A z-score tells you how many standard deviations a value lies above or below the mean. Enter a value, the mean and the standard deviation to get the z-score and the percentile it corresponds to.
Enter the value, mean and standard deviation.
Read the z-score and its percentile.
Z-score: z = (x − μ) ÷ σ
Percentile: P = Φ(z) × 100
A z-score tells you how many standard deviations a value sits above or below the mean. A z of 0 is exactly average, +2 is two standard deviations above, and −1 is one below. This calculator takes a value, the mean and the standard deviation, gives the z-score, and then converts it into a percentile using the normal curve.
The z-score is the simplest way to compare numbers measured on different scales. Suppose a student scored 74 in one exam and 81 in another. Which is better depends on how everyone else did. Standardising both marks puts them on a common ruler. Psychometric tests, growth charts, quality control limits and many entrance exam normalisation methods use this idea.
1. Find the mean μ and the standard deviation σ of the group the value belongs to.
2. Subtract the mean from the value: x − μ.
3. Divide by the standard deviation: z = (x − μ) ÷ σ.
4. Read the sign: positive means above the mean, negative below.
5. Look up Φ(z), the area to the left of z under the standard normal curve, in a z-table.
6. Multiply by 100 to express it as a percentile, the percentage of values expected below x.
Subtracting the mean moves the centre of the data to zero. Dividing by the standard deviation rescales the spread so that one unit equals one standard deviation. Whatever the original units, marks, centimetres or rupees, the standardised values always have mean 0 and standard deviation 1. That is why a z-score has no units and why values from completely different scales can be compared. The shape of the distribution does not change, though: standardising a skewed distribution still leaves it skewed.
If the underlying data is roughly normal, the z-score maps to a percentile through the standard normal cumulative function Φ(z). This function has no simple formula, so tables and software approximate it numerically; the calculator does this for you. Some reference points: z = 0 is the 50th percentile, z = 1 about the 84th, z = 1.645 the 95th, z = 1.96 the 97.5th and z = −1 about the 16th. The percentile is only as good as the normality assumption, so treat it as approximate for skewed data such as incomes.
Strictly, z uses the population mean and standard deviation. In practice you often plug in the class average and the sample SD, which works well for large groups. When you standardise a sample mean rather than a single value, the divisor becomes the standard error σ ÷ √n, which is the basis of the z-test. If σ is estimated from a small sample, the t distribution replaces the normal curve, because the estimate itself adds uncertainty.
Sanjay scored 74 in a college statistics test where the class mean was 61 and the standard deviation was 9. He wants to know how far above average he is and roughly what share of the class scored below him.
z = (x − μ) ÷ σ: z = (74 − 61) ÷ 9 = 1.4444
Area to the left under the standard normal curve: Φ(1.4444) = 0.9257 → 92.57th percentile
Answer: Z-score 1.4444; Percentile 92.57%
Dividing by the variance instead of the standard deviation.
Reversing the subtraction, μ − x, which flips the sign of z.
Reading the percentile as a percentage score; the 90th percentile means doing better than about 90% of people, not scoring 90%.
Trusting the percentile for strongly skewed or small data sets, where the normal curve is a poor fit.
Comparing z-scores from groups that are not comparable, such as a small coaching batch and a national exam.
Comparing a student's performance across subjects with different averages and spreads.
Converting raw test scores into standard scores in psychology and aptitude testing.
Flagging unusual readings in quality control, often values beyond z = ±3.
Interpreting child growth measurements expressed as z-scores against reference charts.
Detecting outliers in data science before building models.
Can a z-score be negative?
Yes, a negative z means the value is below the mean.
Does the percentile assume a normal distribution?
Yes. For skewed data the percentile will only be approximate.