Z-score Calculator
Values
Fill in the values and click Calculate to see results
Example 1
A student scores 85 on a mathematics exam. The class mean is 72 with a standard deviation of 6. Convert the score into a z-score and find what percentage of the class scored below it.
Solution
The percentile is the area under the standard normal curve left of z = 2.1667:
The one-tailed p-value is the area beyond the score: 1 − 0.9849 = 0.0151.
Answer
The z-score is 2.1667. About 98.49% of the class scored below 85, and only 1.51% scored at or above it – the student stands roughly two standard deviations above the class average.
Source: OpenStax Introductory Statistics, ch. 6
Example 2
Eight fasting glucose readings are collected from one patient over a week (in mmol/L): 5.2, 5.4, 4.9, 5.1, 5.6, 5.3, 5.0, 5.5. The next reading is 6.1 mmol/L. Using the sample standard deviation, find the z-score of the new reading.
Solution
The sum of the eight readings is
Dividing by the count gives the mean:
Deviations from 5.25: −0.05, +0.15, −0.35, −0.15, +0.35, +0.05, −0.25, +0.25. Squared and summed: 0.0025 + 0.0225 + 0.1225 + 0.0225 + 0.1225 + 0.0025 + 0.0625 + 0.0625 = 0.42.
The sample variance is
Taking the square root gives the sample standard deviation:
The z-score of the new reading is
Answer
The new reading has a z-score of 3.47 – more than three standard deviations above the patient's weekly mean, a reading that would occur by chance less than once in a thousand under a normal model. That warrants repeating the test.
Source: NIST/SEMATECH e-Handbook of Statistical Methods
Example 3
A graduate programme admits only the top 5% of applicants on a standardised test. What z-score marks that cutoff? And what z-score would bound the middle 95% of all test takers?
Solution
The top 5% means 95% of the area lies below the cutoff:
The middle 95% leaves 2.5% in each tail, so the boundaries are the 2.5th and 97.5th percentiles:
By symmetry the lower boundary is −1.96.
Answer
The top-5% cutoff sits at z = 1.645, and the middle 95% of test takers fall between z = −1.96 and z = +1.96 – the two most famous numbers in statistics.
Source: PennState STAT 500, Applied Statistics
Example 4
Using the z-score table, find the area under the standard normal curve to the left of z = 1.23. Then determine the area between z = −1.23 and z = 1.23.
Solution
From the table, Φ(1.2) = 0.8849 and Φ(1.3) = 0.9032. The gap between the two table rows is:
Interpolating three-tenths of the way from Φ(1.2) toward Φ(1.3):
The exact value is Φ(1.23) = 0.8907 – interpolation was off by only 0.0003.
By symmetry, Φ(−1.23) = 1 − 0.8907 = 0.1093. The area between the two z-scores is:
Answer
About 89.07% of the curve lies left of z = 1.23, and the band between ±1.23 holds 78.14% of the total area. This page's calculator gives Φ(1.23) = 0.8907 directly.
Source: OpenStax Introductory Statistics, ch. 6
Published:
Updated:
References
- [1]NIST, NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook
- [2]PennState Eberly College of Science, STAT 500: Applied Statistics. https://online.stat.psu.edu/stat500
- [3]OpenStax, Rice University, OpenStax Introductory Statistics. https://openstax.org/details/introductory-statistics
- [4]Khan Academy, Normal distributions and the empirical rule. https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data
- [5]NIST (Abramowitz & Stegun), Handbook of Mathematical Functions, ch. 7: Error Function and Fresnel Integrals, 1964. https://www.nist.gov/history/mathematical-tables-handbook-mathematical-functions
Glossary
- Z-score – The number of standard deviations a value lies from the mean, computed as (x − μ) / σ. Also called a standard score or standardised value.
- Standard normal distribution – The bell-shaped probability distribution with mean 0 and standard deviation 1. Every z-score is a location on this curve.
- Φ (cumulative distribution function) – The function returning the area under the standard normal curve to the left of a given z-score. Φ(1.96) = 0.975 means 97.5% of the area lies below z = 1.96.
- Percentile – The percentage of a distribution falling below a value. For normal data it equals Φ(z) × 100.
- P-value – The probability, under a null model, of a result at least as extreme as the one observed. Computed from the tail area of the distribution beyond the z-score.
- One-tailed test – A significance test concerned with deviation in one direction only. Its p-value is the single tail area beyond z in that direction.
- Two-tailed test – A significance test concerned with deviation in either direction. Its p-value doubles the smaller tail area, so ±1.96 corresponds to p = 0.05.
- Critical value – The z-score cutoff for a conventional significance level: 1.645 (one-tailed) or 1.96 (two-tailed) at 5%, 2.576 at 1% two-tailed.
- Z table – A printed table of Φ(z) values used to convert z-scores into areas by hand, typically covering −3.4 to +3.4.
- Standardisation – The transformation of measurements from different scales onto the common z-score scale by subtracting a reference mean and dividing by a reference standard deviation.
- Sigma level – A quality metric expressing how many standard deviations fit between a process mean and its specification limit. Six sigma corresponds to about 3.4 defects per million.
- Inverse normal (percentile to z) – The reverse lookup that finds the z-score with a specified area below it, implemented in this calculator by bisection on Φ.
How to Use?
- 1
Choose your starting point
Select Raw Value if you already know the mean and standard deviation, Data Set if you have a list of measurements, or Percentile if you want the z-score for a given area.
- 2
Enter a raw value with its reference
In Raw Value mode, type the value, the mean μ, and the standard deviation σ. For a student scoring 85 against a class mean of 72 with σ = 6, enter 85, 72, and 6.
- 3
Or paste a data set
In Data Set mode, set the number of values, enter each measurement, pick Sample or Population, and enter the value to score. The calculator computes the mean and standard deviation first, then scores your value against them.
- 4
Or start from a percentile
In Percentile mode, enter an area from 0 to 100. The calculator finds the z-score with exactly that much curve below it – 95, for instance, returns 1.645.
- 5
Read the results
Raw Value mode shows the z-score with its percentile and one- and two-tailed p-values. Data Set mode additionally shows the computed mean, standard deviation, count, and a table scoring every value you entered. Percentile mode shows the z-score with its one- and two-tailed tail areas.
- 6
Check the shaded curve
The chart shades the area below your z-score on the standard normal curve. The shaded share is your percentile made visible – a quick visual check that the numbers match intuition.
- 7
Cross-check with the z-score table
Use the z-score tables on this page to verify the percentile by hand, interpolating between rows for two-decimal z-scores. Agreement within a few tenths of a percent confirms both the table and the calculator.
Frequently Asked Questions
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. For example, a value of 85 against a mean of 72 and a standard deviation of 6 gives z = (85 − 72) / 6 = 2.17. Enter the same three numbers in the Raw Value mode of this calculator and you also get the percentile and p-values alongside the z-score.
It tells you how many standard deviations your value sits from the mean, in a unit that works for any measurement. A z of 0 is exactly average; a z of +2 is two standard deviations above average, which is roughly the 98th percentile; a z of −1.5 is one and a half standard deviations below, near the 7th percentile. Because the unit is standard deviations, a z of 2 means the same rarity for an exam score, a blood test, or a manufacturing reading.
It depends on the direction you care about. On a test or any "higher is better" measure, z = +2 or above is excellent (top 2.3%). On a defect count or a risk measure, the same magnitude is alarming. Between −1 and +1 lies the ordinary middle 68% of any bell-shaped data. Values beyond ±3 occur in less than 0.3% of a normal distribution and usually justify a second look, whether celebratory or investigative.
z = (x − μ) / σ, where x is the value, μ is the population mean, and σ is the population standard deviation. When you only have a sample, use the sample mean and sample standard deviation as estimates of μ and σ. The result expresses the gap between your value and the mean in units of standard deviation.
Yes. A negative z-score simply means the value is below the mean. A z of −1.8 sits 1.8 standard deviations below average, around the 4th percentile. Negative z-scores are common in quality control, growth monitoring, and finance, where values below a reference matter as much as values above it. The sign carries the direction; the magnitude carries the rarity.
z = 1.645. The 95th percentile leaves 95% of the curve below it and 5% above, and that area split occurs 1.645 standard deviations above the mean. Enter 95 in the Percentile mode of this calculator to confirm, or interpolate between Φ(1.6) = 0.9452 and Φ(1.7) = 0.9554 in the z-score table.
It depends on the tail. For a one-tailed test, p = 0.05 corresponds to z = 1.645 (or −1.645 in the other direction). For a two-tailed test, p = 0.05 corresponds to z = ±1.96, because the 5% is split into two 2.5% tails. This calculator reports both p-values with every z-score.
Because it is the two-tailed 5% cutoff. Roughly 95% of a normal distribution lies between −1.96 and +1.96, so a z beyond those bounds is unusual enough to reject a null hypothesis at the conventional 0.05 significance level. That single convention makes ±1.96 the threshold behind 95% confidence intervals, two-tailed significance tests, and most published "statistically significant" claims.
The z-score is undefined. A standard deviation of zero means every value in the reference set is identical, so there is no spread to measure distance against – division by zero. This calculator flags the input as an error rather than returning a number. If you see this with real data, check for a duplicated single value or a data entry mistake.
Find your z-score's row and take the area from the column. With 0.1-step tables like the ones on this page, interpolate between neighbouring rows for a second decimal: for z = 1.96, average the gap between Φ(1.9) = 0.9713 and Φ(2.0) = 0.9772 to get 0.975. For negative z, either use the negative table directly or apply symmetry: Φ(−z) = 1 − Φ(z).
Both standardise a value against a mean and spread, but a t score uses the t distribution, which is wider than the normal when the sample is small. With 30 or more observations the two are nearly identical. Below that, the t cutoffs for significance grow larger than the z cutoffs, correctly demanding more evidence from less data. Use z when σ is known or the reference set is large; use t when σ is estimated from a small sample.
Use the Percentile mode of this calculator: enter the area as a percentage and it inverts the normal curve to find the matching z. The 90th percentile gives z = 1.282, the 95th gives 1.645, the 97.5th gives 1.960, and the 99th gives 2.326. For areas below 50% the z is negative: the 5th percentile is −1.645.
Yes. The z-score itself is pure arithmetic and valid for any distribution, but converting it to a percentile or p-value assumes the standard normal shape. If your data is skewed, the true fraction of values below yours can differ from the normal-based estimate. For roughly symmetric data the approximation is excellent; for heavily skewed data, report the observed data percentile instead.
The standard deviation is the unit in which z-scores are measured. The z formula divides by σ precisely so the answer reads "how many standard deviations from the mean". Our Standard Deviation Calculator computes σ from a raw list of values; this page then uses that σ to score any value you choose. The two tools together cover the full workflow from raw data to standardised position.
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