Z-Scores Explained with Worked Examples

What a z-score measures

A raw score on its own says little. Scoring 85 on a test sounds good, but if everyone else scored 95 it is actually weak. A z-score, also called a standard score, fixes this by measuring a value relative to its group. It answers one question: how many standard deviations is this value from the mean?

Here $x$ is the value, $\mu$ is the mean and $\sigma$ is the standard deviation. A positive z-score means the value is above the mean, a negative one means below, and zero means exactly average. A z-score of 2 means two standard deviations above the mean, which is unusually high in most data sets. If you need a refresher on computing the standard deviation itself, see how to calculate standard deviation.

Calculating a z-score step by step

The units always cancel, because the numerator and denominator are both measured in the original units. That is what makes z-scores unitless and comparable across completely different measurements.

Comparing results on different scales

The most practical use of z-scores is fair comparison. Suppose a student scores 82 on a chemistry exam where the mean is 75 and the standard deviation is 5, and 88 on a history exam where the mean is 80 and the standard deviation is 8. The history mark is higher, but which result is more impressive?

Relative to their classmates, the chemistry result is stronger: 1.4 standard deviations above average compared with 1.0. The same idea lets scientists compare measurements from different instruments, and lets analysts compare a company's performance against industries with very different typical values.

Working backwards from a z-score

Rearranging the formula gives the raw value that corresponds to a given z-score:

On the test with mean 70 and standard deviation 10, a z-score of $-1.2$ corresponds to $70 + (-1.2)(10) = 58$. This form is useful for setting cut-offs, for example deciding which raw score a student needs to land in the top 10 percent, as Example 4 below shows.

From z-scores to percentiles

When data follow a normal distribution, the bell curve, a z-score can be converted into a percentile, the proportion of values that fall below it. The standard normal distribution has mean 0 and standard deviation 1, and its cumulative function, written $\Phi(z)$, gives that proportion. A few landmark values are worth remembering:

• $\Phi(0) = 0.5$: the mean sits at the 50th percentile.
• $\Phi(1) \approx 0.8413$ and $\Phi(-1) \approx 0.1587$.
• $\Phi(1.5) \approx 0.9332$, so the student in Example 1 scored higher than about 93 percent of the class.
• $\Phi(-0.8) \approx 0.2119$, so the student in Example 2 is near the 21st percentile.
• $\Phi(1.96) \approx 0.975$, which is why 1.96 appears in 95 percent confidence intervals.

The empirical rule summarises the bell curve: about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two and about 99.7 percent within three. Values with $|z| > 3$ are therefore rare and are often investigated as possible outliers.

Z-scores for sample means

Averages vary less than individual values. If samples of size $n$ are drawn from a population with mean $\mu$ and standard deviation $\sigma$, the sample mean has standard deviation $\frac{\sigma}{\sqrt{n}}$, called the standard error. The z-score of a sample mean is

For example, if 25 students from the test above average 73, the standard error is $\frac{10}{\sqrt{25}} = 2$ and $z = \frac{73 - 70}{2} = 1.5$. A three-point lift is ordinary for one student but noticeably unusual for an average of 25 students. This formula is the starting point for z-tests and confidence intervals in introductory statistics.

Cautions when using z-scores

• Converting z-scores to percentiles with $\Phi$ is only valid when the data are approximately normal. For strongly skewed data, such as incomes, percentiles must come from the data themselves.
• Use the population standard deviation when you know it and the sample standard deviation as an estimate otherwise; small samples call for the t-distribution instead.
• A z-score describes position within one group. Comparing z-scores from groups with very different ability levels can still mislead.
• Keep the sign. A z-score of $-2$ and one of $2$ are equally unusual but mean opposite things.

The z-score calculator computes z-scores, percentiles and the area between two values, and the standard deviation calculator gives the mean and standard deviation you need first.

Where z-scores appear in practice

Standardised test reports often convert raw marks into scaled scores or percentiles that are built on the same idea as a z-score. Paediatricians track a child's height and weight against growth charts, and some clinical tools report those measurements directly as z-scores for the child's age. Bone density scans report a T-score, which is a z-score measured against healthy young adults. Quality engineers flag parts whose measurements drift more than three standard deviations from the target, and finance analysts standardise returns so that funds with different risk levels can be compared on one scale.

Practice problems

1. A data set has mean 50 and standard deviation 4. Find the z-score of 56. Answer: $z = \frac{56 - 50}{4} = 1.5$.
2. Scores have mean 100 and standard deviation 15. Which score has $z = -2$? Answer: $100 + (-2)(15) = 70$.
3. Values are normal with mean 500 and standard deviation 100. What proportion lie below 650? Answer: $z = 1.5$, so about $\Phi(1.5) \approx 0.9332$, or 93 percent.

Frequently asked questions