Use this calculator to compute the z-score of a normal distribution.
Please provide any one value to convert between z-score and probability. This is the equivalent of referencing a z-table.
Use this calculator to find the probability (area P in the diagram) between two z-scores.
A z-score calculator answers a question that sits at the heart of applied statistics: given a single value drawn from a normal distribution, how unusual is it? The z-score, also called the standard score or normal score, expresses that answer as a single dimensionless number - the count of standard deviations separating your value from the mean. A z-score of 2 means the value sits two standard deviations above average; a z-score of -1.5 means it sits one and a half below. Once you have that number, a normal distribution table (or this calculator) converts it directly into a probability, telling you what proportion of the population falls above, below, or between any values you care about.
This page provides the three tools that cover essentially every z-score task. The first computes a z-score from a raw score, population mean, and standard deviation, showing the full working and the associated probabilities. The second is a complete z-table replacement: enter a z-score and get every related probability, or enter any one probability and get the z-score that produces it. The third finds the area under the normal curve between two z-scores - the calculation behind confidence intervals and most "what fraction falls between" questions.
The z-score is a dimensionless quantity indicating the signed, fractional number of standard deviations by which a value lies above or below the mean. Values above the mean have positive z-scores; values below have negative ones; a value exactly at the mean has a z-score of zero.
Its power comes from standardization. Raw measurements arrive in incompatible units - a test score out of 800, a height in centimeters, a response time in milliseconds - and cannot be compared directly. Converting each to a z-score places them all on a single common scale where the mean is 0 and the standard deviation is 1. A student who scores 1.8 standard deviations above the mean on a chemistry exam has done better, relative to peers, than one who scored 1.2 above on a physics exam, regardless of the raw numbers or how the two tests were graded.
The z-score is calculated by subtracting the population mean from the raw score, then dividing by the population standard deviation:
z = (x − μ) ÷ σ
where x is the raw score (the data point in question - a test score, height, age, and so on), μ is the population mean, and σ is the population standard deviation.
Working through the calculator's default example: a raw score of 5 drawn from a population with mean 3 and standard deviation 2 gives z = (5 − 3) ÷ 2 = 1. The value sits exactly one standard deviation above the mean. Consulting the normal distribution, P(x < 5) = 0.84134 - about 84% of the population falls below this value, and roughly 16% above it.
When you are working with a sample rather than a full population, the same formula is used with the sample mean and sample standard deviation substituted in. And when the z-score describes a sample mean rather than an individual observation, the standard deviation is replaced by the standard error, σ ÷ √n, because averages of n observations vary less than individual observations do. Forgetting that √n is one of the most common mistakes in introductory hypothesis testing.
Enter the raw score, the population mean, and the standard deviation. The calculator returns the z-score along with three probabilities: the proportion of the distribution below your value, the proportion above it, and the proportion lying between the mean and your value. It also shows the arithmetic step by step, so you can check the working or reproduce it by hand.
This is the z-table in interactive form. Provide any one of the six fields and the calculator fills in the rest:
Working backwards from a probability is what makes this genuinely useful. Entering 0.95 in the P(-Z < x < Z) field immediately returns Z = 1.96, the critical value behind every 95% confidence interval you have ever seen.
Enter a left bound and a right bound and the calculator returns the area between them, the combined area outside them, and each individual tail. If you enter the bounds in the wrong order, they are swapped automatically. This tool answers questions of the form "what proportion of the population falls between these two values?" - the natural question once you have converted two raw scores into z-scores.
Behind every z-score lookup is the standard normal distribution: a bell curve with mean 0 and standard deviation 1, whose total area equals 1. The probability that a value falls in any interval is the area under the curve over that interval. A z-table - and this calculator - reports those areas.
The area cannot be written in elementary functions; it is defined through the error function and evaluated numerically. That is precisely why z-tables existed in the first place, and why a calculator is more convenient than one: tables are typically printed to two decimal places of z, forcing you to interpolate, whereas this calculator accepts any z-score and evaluates the integral directly.
A few landmark values are worth committing to memory, because they recur constantly:
The 68-95-99.7 rule, also called the empirical rule, is the quickest way to sanity-check any z-score. For a normal distribution, approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
The precise figures, which you can confirm with the converter above, are 68.269%, 95.450%, and 99.730%. The rule provides an immediate reality check: if you compute a z-score of 4 and the empirical rule says 99.7% of data lies within three standard deviations, you know at once that you are looking at something genuinely rare - roughly one observation in thirty thousand - and it is worth asking whether the value is a real extreme or a data-entry error.
Standardized testing. Percentile ranks on the SAT, GRE, IQ tests, and similar instruments are computed from z-scores. An IQ test with mean 100 and standard deviation 15 assigns a score of 130 a z-score of 2, placing it above about 97.7% of the population.
Hypothesis testing. The z-test compares an observed statistic against its null distribution. The resulting z-statistic converts into a p-value using exactly the tail probabilities this calculator produces, and that p-value is compared against the chosen significance level.
Quality control. Manufacturing uses z-scores to detect drift. Control charts flag any measurement beyond ±3 standard deviations, and the "six sigma" methodology takes its name directly from this scale.
Finance. Risk models express returns in standard deviations from the mean, and the Altman Z-score - a different but related construct - combines several financial ratios into a single bankruptcy-risk indicator.
Medicine and growth monitoring. Paediatric growth charts report height, weight, and head circumference as z-scores against reference populations, and bone density is reported as a T-score and Z-score derived the same way.
Outlier detection. A common screening rule flags any observation with |z| > 3 for review, on the grounds that such values occur naturally in only about 0.3% of cases.
The sign tells you direction and the magnitude tells you how unusual the value is. A z-score of 0 is exactly average. Values between -1 and 1 are entirely typical, covering roughly two-thirds of the distribution. Beyond ±2 you are in the outer 5%, and beyond ±3 in the outer 0.3%.
Two cautions are worth stating plainly. First, a z-score is not automatically a percentile. It converts to one only if the underlying distribution is normal. For a heavily skewed distribution - income is the classic example - a z-score of 2 does not correspond to the 97.7th percentile, and treating it as though it does will mislead you.
Second, a large z-score is not by itself evidence of an error. In a large enough dataset, extreme values are expected. In a sample of 10,000 normal observations you should see roughly 27 values beyond ±3 standard deviations purely by chance. Discarding them as outliers without investigation throws away real data.
The z-score assumes the population standard deviation is known. In practice it usually is not, and it must be estimated from the sample - which introduces extra uncertainty. The t-distribution accounts for that additional uncertainty with heavier tails than the normal distribution, and how much heavier depends on the degrees of freedom (sample size minus one).
The practical guidance: use the z-distribution when the population standard deviation is known, or when the sample is large (conventionally n > 30). Use the t-distribution when the standard deviation is estimated from a small sample. As the sample size grows, the t-distribution converges on the normal, which is why the distinction stops mattering for large samples.
Yes, and it simply means the value falls below the mean. A z-score of -1.5 sits one and a half standard deviations below average. The sign carries no judgement about good or bad - for a golf score or an error rate, a negative z-score is the desirable outcome.
It depends entirely on context. Statistically, |z| > 2 is uncommon (outer 5%) and |z| > 3 is rare (outer 0.3%). Whether that is good or bad depends on what is being measured. On an exam a z-score of 2 is excellent; on a manufacturing tolerance it signals a process that needs attention.
Rearrange the formula: x = μ + zσ. For a distribution with mean 100 and standard deviation 15, a z-score of 1.5 corresponds to a raw score of 100 + 1.5 × 15 = 122.5.
P(x < Z) is the entire area to the left of Z, including everything below the mean. P(0 < x < Z) is only the slice between the mean and Z. Because the normal distribution is symmetric and half its area lies below the mean, the two differ by exactly 0.5 for positive Z. Different textbooks tabulate different conventions, which is why the converter reports both.
Because 1.96 is the exact value. For 95% of the area to fall within ±Z, each tail must contain 2.5%, and the z-score with P(x < Z) = 0.975 is 1.959964, which rounds to 1.96. Two standard deviations actually capture 95.45%, slightly more than 95%. The approximation is fine for mental arithmetic but 1.96 is the figure used in published work.
Calculating one does not - the formula works on any data with a mean and standard deviation, and standardizing is often useful regardless. But converting a z-score into a probability or percentile does require approximate normality, because that conversion reads areas from the normal curve. For strongly non-normal data, use a distribution-free method or transform the data first.
No. The standard deviation is a property of the whole distribution, describing its typical spread in the original units. The z-score is a property of an individual value, describing how many of those standard deviations that value lies from the mean. The standard deviation is an input to the z-score calculation, not the same thing as its result.
It evaluates the standard normal cumulative distribution function numerically, using a high-precision series expansion of the error function for moderate z-scores and a continued-fraction expansion in the tails. This is more accurate than interpolating a printed z-table and gives results identical to statistical software.
This Z-score Calculator is provided for educational and general informational purposes only. Results assume the underlying data follows a normal distribution and that the mean and standard deviation supplied are accurate. Statistical conclusions should be interpreted in the context of study design, sample size, and distributional assumptions. Consult a qualified statistician for analyses that inform significant decisions.