Free z-score calculator. Enter a raw score, mean, and standard deviation to get the z-score, percentile rank, and the probability of scoring above or below it.
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Enter a raw score x, the mean of its distribution, and the standard deviation. The calculator standardizes the score with z = (x - mean) / sd, which says how many standard deviations the value sits above or below the mean. It then feeds z through the standard normal cumulative distribution to get the percentile rank (the share of values below yours) and the matching tail probabilities. The empirical rule says about 68 percent, 95 percent and 99.7 percent of values fall within one, two and three standard deviations of the mean. A z-score of 1.96 corresponds to the 95 percent confidence interval. To convert a raw score to a percentile, first calculate the z-score, then look up its area under the normal curve.
A z-score restates any value as a distance from the mean, measured in standard deviations. The formula is z = (x - mean) / standard deviation.
Subtracting the mean centers the value on zero; dividing by the standard deviation puts every distribution on the same scale. A z-score of 1 always means one standard deviation above average, whether the raw numbers are exam marks, heights, or blood pressure readings.
Positive z-scores sit above the mean, negative z-scores sit below it, and a z-score of 0 is exactly average.
Enter the raw score x, the mean of the distribution it came from, and the standard deviation (which must be greater than zero). The calculator returns the z-score, the percentile rank, and the probability of a value scoring above or below yours, plus a plain-words interpretation.
Use the population values when you have them. If you only have a sample, the same formula works as an approximation, but say so when you report the result.
To go the other way, from a percentile back to a raw score, rearrange the formula: × = mean + z × standard deviation (the inverse calculation).
The percentile is the area under the normal curve to the left of your z-score, a quantity written Phi(z) and once looked up in a printed z-table. A z-score of 0 is the 50th percentile, 1 is about the 84.13th percentile, 2 is about the 97.72nd percentile, and -1 is about the 15.87th percentile.
Do not confuse percentile with the percent above the score: they are complements. Scoring at the 84th percentile means about 16 percent scored above you, not 84 percent.
In a normal distribution, about 68.27 percent of values fall within one standard deviation of the mean (z between -1 and 1), about 95.45 percent fall within two standard deviations, and about 99.73 percent fall within three.
This 68-95-99.7 rule is a quick sanity check: if your z-score is 2.5, you already know you are beyond roughly 99 percent of the distribution before computing anything.
Z-scores use the normal distribution and fit when you know the population standard deviation or your sample is large. T-scores use the t-distribution, which has heavier tails, and fit small samples where the standard deviation itself is estimated from the data.
With large samples the two give nearly identical answers. With a class of 12 students and an estimated standard deviation, the t-score is the honest choice.
The classic error is reading the wrong tail: treating a left-tail percentile as the probability of scoring above. Another is forgetting that a negative z-score is perfectly valid: it only means below average, not a broken calculation.
Rounding the z-score before looking up the percentile can shift the answer, and applying normal-curve percentiles to skewed data produces percentiles that look precise but are not. Finally, keep sample and population standard deviations straight: they answer different questions.
A student scores 90 on an exam where the class mean is 80 and the standard deviation is 5. The calculation runs in three steps.
z = (90 - 80) / 5 = 10 / 5 = 2.00. The left-tail area Phi(2.00) is 0.9772, so the percentile rank is 97.72: the student outperformed about 97.72 percent of the class, and only about 2.28 percent scored higher.
This is exactly what this calculator returns for × = 90, mean = 80, standard deviation = 5.
A z-score, also called a standard score, tells you how many standard deviations a value is from the mean of its distribution. It is computed as z = (x - mean) / standard deviation, and it lets you compare values from completely different scales.
Subtract the mean from the raw score, then divide by the standard deviation. For a score of 85 with a mean of 75 and a standard deviation of 10: z = (85 - 75) / 10 = 1.0.
The value is below the mean. A z-score of -1 means one standard deviation below average. Negative z-scores are normal and valid, not errors.
The value equals the mean exactly. On the normal curve that is the 50th percentile: half the distribution sits below and half above.
Find the area under the normal curve to the left of the z-score and multiply by 100. A z-score of 1.28 has a left-tail area of about 0.8997, so it is about the 89.97th percentile.
For a two-sided 95% confidence interval the critical z-scores are plus and minus 1.96, leaving 2.5% in each tail. For a one-sided 95% cutoff the critical value is about 1.645. The 90% and 99% two-sided values are 1.645 and 2.576.
Yes. Any value below the mean produces a negative z-score. For example, a score of 60 with a mean of 70 and a standard deviation of 10 gives z = -1.0.
Also called the 68-95-99.7 rule: in a normal distribution about 68.27% of values lie within one standard deviation of the mean, 95.45% within two, and 99.73% within three.
Left-tail probability is the share of the distribution at or below your value; right-tail is the share at or above it. For the same z-score the two always add to 1.
A z-score uses the normal distribution and a known population standard deviation. A t-score uses the t-distribution with heavier tails and fits small samples where the standard deviation is estimated from the data.
Use a z-score when you know the population standard deviation or your sample is large (roughly 30 or more). Use a t-score for small samples with an estimated standard deviation.
Not exactly. Percentiles from z-scores assume a normal, bell-shaped distribution. For skewed or heavy-tailed data, treat the percentile as a rough approximation and consider methods built for your data's actual shape.
Values with very large absolute z-scores are rare under a normal model: beyond plus or minus 3 covers only about 0.27% of values, so |z| > 3 is a common flag for a potential outlier worth investigating.
It measures how extreme a value is in either direction: P(|Z| >= |z|). It is the standard quantity in two-sided hypothesis tests and confidence intervals, where an unusually low value is just as interesting as an unusually high one, and it doubles as a p-value.
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