Find a z-score from a raw value, convert a z-score into an area or p-value under the standard normal curve, and look up critical values for hypothesis testing — with the full working shown.
Quick Answer: How Do You Find a Z-Score?
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. The result tells you how many standard deviations x sits from the mean — positive means above average, negative means below. A z-score of 0 is exactly average; a z-score beyond ±2 is unusual; beyond ±3 is rare. Once you have z, the calculator below converts it into a percentile, an area under the normal curve, and a p-value for hypothesis testing.
Calculate Z-Score
Z = (X − μ) / σ
Value (X)
Mean (μ)
Std Dev (σ)
Find area from Z
Z-score
Area / probability
Result
Z-Score
1.00
X=75 is 1.00 SD above mean
Metric
Value
Standard Normal Table / Z-Table (P(Z < z))
This z-table covers both negative and positive z-scores, from z = −3.4 to z = +3.45. Rows give the whole number and first decimal of z; columns give the second decimal — read across and down, then multiply by 100 for a percentage.
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
Critical Z-Values for Hypothesis Testing
Significance (α)
One-tailed z
Two-tailed z
0.10 (90% CI)
±1.282
±1.645
0.05 (95% CI)
±1.645
±1.960
0.025
±1.960
±2.240
0.01 (99% CI)
±2.326
±2.576
0.001 (99.9% CI)
±3.090
±3.291
Quick Formula Summary
Z-score (sample): z = (x − x̄) / s
Z-score (population): z = (x − μ) / σ
Raw value from z: x = μ + z × σ
Standard error of the mean: SE = σ / √n
Z-score of a sample mean: z = (x̄ − μ) / (σ / √n)
P-value (two-tailed) = 2 × P(Z > |z|)
What Is a Z-Score?
A z-score, also called a standard score, tells you how many standard deviations a single value sits from the mean of its data set. It converts a raw number — a test mark, a height, a return, a lab result — into a common, unitless scale that can be compared directly against other values, even ones measured on completely different scales.
The sign carries meaning: a positive z-score means the value is above the mean, a negative z-score means it is below the mean, and a z-score of exactly 0 means the value is the mean. The size tells you how unusual the value is. A z-score of 0.5 is unremarkable — well within the crowd. A z-score of 3 is rare, sitting in the extreme tail of a normal distribution.
💡 Did You Know? The term "z-score" comes from the letter Z traditionally used for the standard normal distribution — the bell curve with a mean of 0 and a standard deviation of 1. Any normally distributed value can be "standardised" into this Z distribution, which is exactly what the z-score formula does.
Why Z-Scores Matter
They allow fair comparison across scales. A mark of 78/100 and a mark of 132/200 cannot be compared directly, but their z-scores can.
They flag outliers. Values beyond roughly ±2 or ±3 standard deviations are statistically unusual and worth a second look.
They underpin hypothesis testing. Z-tests convert a sample result into a z-score, then use the standard normal distribution to calculate a p-value.
They standardise scores across cohorts. University entrance scaling, standardised testing and growth charts all rely on z-scores to make results comparable across groups.
They convert into percentiles. Once you have z, the standard normal table converts it into "percentage of the population below this value."
Z-Score Formula
The formula depends on whether you know the population parameters or only sample statistics.
Population z-score: z = (x − μ) / σ
Sample z-score: z = (x − x̄) / s x = the value | μ, x̄ = mean | σ, s = standard deviation
A related version standardises a sample mean rather than a single observation — this is the basis of the one-sample z-test used in hypothesis testing:
z = (x̄ − μ₀) / (σ / √n) x̄ = sample mean | μ₀ = hypothesised population mean | σ/√n = standard error of the mean
Step-by-Step Calculation Examples
Example 1 — a single test score
A class test has a mean of 75 and a standard deviation of 6. A student scores 82.
z = (82 − 75) / 6 = 7 / 6 = 1.1667
P(Z < 1.1667) ≈ 0.8783, so the student scored better than about 87.8% of the class.
Example 2 — a below-average result
Mean = 70, SD = 8, observed value = 58.
z = (58 − 70) / 8 = −12 / 8 = −1.5
P(Z < −1.5) ≈ 0.0668, so this value sits in roughly the bottom 6.7% — a below-average but not extreme result.
Example 3 — comparing two different scales
Maths: score 78, class mean 65, SD 10 → z = (78 − 65)/10 = 1.30
English: score 132/200, class mean 110, SD 20 → z = (132 − 110)/20 = 1.10
Even though 132 is a bigger raw number than 78, the Maths result is relatively stronger — a higher z-score means it sits further above its own class average.
Example 4 — z-score of a sample mean (hypothesis test)
Claimed population mean μ₀ = 50. A sample of size n gives a sample mean x̄ = 52, with standard error SE = 1.2.
Because 0.0956 > 0.05, this result is not statistically significant at the conventional 5% level — there isn't strong enough evidence to reject the claim that the true mean is 50.
Normal Distribution and the 68–95–99.7 Rule
Z-scores are most informative for data that is approximately normally distributed — the symmetric bell curve where values cluster near the mean and thin out towards the extremes. For such data, the empirical rule gives an instant sense of scale.
Range
Proportion of data
Interpretation
z between −1 and +1
≈ 68.27%
About two in three values fall this close to the mean
z between −2 and +2
≈ 95.45%
Outside this range is genuinely uncommon
z between −3 and +3
≈ 99.73%
Beyond this is rare — often flagged as an outlier
Z-Score, Percentile and P-Value
These three ideas are closely linked but answer slightly different questions, and mixing them up is one of the most common statistics mistakes.
Term
What it answers
Example
Z-score
How many standard deviations from the mean?
z = 1.5
Percentile
What percentage of values fall below this one?
z = 1.5 → 93.3rd percentile
P-value (one-tailed)
Probability of a result this extreme or more, in one direction
P(Z > 1.5) ≈ 0.0668
P-value (two-tailed)
Probability of a result this extreme in either direction
2 × P(Z > 1.5) ≈ 0.1336
⚡ Pro Tip: Use a two-tailed p-value whenever you only care whether a result differs from expectation, in either direction. Use a one-tailed p-value only when the direction of the effect is specified in advance — for example, testing whether a new process specifically increases output, not just changes it.
How to Find a Z-Score, Percentile or P-Value on a TI-84
The TI-84 has two built-in functions for this, both under 2nd → VARS (DISTR), and they do opposite jobs.
normalcdf — z-score or raw value to percentile / p-value
Press 2nd → VARS, select 2:normalcdf(
For a z-score: enter normalcdf(-1E99, 1.5) to find P(Z < 1.5) — this returns 0.9332, the 93.32nd percentile
For raw values: normalcdf(lower, upper, μ, σ) works directly on the original scale without converting to z first
invNorm — percentile back to a z-score
Press 2nd → VARS, select 3:invNorm(
invNorm(0.95) returns 1.645 — the z-score below which 95% of the standard normal distribution falls
This is the TI-84 equivalent of reading a z-table backwards, and it's exactly how the critical values in the table above are generated
Common Mistakes
Mistake
Why it goes wrong
How to avoid it
Dividing by variance instead of standard deviation
Variance is in squared units, so the z-score comes out wrong
Always divide by σ or s, never σ² or s²
Confusing a z-score with a percentile
A z-score of 1 is not "1%" or "the 1st percentile"
Convert through the standard normal table — z = 1 is about the 84th percentile
Using a one-tailed p-value when the test should be two-tailed
Halves the reported p-value, making results look more significant than they are
Default to two-tailed unless the direction of the effect was specified beforehand
Applying z-scores to strongly skewed data
The empirical rule and normal-table percentiles assume roughly normal data
Check the shape of the distribution first, or use percentile ranks directly
Mixing up population and sample formulas
μ/σ and x̄/s are not interchangeable when the sample is small
Use μ and σ only when you have full population data; otherwise use x̄ and s
⚠️ Common Mistake: Treating a z-score near 0 as "wrong" or an error. A z-score of 0 simply means the value equals the mean — it is the most typical result possible, not a problem with the data.
Practical Applications
Education
Standardised test results — and Australian university entrance scaling such as ATAR — convert raw subject scores into a common scale so that a mark in one subject can be fairly compared with a mark in another, even when the subjects have very different difficulty levels and mark distributions.
Quality control and manufacturing
Process control charts plot measurements as z-scores (or as multiples of standard deviation) from a target value, flagging anything more than two or three standard deviations away for investigation — the statistical basis of six sigma quality programs.
Finance
Z-scores measure how unusual a return or price move is relative to its typical volatility, and a specific weighted z-score formula — the Altman Z-score — is used to estimate a company's bankruptcy risk from its financial ratios.
Health and growth charts
Paediatric growth charts report height, weight and head circumference as z-scores relative to a reference population, making it possible to track whether a child is tracking normally regardless of age or sex-specific differences in the raw numbers.
Z-Score Cheat Sheet
Quick Reference
Population z-score: z = (x − μ) / σ
Sample z-score: z = (x − x̄) / s
Z-score of a sample mean: z = (x̄ − μ₀) / (σ/√n)
Raw value from z: x = μ + z × σ
90% CI critical value: z = ±1.645 | 95%: ±1.960 | 99%: ±2.576
Empirical rule: 68% / 95% / 99.7% within 1 / 2 / 3 standard deviations
Two-tailed p-value: p = 2 × P(Z > |z|)
Excel: =STANDARDIZE(x, mean, sd), =NORM.S.DIST(z, TRUE), =NORM.S.INV(probability)
Practice Questions
Beginner (with answers)
A data set has mean 40 and standard deviation 5. Find the z-score of the value 50.
A data set has mean 100 and standard deviation 15. Find the z-score of the value 85.
What value corresponds to a z-score of 2 in a distribution with mean 60 and SD 4?
Is a z-score of −0.2 above or below the mean?
Roughly what percentage of a normal distribution lies within 2 standard deviations of the mean?
Show answers
1) z = (50−40)/5 = 2.0 2) z = (85−100)/15 = −1.0 3) x = 60 + 2×4 = 68 4) Below the mean (negative z) 5) About 95%
Advanced (with answers)
A sample of 36 items has mean 48 and the population SD is known to be 9. Find the z-score for testing whether the true mean differs from 45.
Using the result from Q1, is the result significant at α = 0.05 (two-tailed)?
A student scores 70 on Test A (mean 60, SD 8) and 78 on Test B (mean 70, SD 6). Which result is relatively stronger?
Find the two values that bound the middle 90% of a normal distribution with mean 50 and SD 10.
If P(Z < z) = 0.90, what is z (to 3 decimal places)?
Show answers
1) SE = 9/√36 = 1.5, z = (48−45)/1.5 = 2.0 2) Yes — |z| = 2.0 > 1.96, so it is significant at the 5% level 3) Test A: z = (70−60)/8 = 1.25; Test B: z = (78−70)/6 = 1.33 — Test B is relatively stronger 4) 50 ± 1.645×10 = 33.55 to 66.45 5) z ≈ 1.282
🔑 Key Takeaways
Z-score = (x − mean) / standard deviation — it measures how many standard deviations x sits from the mean.
z = 0 is exactly average; z = +1 is one SD above; z = −2 is two SDs below.
A z-score converts to a percentile and a p-value using the standard normal cumulative distribution function.
About 95% of a normal distribution falls between z = −1.96 and z = +1.96 — the basis of the 95% confidence level.
Z-scores let you compare values measured on entirely different scales, such as two different test results.
Use two-tailed p-values by default; use one-tailed only when the direction of an effect is specified in advance.
Frequently Asked Questions
A z-score, or standard score, measures how many standard deviations a value sits above or below the mean of its distribution. It is calculated as z = (x − mean) ÷ standard deviation. A z-score of +2 means the value is two standard deviations above the mean — higher than about 97.7% of the distribution. A negative z-score means the value is below the mean.
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. For example, with mean 70 and SD 5, a value of 75 gives z = (75 − 70) / 5 = 1.0. If you only have sample data, use the sample mean x̄ and sample standard deviation s in place of μ and σ.
A z-score of 1 means the value is one standard deviation above the mean, roughly the 84th percentile. A z-score of 2 is two standard deviations above, roughly the 97.7th percentile — the beginning of "unusual." A z-score of 3 is three standard deviations above, roughly the 99.87th percentile — genuinely rare, and often treated as an outlier.
Look up the z-score in a standard normal table, or use this calculator's "Find area from Z" tool, to get P(Z < z). That probability, expressed as a percentage, is the percentile — the proportion of the distribution that falls below your value. For example, z = 1.645 corresponds to P(Z < z) ≈ 0.95, the 95th percentile.
A p-value is the probability of observing a result at least as extreme as the calculated z-score, assuming the null hypothesis is true. For a two-tailed test, p = 2 × P(Z > |z|). A z-score of 1.96 gives a two-tailed p-value of exactly 0.05; a z-score of 2.576 gives p ≈ 0.01. This calculator computes the CDF using an error-function approximation to find these areas directly.
A critical z-value is the cutoff z-score for a chosen significance level or confidence level. For a 95% confidence interval, the two-tailed critical value is ±1.96. For a one-tailed test at α = 0.05, the critical value is 1.645. Results with a z-score beyond the critical value are considered statistically significant at that level.
Yes. A negative z-score simply means the value is below the mean. Unlike standard deviation, which can never be negative, a z-score can be any real number — positive above the mean, negative below it, and zero exactly at the mean.
A z-score uses the standard normal distribution and assumes the population standard deviation is known, or that the sample is large (n ≥ 30). A t-score uses the t-distribution, which has heavier tails to account for the extra uncertainty of estimating the standard deviation from a small sample. As sample size grows, the t-distribution converges to the normal distribution and the two become nearly identical.
Rearrange the formula: x = μ + z × σ. For a distribution with mean 50 and standard deviation 8, a z-score of 1.5 corresponds to x = 50 + 1.5 × 8 = 62.
The conversion from z-score to percentile relies on the shape of the normal (bell) curve. The z-score itself can be calculated for any data set, but the percentile and p-value it implies are only accurate when the underlying data is roughly normally distributed. For strongly skewed data, use percentile ranks calculated directly from the sorted data instead.
A one-sample z-test standardises the sample mean using z = (x̄ − μ₀) / (σ/√n), where μ₀ is the value being tested against. The resulting z-score is compared to a critical value (for example ±1.96 at the 5% level) or converted into a p-value. If |z| exceeds the critical value, or the p-value is below the significance level, the result is considered statistically significant.
Yes. Enter a value, mean and standard deviation to see the z-score, the exact area to the left and right of it, the two-tailed probability, the percentile rank and whether the result is significant at the 5% level — plus a chart of where it sits on the standard normal curve. The separate area calculator converts any z-score directly into a probability using the four common tail options.
"Standard score" is another name for a z-score, so the conversion is the same: look up P(Z < z) on the standard normal table, or enter the z-score into the "Find area from Z" tool above with "Left tail" selected. The result, expressed as a percentage, is the percentile — for example, a standard score of 1.5 corresponds to roughly the 93rd percentile.
Use 2nd → VARS to reach the distribution menu. normalcdf(lower, upper) converts a z-score or raw value range into an area — a percentile or p-value. invNorm(area) does the reverse, converting a percentile back into a z-score. For example, normalcdf(-1E99, 1.5) returns 0.9332, and invNorm(0.95) returns 1.645.
Yes. The standard normal table above runs from z = −3.4 to z = +3.45, covering both negative z-scores (values below the mean) and positive ones (values above the mean) in a single table, rather than requiring a separate negative z-score table.
Last updated: August 2026
Reviewed by Mohsin Iqbal using standard statistical definitions, with every worked example independently verified. This page is for educational purposes.