Z Score Calculator

A z score says how many standard deviations a value sits from the mean. The z score formula is one subtraction and one division: z = (x - μ) / σ, where x is the value, μ is the mean and σ is the standard deviation. A test score of 85 in a class with a mean of 70 and a standard deviation of 10 has z = (85 - 70) / 10 = 1.50. A value below the mean gets a negative z, and the mean itself scores 0.

The z score becomes a percentile through the standard normal curve. If the scores are roughly normal, the area below z = 1.50 is 0.9332, so 85 beats about 93.32% of the class and sits at the 93rd percentile. The area to the right is what is left over: 0.0668.

A z-score table lists the area to the left of z. Round z to two decimals. The row gives the whole number and first decimal, the column gives the second decimal, and the cell where they cross is the area. For z = 1.96 that is row 1.9 and column 0.06, which reads 0.9750. Negative values have their own half of the table, and the curve is symmetric, so the area left of -z equals the area right of z. The table under the result switches halves to match the sign of your z.

Areas between two values come from the same curve. Between z = -1 and z = 1 lies 68.27% of it, between -2 and 2 lies 95.45%, and between -3 and 3 lies 99.73%. Those are the exact figures behind the 68-95-99.7 rule. For any two z values, the area between them is the left area at the larger one minus the left area at the smaller one.

Going the other way, from a percentile to z, is the inverse normal. The 95th percentile is z = 1.6449, and the middle 95% runs from -1.96 to 1.96. That 1.96 is the critical value behind every 95% confidence interval.

The percentile reading holds only for data close to normal. Incomes and waiting times are skewed, and for them the share of values below a z score has to be counted from the data.