What does a z-score mean?

A z-score expresses how many standard deviations a value is above or below a mean.

What is the percentile for a z-score?

For a standard normal model, the percentile below z is the cumulative distribution function Φ(z).

Z-Score Calculator — Standard Score, Percentile & Normal Curve

Q: Can a z-score be negative?

Yes. A negative z-score means the value is below the mean.

THE CONCEPT

What is a z-score?

A z-score converts a raw value into a standardized distance from the mean. Instead of saying only “85 points” or “180 cm”, it tells you whether the value is at the center of the distribution, above it, below it, and by how many standard deviations.

The key idea is simple: first measure the gap between the value and the mean; then divide that gap by the standard deviation. Because the units cancel, z-scores let you compare values that originally live on different scales.

Standardization removes the original unit. In this example, 85 is 15 points above a mean of 70; with σ = 10, that becomes z = 1.5.

−3σfar below

−2σlow

−1σbelow

0mean

+1σabove

+2σhigh

+3σfar above

WHY IT MATTERS

Where are z-scores used?

Z-scores are useful whenever the question is about relative position rather than the raw unit itself. They appear in education, measurement, quality control, data analysis and many statistical procedures.

Exam and test scores

A 90 is meaningful only in context. A z-score tells you how far that result sits from the class or population mean.

Heights and measurements

Centimeters, kilograms or other units can be standardized so observations from different scales can be compared.

Quality control

Manufacturing measurements can be expressed as standardized distances from a target distribution to flag unusually high or low observations.

Data analysis

Standardized values are useful when comparing variables with different means and standard deviations or preparing for normal-model calculations.

HOW TO USE IT

How to use the z-score calculator

Choose what you want to solve for, enter the other known quantities, and read the standardized result together with its normal-model percentile and tail areas.

Choose the unknown

Use “Solve for” to calculate z, x, μ, or σ.

Enter the known values

For a z-score, enter x, μ and a positive σ.

Calculate

The calculator applies the selected rearrangement and updates the live visuals.

Interpret

Read the sign, distance from the mean, percentile and tail area together.

THE MATHEMATICS

Z-score formula and rearrangements

The standard population form is the difference between the observation and the mean, divided by the population standard deviation. For a sample, the analogous expression uses the sample mean and sample standard deviation.

z = (x − μ) / σ

z: standard score
x: raw observed value
μ: population mean
σ: population standard deviation, σ > 0

Solve for x

x = μ + zσ

Use this when you know the desired standardized position and want the corresponding raw value.

Solve for μ

μ = x − zσ

Use this when the raw value, z-score and spread are known.

Solve for σ

σ = (x − μ) / z

Requires z ≠ 0 and a positive resulting standard deviation.

Sample version

z = (x − x̄) / s

When working from a sample, use the sample mean x̄ and sample standard deviation s as appropriate.

Units disappear. If x and μ are both measured in centimeters and σ is also in centimeters, the units cancel. That is why a z-score is dimensionless.

INTERPRETATION

How to interpret a z-score

The sign tells you the side of the mean. The magnitude tells you the distance. A z-score of +2 and a z-score of −2 are equally far from the mean, but on opposite sides.

z range	Position	Plain-language reading	Normal-model intuition
z = 0	At the mean	Exactly centered	50th percentile
0 < z < 1	Above mean	Moderately above average	Above the 50th percentile
1 ≤ z < 2	Clearly above	Higher than most observations	Roughly 84th–97.5th percentile
z ≥ 2	Far above	Unusually high in a normal model	At or above about the 97.5th percentile
−1 < z < 0	Below mean	Moderately below average	Below the 50th percentile
z ≤ −2	Far below	Unusually low in a normal model	At or below about the 2.5th percentile

68.27%About this much lies within ±1σ of the mean in a normal distribution.

95.45%About this much lies within ±2σ.

99.73%About this much lies within ±3σ.

AREA UNDER THE CURVE

Z-scores and normal probabilities

For a standard normal random variable Z, the cumulative distribution function is written as Φ(z). It gives the area to the left of z. The right-tail probability is 1 − Φ(z), and the two-sided area beyond ±|z| is 2[1 − Φ(|z|)].

The same curve can answer “what percentage is below this value?” once the raw value has been converted to z.

The empirical rule is a quick normal-distribution reference. It is an approximation, not a substitute for Φ(z).

Normal distribution chart showing standard deviations, percentiles and z-scores — A real educational normal-distribution chart showing how standard deviations, percentiles and z-scores line up on the same bell curve. The image is in the public domain.

WORKED EXAMPLES

Z-score examples, step by step

EXAMPLE 01 · EXAM

Score of 85

Suppose x = 85, μ = 70 and σ = 10.

z = (85 − 70) / 10 = 1.50

≈ 93.32nd percentile

The score is 1.5 standard deviations above the mean.

EXAMPLE 02 · HEIGHT

Height of 180 cm

Suppose μ = 170 cm and σ = 7 cm.

z = (180 − 170) / 7 ≈ 1.43

≈ 92.35th percentile

The original unit is centimeters; the standardized score is unit-free.

EXAMPLE 03 · LAB

Measurement of 3.2

Suppose μ = 4.0 and σ = 0.5.

z = (3.2 − 4.0) / 0.5 = −1.60

≈ 5.48th percentile

The negative sign indicates a result below the mean.

EXAMPLE 04 · REVERSE

Find x for z = 1.96

With μ = 70 and σ = 10:

x = 70 + 1.96 × 10 = 89.6

x = 89.6

This is the raw value that sits 1.96 standard deviations above the mean.

Visual check: a positive z places the point to the right of the mean. The farther the point moves, the smaller the right-tail probability becomes.

Z TABLE

Z-table: area from 0 to z

A traditional z-table often reports the area between z = 0 and a positive z-value. To recover the left-tail probability for positive z, add 0.5000 to the table entry. For a negative z, use symmetry.

Z-table (area from 0 to z)

Row is the first decimal of z, column is the second. The cell is the area between 0 and that z under a standard normal — not the left tail. Left tail = 0.5 + cell. Example: z = 1.53 → row 1.5, column 0.03 → 0.43699. Left tail ≈ 0.937.

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0	0.00399	0.00798	0.01197	0.01595	0.01994	0.02392	0.02790	0.03188	0.03586
0.1	0.03983	0.04380	0.04776	0.05172	0.05567	0.05962	0.06356	0.06749	0.07142	0.07535
0.2	0.07926	0.08317	0.08706	0.09095	0.09483	0.09871	0.10257	0.10642	0.11026	0.11409
0.3	0.11791	0.12172	0.12552	0.12930	0.13307	0.13683	0.14058	0.14431	0.14803	0.15173
0.4	0.15542	0.15910	0.16276	0.16640	0.17003	0.17364	0.17724	0.18082	0.18439	0.18793
0.5	0.19146	0.19497	0.19847	0.20194	0.20540	0.20884	0.21226	0.21566	0.21904	0.22240
0.6	0.22575	0.22907	0.23237	0.23565	0.23891	0.24215	0.24537	0.24857	0.25175	0.25490
0.7	0.25804	0.26115	0.26424	0.26730	0.27035	0.27337	0.27637	0.27935	0.28230	0.28524
0.8	0.28814	0.29103	0.29389	0.29673	0.29955	0.30234	0.30511	0.30785	0.31057	0.31327
0.9	0.31594	0.31859	0.32121	0.32381	0.32639	0.32894	0.33147	0.33398	0.33646	0.33891
1.0	0.34134	0.34375	0.34614	0.34849	0.35083	0.35314	0.35543	0.35769	0.35993	0.36214
1.1	0.36433	0.36650	0.36864	0.37076	0.37286	0.37493	0.37698	0.37900	0.38100	0.38298
1.2	0.38493	0.38686	0.38877	0.39065	0.39251	0.39435	0.39617	0.39796	0.39973	0.40147
1.3	0.40320	0.40490	0.40658	0.40824	0.40988	0.41149	0.41308	0.41466	0.41621	0.41774
1.4	0.41924	0.42073	0.42220	0.42364	0.42507	0.42647	0.42785	0.42922	0.43056	0.43189
1.5	0.43319	0.43448	0.43574	0.43699	0.43822	0.43943	0.44062	0.44179	0.44295	0.44408
1.6	0.44520	0.44630	0.44738	0.44845	0.44950	0.45053	0.45154	0.45254	0.45352	0.45449
1.7	0.45543	0.45637	0.45728	0.45818	0.45907	0.45994	0.46080	0.46164	0.46246	0.46327
1.8	0.46407	0.46485	0.46562	0.46638	0.46712	0.46784	0.46856	0.46926	0.46995	0.47062
1.9	0.47128	0.47193	0.47257	0.47320	0.47381	0.47441	0.47500	0.47558	0.47615	0.47670
2.0	0.47725	0.47778	0.47831	0.47882	0.47932	0.47982	0.48030	0.48077	0.48124	0.48169
2.1	0.48214	0.48257	0.48300	0.48341	0.48382	0.48422	0.48461	0.48500	0.48537	0.48574
2.2	0.48610	0.48645	0.48679	0.48713	0.48745	0.48778	0.48809	0.48840	0.48870	0.48899
2.3	0.48928	0.48956	0.48983	0.49010	0.49036	0.49061	0.49086	0.49111	0.49134	0.49158
2.4	0.49180	0.49202	0.49224	0.49245	0.49266	0.49286	0.49305	0.49324	0.49343	0.49361
2.5	0.49379	0.49396	0.49413	0.49430	0.49446	0.49461	0.49477	0.49492	0.49506	0.49520
2.6	0.49534	0.49547	0.49560	0.49573	0.49585	0.49598	0.49609	0.49621	0.49632	0.49643
2.7	0.49653	0.49664	0.49674	0.49683	0.49693	0.49702	0.49711	0.49720	0.49728	0.49736
2.8	0.49744	0.49752	0.49760	0.49767	0.49774	0.49781	0.49788	0.49795	0.49801	0.49807
2.9	0.49813	0.49819	0.49825	0.49831	0.49836	0.49841	0.49846	0.49851	0.49856	0.49861
3.0	0.49865	0.49869	0.49874	0.49878	0.49882	0.49886	0.49889	0.49893	0.49896	0.49900
3.1	0.49903	0.49906	0.49910	0.49913	0.49916	0.49918	0.49921	0.49924	0.49926	0.49929
3.2	0.49931	0.49934	0.49936	0.49938	0.49940	0.49942	0.49944	0.49946	0.49948	0.49950
3.3	0.49952	0.49953	0.49955	0.49957	0.49958	0.49960	0.49961	0.49962	0.49964	0.49965
3.4	0.49966	0.49968	0.49969	0.49970	0.49971	0.49972	0.49973	0.49974	0.49975	0.49976
3.5	0.49977	0.49978	0.49978	0.49979	0.49980	0.49981	0.49981	0.49982	0.49983	0.49983
3.6	0.49984	0.49985	0.49985	0.49986	0.49986	0.49987	0.49987	0.49988	0.49988	0.49989
3.7	0.49989	0.49990	0.49990	0.49990	0.49991	0.49991	0.49992	0.49992	0.49992	0.49992
3.8	0.49993	0.49993	0.49993	0.49994	0.49994	0.49994	0.49994	0.49995	0.49995	0.49995
3.9	0.49995	0.49995	0.49996	0.49996	0.49996	0.49996	0.49996	0.49996	0.49997	0.49997
4.0	0.49997	0.49997	0.49997	0.49997	0.49997	0.49997	0.49998	0.49998	0.49998	0.49998

COMMON MISTAKES

Common z-score mistakes

Using the wrong standard deviation

Do not mix a population mean with an unrelated standard deviation. The numerator and denominator must describe the same distribution or modeling context.

Forgetting the sign

A z-score of −1.8 is not the same interpretation as +1.8. The magnitude gives distance; the sign gives direction.

Calling z a percentile

z is a standardized distance. The percentile is Φ(z) under a standard normal model. They are related, but they are not the same number.

Assuming normality automatically

The z-score formula itself can be computed without a normal distribution, but percentile and tail interpretations require a probability model. The normal model is an assumption, not a guarantee.

Student rule: always read a z-score in three steps: sign → distance → probability. Ask which side of the mean the value is on, how many standard deviations away it is, and only then interpret the percentile or tail area.

QUESTIONS STUDENTS ASK

Z-score calculator FAQ

Can a z-score be negative?

Yes. A negative z-score means the raw value is below the mean. A positive z-score means it is above the mean, while z = 0 is exactly at the mean.

Is a z-score the same as a standard score?

Yes. “Z-score” and “standard score” commonly describe the standardized value obtained by subtracting a mean and dividing by a standard deviation.

What is the difference between z-score and percentile?

The z-score is a standardized distance. Under a standard normal model, the percentile below that z is Φ(z), which converts the position into a cumulative percentage.

Do I use σ or s?

Use σ when the population standard deviation is the quantity in your model. For a sample, the analogous sample standardization uses the sample mean x̄ and sample standard deviation s.

Why must σ be greater than zero?

A z-score divides by the standard deviation. If the standard deviation is zero, there is no spread and division by zero is undefined.

Does a z-score prove that a value is unusual?

Not by itself. “Unusual” depends on the distributional model and context. In a normal model, values around ±2 and ±3 standard deviations are increasingly far into the tails.

Z-score calculator

Bell curve position

Distance from the mean

What this standardized value is telling you

Rearranged equations

Your values