Z-Score Calculator

The Z-score (also called the standard score) indicates how many standard deviations a value is above or below the mean. It's a dimensionless number: positive values indicate that the value is above the mean, negative values indicate it's below the mean, and zero means it's exactly equal to the mean.

Because the z-score removes the original unit of measurement, you can compare very different values on the same scale. Once they are expressed as z-scores, test results and physical measurements can be directly compared with each other.

Formula for z-score:

For a single value from a population whose mean and standard deviation are known, the standardized value is:

In this formula x is the original value, μ (the Greek letter mu) is the population mean and σ (sigma) is the population standard deviation. Subtracting the mean measures how far the value is from the center, and dividing by the standard deviation expresses that distance in units of the standard deviation.

The same equation can be used to calculate four different quantities.

The identity above links four quantities together so that if you know three of them, you can calculate the fourth. This calculator solves the equation for the quantity you select:

Computing x from a z-score is an inverse lookup that is required when a particular cut-off point is given as a z-score. For example, an admissions rule that takes the top percent actually refers to a z-score, and converting it back to the original value will determine who has met the admission threshold.

An example:

Suppose the test score is 85, the mean is 75 and the standard deviation is 10. These are the values that the calculator will load by default.

A z-score of 1 means that the value is exactly one standard deviation above the mean. In a normal distribution this position is approximately at the 84th percentile, meaning it exceeds about 84% of the values in the group.

Version for sample means.

When you are looking at the mean of a sample and not individual values, the spread in the mean is less than the spread in individual values. You need to divide by the standard error; as your sample size goes up, this standard error reduces that spread.

Suppose a lot of 16 bolts has a target mean of 10 mm and a standard deviation of 0.4 mm. If the actual mean for this lot is 10.2 mm, with a standard error of 0.1, then the z-score would be 0.2 divided by 0.1, or 2. A seemingly small difference in raw values corresponds to a change of two standard errors when sample size is taken into account.

When you switch to the sample mean mode, the calculator adds a field for the sample size and uses this formula. The t-distribution is more accurate for small samples with unknown population standard deviation because it accounts for the extra uncertainty that comes from estimating the spread from the sample itself.

From Z-scores to percentiles.

Z-scores and percentiles describe the same position from two different perspectives. A percentile is the area under the standard normal distribution to the left of a z-score. The calculator gives probabilities for the left, right, or both sides.

There are some important values to note: a z-score of 0 is the 50th percentile, 1 is approximately the 84th percentile, 2 is approximately the 98th percentile and -1 is approximately the 16th percentile. A complete lookup table on the calculator lists the most commonly used values.

Rules of thumb

For distributions that are approximately normal, the "68–95–99.7 rule" can quickly give an idea of a range of values. Roughly 68% of values lie within one standard deviation of the mean, about 95% within two standard deviations, and roughly 99.7% within three standard deviations.

You can also check any arbitrary range by enabling the option for area between two z-scores. The calculator will load the range of -1 to 1 by default and return the known value of 0.6827.

Interpretation of z-scores

The sign indicates the direction while the magnitude indicates how unusual a value is. The following table lists some commonly used thresholds.

Application areas for z-values

In education, standardized tests are reported with z-scores and percentiles so that the results can be compared to all other test takers rather than just being a raw score.

In quality control, the z-score indicates when a measured value deviates from the target value outside of the range of normal fluctuations.

In data analysis, standardizing each column to a z-score allows all features to be on the same scale. Many statistical and machine learning procedures require this before comparing or combining features.

Situations where a z-score can be misleading.

Distorted data.Interpretation of percentiles is complicated by the fact that the data may not be normally distributed. For highly skewed distributions such as incomes or reaction times, the percentile calculated in this manner may misrepresent the "typical" value shown by the data.

The sample size is very small.If the number of observations is small, say less than 30, then the sample standard deviation can only be considered as a rough estimate. The t-distribution takes additional uncertainties into account compared to normal z-values.

The standard deviation is wrong.Confusing sample standard deviations with population standard deviations or using a standard deviation from the wrong reference group will result in a z-score that looks impressive but is actually flawed.

Extreme outliers.A single extreme value can increase the standard deviation and thus decrease the z-scores of all others, possibly masking outliers you are looking for.

Symbols used

The table below lists the symbols used in the above formulas.

This calculator is a tool for general educational purposes and analysis. It assumes normal distribution when converting Z-scores to percentiles, so the percentiles should be considered as approximations for small samples or highly skewed data.