Z-scores explained: how far from average is a value?

A z-score tells you how many standard deviations a value lies above or below the mean: z = (x − μ) ÷ σ. A z-score of 2 means two standard deviations above average; −1 means one below. For data that follow a normal distribution, the z-score converts straight into a percentile. The z-score calculator does both steps and draws the bell curve, and this guide explains the reasoning.

What a z-score tells you

Raw numbers are hard to judge on their own. A test mark of 72 could be excellent or ordinary depending on how everyone else did. A z-score fixes that by measuring distance from the mean in units of spread. Its sign tells you the direction (positive is above average, negative below), and its size tells you how unusual the value is. Because every z-score is on the same scale, values from completely different measurements, such as heights, exam marks and reaction times, can be compared directly.

Calculating a z-score

You need three numbers: the value x, the mean μ and the standard deviation σ. Subtract the mean, then divide by the standard deviation. On an IQ scale with mean 100 and standard deviation 15, a score of 130 gives z = (130 − 100) ÷ 15 = 2, and a score of 85 gives z = (85 − 100) ÷ 15 = −1. If you have raw data rather than a known mean and spread, the standard deviation calculator finds μ and σ first. Strictly, the formula uses the population mean and standard deviation; with a sample, you are estimating them, which matters most when the sample is small.

From z-score to percentile

If the data are normally distributed, the area under the bell curve to the left of z is the proportion of values below it, which is the percentile. That area comes from the normal cumulative distribution function, which has no simple formula, so it is read from tables or computed:

• z = 0 is the 50th percentile, exactly average;
• z = 1 is about the 84.13th percentile, and z = −1 about the 15.87th;
• z = 1.5 is about the 93.32nd percentile;
• z = 2 is about the 97.72nd percentile, so an IQ of 130 is higher than roughly 97.7% of scores.

It also works backwards. To find the cut-off for the top 10%, look up the z-score of the 90th percentile, about 1.2816, and convert: 100 + 1.2816 × 15 ≈ 119.2. The 95th percentile is z ≈ 1.645, and in hypothesis tests z = 1.96 marks the familiar two-tailed p-value of 0.05. For percentiles of raw data that are not normal, the percentile calculator ranks the actual values instead.

Comparing scores on different scales

Suppose a student scores 72 in maths, where the class mean is 60 and the standard deviation 8, and 80 in English, where the mean is 70 and the standard deviation 10. The English mark looks better, but the z-scores say otherwise: maths is (72 − 60) ÷ 8 = 1.5, about the 93rd percentile, while English is (80 − 70) ÷ 10 = 1.0, about the 84th. Relative to classmates, the maths result was stronger. This is why standardised tests and many grading schemes convert raw marks to z-scores or to scales built from them.

The 68–95–99.7 rule

In a normal distribution about 68.27% of values lie within one standard deviation of the mean (z between −1 and 1), 95.45% within two and 99.73% within three. So a value with |z| above 2 happens about 1 time in 22, and one above 3 about 1 time in 370. That makes z-scores a handy first check for outliers in measurements: a reading with z = 4 is either a genuine rarity or, more often, a recording error worth checking. Standard deviation explained: what it measures, how it's calculated, and the 68–95–99.7 rule goes deeper into the spread itself, and the confidence interval calculator uses the same z values to build ranges around an estimate.

When z-scores mislead

The percentile conversions are only as good as the normal assumption. Incomes, house prices and waiting times are skewed, with long tails on one side, so a z-score of 2 might be the 90th percentile rather than the 98th. Small samples make the mean and standard deviation themselves uncertain, and a single extreme value can inflate σ and shrink every other z-score. Look at the data first: if the median and mean are far apart, the distribution is probably skewed, as Mean, median and mode: which average to use, and when the mean lies explains, and ranking-based percentiles are safer.