Z-Score Calculator
Enter a raw value, the mean, and the standard deviation to get the z-score — the single number that says how far a value sits from average, measured in standard deviations.
Three numbers in, one z-score out
Enter the raw value, the mean, and the standard deviation and the calculator returns the z-score, z = (x − μ) ÷ σ.
Same scale matters
The value, mean, and standard deviation must come from the same scale — mixing units gives a meaningless z-score.
What is a z-score?
Distance from the mean, in standard deviations
A z-score, also called a standard score, expresses how far a single value lies from the mean of its distribution — not in raw units, but in standard deviations. A z-score of 2 means the value is two standard deviations above average; a z-score of −1 means it is one standard deviation below. Because the original units cancel out, z-scores put very different measurements on one common footing: you can compare a test score, a height, and a reaction time directly once each is converted to a z-score. That makes the z-score the workhorse of statistics for spotting outliers, standardizing data, and reading probabilities off the normal curve.
Enter the raw value, the mean, and the standard deviation to get the z-score instantly.
One short formula: subtract the mean, then divide by the standard deviation.
z = (x − μ) ÷ σHere x is the raw value, the Greek letter μ (mu) is the mean, and σ (sigma) is the standard deviation. The numerator (x − μ) is the raw distance from the mean, and dividing by σ rescales that distance into standard-deviation units. The standard deviation must be greater than zero, otherwise the formula would divide by zero. A value exactly at the mean gives a z-score of 0.
Suppose a value of 85 comes from a distribution with a mean of 75 and a standard deviation of 5.
Note the value, mean, and spread
The raw value is 85, the mean is 75, and the standard deviation is 5 — all on the same scale.
Subtract the mean
85 − 75 = 10 — the raw distance above the mean.
Divide by the standard deviation
10 ÷ 5 = 2 — the value sits two standard deviations above the mean.
The single z-score figure carries a lot of meaning once you know how to read it. The sign tells you the direction: a z-score of 0 is exactly average, a positive z-score sits above the mean, and a negative one sits below — so −1.5 is one and a half standard deviations under average. The size tells you how unusual the value is. In a roughly normal distribution about 68% of values fall within ±1, about 95% fall within ±2, and about 99.7% fall within ±3, so a value with a z-score near 0 is utterly ordinary while one beyond about ±3 is rare — fewer than 1 in 300 values reach that far. Many fields treat a z-score past ±2 as noteworthy and past ±3 as a likely outlier worth a second look. Because z-scores strip out the original units, they also let you compare across scales: a student who is +1.8 in maths but −0.4 in reading is clearly stronger in maths relative to peers, even though the two tests use entirely different point systems. Always judge a z-score against the shape of your data — the percentage rules above hold tightly only when the distribution is close to normal.
The formula is exact, but a few practical points are worth keeping in mind.
Same scale, valid spread, and a near-normal shape
A z-score only makes sense when the value, mean, and standard deviation all come from the same distribution and the same scale — never mix units. The standard deviation must be greater than zero, since a spread of zero means every value is identical and no z-score is defined. The familiar 68–95–99.7 percentages apply only when the data is roughly normal; for skewed or heavy-tailed distributions a given z-score can be far more or less common than the normal curve suggests, so use it as a guide, not a guarantee.