What you will learn
- Compute a z-score and explain what it measures
- Use z-scores to compare values on different scales
- Understand what a percentile tells you about a value
- Apply these tools to spot unusual outcomes in finance
The empirical rule kept counting standard deviations away from the mean. The z-score turns that counting into an exact, formal number, and it is one of the handiest tools in statistics. It lets you say how unusual any single value is, and it lets you compare values that come from completely different scales. This lesson makes that concrete.
Z-scores and standardizing (Khan Academy)
Introduces the z-score as the number of standard deviations a value sits from the mean. Focus on the simple formula and what a positive or negative z means.
What a z-score is
A z-score answers one question: how many standard deviations is this value from the mean? You compute it by taking the value, subtracting the mean, and dividing by the standard deviation. The sign tells you the direction. A positive z-score means the value is above the mean, and a negative one means it is below. The size tells you how far. A z-score of 0 sits exactly at the mean, a z-score of 1 is one standard deviation above, and a z-score of negative 2 is two standard deviations below. This is called standardizing, because it re-expresses any value in the universal language of standard deviations.
The formula in symbols
- x = the value
- mean = the average
- SD = the standard deviation
You take the value, subtract the mean, and divide by the standard deviation. The subtraction measures how far the value sits from the center, and dividing by the standard deviation rescales that distance into a count of standard deviations. The worked example below applies it to a test score.
Computing a z-score
A student scores 85 on a test where the class mean is 70 and the standard deviation is 5. What is the z-score?
- Subtract the mean from the value. 85 minus 70 is 15.
- Divide by the standard deviation. 15 divided by 5 is 3.
- Interpret it. A z-score of 3 means the score is three standard deviations above the mean.
Why it matters: By the empirical rule, only about 0.3 percent of values sit beyond three standard deviations, so this student did exceptionally well relative to the class.
Calculate a z-score
A stock returns 60 in a period where similar stocks averaged 75 with a standard deviation of 5. What is the z-score of this stock's return?
Why standardizing is so useful
The real power of the z-score is comparison. Say one stock's return has a z-score of 2 within its peer group, and a completely different stock's return has a z-score of 0.5 within its own peer group. Even though the raw returns and the two groups are totally different, you can immediately say the first stock's result was far more unusual relative to its peers. Z-scores strip away the original units and scale, so numbers that were not comparable become directly comparable. This is exactly why standard deviation and z-scores show up everywhere risk and performance are measured.
Percentiles
A percentile is a closely related, more intuitive way to place a value. The percentile of a value is the percentage of the data that falls below it. If your test score is at the 90th percentile, you scored higher than 90 percent of everyone who took the test. The median, which you met earlier, is simply the 50th percentile: half the data is below it. Percentiles are popular because they are easy to explain to anyone, no standard deviations required. They are also widely used in finance, for instance in Value at Risk, which asks about a low percentile of the distribution of possible losses to describe a worst-case scenario.
Z-scores and percentiles (Simple Learning Pro)
Connects z-scores to percentiles and the normal curve. Watch how a z-score maps to the percentage of data below a value.
| Z-score | Roughly what it means |
|---|---|
| 0 | Right at the mean, the 50th percentile |
| +1 | One SD above the mean, about the 84th percentile |
| +2 | Two SD above, about the 97.5th percentile, unusual |
| -2 | Two SD below, about the 2.5th percentile, unusual |
| +3 or beyond | Extremely rare, top 0.15 percent |
A z-score turns any number into how many standard deviations it is from normal. It is how you compare things that were never on the same scale.
Compare across scales
Analyst A's fund beat its benchmark with a z-score of 1.0 relative to peer funds. Analyst B's fund beat its benchmark with a z-score of 2.5 relative to a different peer group. Whose result was more unusual relative to their peers?
A z-score of 2.5 sits farther into the tail than 1.0, so Analyst B's result is more unusual relative to their peers. Z-scores make different scales directly comparable.Explain standardizing
In your own words, explain why converting values to z-scores lets you compare results that come from different scales or groups.
Write an answer before comparing it with the model response.
Model answer
A z-score expresses a value as the number of standard deviations it sits from its own mean, which removes the original units and scale. Once two values are both measured in standard deviations from their own means, they are on the same universal scale, so I can compare how unusual each one is even if the raw numbers and the groups they came from are completely different. A larger z-score, in either direction, always means a more unusual value relative to its group.
You can now describe single values precisely and compare them across scales. Everything so far has assumed you knew the true mean and standard deviation. The next lesson faces reality: you almost never know them, and must estimate them from a sample.