) and 85 on a chemistry test (mean 75, SD 10). Which score is relatively better?
Despite the higher raw score in chemistry, the biology score is relatively better (z = 1.33 vs z = 1.00). The student performed further above the class average in biology.
Example 4: Finding a Percentage from a Z-Score
The weights of apples from an orchard are normally distributed with a mean of 150g and a standard deviation of 20g. What percentage of apples weigh less than 180g?
Looking up z = 1.5 in a standard normal table gives a cumulative probability of 0.9332. This means 93.32% of apples weigh less than 180g.
Using the Standard Normal Table
A z-table (also called the standard normal table) gives the cumulative probability from the far left of the distribution up to a given z-score. Here are some commonly used values:
| Z-score | Cumulative probability | Meaning |
|---|---|---|
| -2.0 | 0.0228 | 2.28% below this value |
| -1.0 | 0.1587 | 15.87% below this value |
| 0.0 | 0.5000 | 50% below (the mean) |
| 1.0 | 0.8413 | 84.13% below this value |
| 1.96 | 0.9750 | 97.5% below (used for 95% CIs) |
| 2.0 | 0.9772 | 97.72% below this value |
Common Mistakes
Confusing z-scores with percentiles
A z-score of 1.0 does not mean the 1st percentile. It corresponds to the 84th percentile. You need a z-table or calculator to convert between them.
Applying z-scores to non-normal data
Z-scores assume the data follows a normal distribution. If the data is heavily skewed, z-scores and the empirical rule will not give accurate results.
Using the sample mean/SD for population z-scores
When working with a sample, be clear about whether you are using the sample statistics or the known population parameters. The interpretation changes accordingly.
Forgetting the sign
A data point below the mean gives a negative z-score. Losing the negative sign will place your value on the wrong side of the distribution.
Try It Yourself
Calculate standard deviations and explore the spread of your own datasets with our free standard deviation calculator. It computes both population and sample statistics instantly.
Open Standard Deviation CalculatorRelated Articles
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