Z Score Calculator

Using the Z score calculator

There are two modes, selected at the top of the form.

• Z score & P from score. Enter the raw score x, the mean μ and the standard deviation σ of the distribution (the defaults, 0 and 1, describe the standard normal curve). The table lists the z score and four probabilities: the area above the score, the area at or below it, the area outside z = ±|z| (the two-tailed p-value) and the area between −|z| and +|z|.
• Z from Probability. Enter a tail probability p. You get the one-tailed z score that leaves p in the upper tail, and the pair of two-tailed z scores that leave p/2 in each tail.

The Precision box sets how many decimals are printed, from 2 to 25. Values that are whole numbers, or that have no more than two decimals after rounding, print compactly, so 0.5 shows as 0.50 and 2 as 2.

In the two-tailed rows the labels echo the raw score you typed, but the probabilities are for z = ±|z|, that is, values at least |z| standard deviations from the mean in either direction.

What is a z score?

A z score (also called a standard score) says how many standard deviations a value lies above or below the mean of its distribution. A z score of 0 is exactly average, +1 is one standard deviation above, and −2.5 is two and a half below.

Because it removes the original units, a z score lets you compare things measured on different scales, such as an exam mark and a reaction time, and it is the bridge to probability: for normally distributed data, each z score corresponds to a known percentile. It is also the test statistic in a z-test and the building block of confidence intervals.

Z score formula

z = (x − μ) ÷ σ

To go the other way and recover the raw score from a z score, solve for x:

x = μ + z × σ

If you only know the variance, take its square root first to get σ. The probabilities then come from the standard normal cumulative distribution function Φ:

P(Z > z) = 1 − Φ(z), two-tailed p = 2 × (1 − Φ(|z|))

Z table

A z table lists the area under the standard normal curve for common values of z. The columns below give the cumulative area to the left, the upper-tail area (a one-tailed p-value) and the two-tailed p-value.

These are the critical z values used most often to define a confidence level or a significance threshold:

Because the curve is symmetric, the area to the left of −z equals the area to the right of z. For more distributions and critical values see the critical value calculator and the normal distribution calculator.

Example calculations

From a raw score. An exam has a mean of 70 and a standard deviation of 8, and you scored 82. Then z = (82 − 70) ÷ 8 = 1.5. The area above is 0.0668, so about 6.7% of students scored higher, and the area at or below is 0.9332, which puts you near the 93rd percentile. The two-tailed probability is 0.1336: that share of students would sit at least 1.5 standard deviations from the mean in either direction.

A low value. A lab result of 3.2 comes from a population with mean 5 and standard deviation 0.6. Here z = (3.2 − 5) ÷ 0.6 = −3. The area at or below is 0.00135 and the two-tailed probability is 0.0027, so a result this low occurs in roughly 1 in 740 cases by chance.

From a probability. For a one-tailed test at p = 0.05 the critical z is 1.6449. For a two-tailed test at p = 0.05, with 0.025 in each tail, the critical values are ±1.96. Convert a critical z back to the scale of your data with x = μ + zσ: with the exam above, the cut-off for the top 5% of marks is 70 + 1.6449 × 8 = 83.2.

Precision and accuracy

Areas under the curve come from the complementary error function, evaluated with rational Chebyshev approximations that are good to about 15 significant digits, including far out in the tails. The inverse (Z from Probability) starts from a rational first guess and sharpens it with one Halley correction step against the exact tail area, which brings it to the same accuracy. A z score of 0, for instance, gives exactly 0.50 on each side, and p = 0.025 gives a critical value of 1.959964 to six decimals.

The Precision box accepts up to 25 decimals, but a computer stores only about 16 significant digits, so anything printed beyond that is just the binary number written out in decimal and carries no extra information. For reporting, four to eight decimals is plenty. Once a score lies more than about 37 standard deviations from the mean the tail area is smaller than the smallest number a computer can hold, and it prints as 0.

Assumptions and common mistakes

• The data must be roughly normal for the probabilities to apply. A z score can be computed for any distribution, but the percentiles only hold for a bell shape.
• Use the t distribution for small samples when σ is estimated from the data rather than known.
• One-tailed versus two-tailed: pick the tail before you look at the data. The two-tailed p-value is exactly twice the one-tailed one.
• Variance is not standard deviation. Enter σ, the square root of the variance, in the standard deviation box.

To see how a z score becomes a p-value in a significance test, try the p-value calculator. The standard deviation itself can be found with the standard deviation calculator.