How to Calculate Z Score

Q: How do I find a z score without a calculator?

The arithmetic is one subtraction and one division, chosen by textbook writers to come out clean (deviations like 15 with σ = 10). Do the subtraction, then ask "how many σ fit into this gap?" — that count is z. Awkward divisions (like 12 / 7 = 1.71) are usually a signal the question intends a table value after rounding to two decimals.

Q: How do I find the z score from a percentile?

That is a reverse lookup, not a formula calculation: turn the percentile into a decimal (95th → 0.95), find it in the body of the table, and read off the row and column — z = 1.645. The finder's probability box on this page automates it.

Q: What do I need to calculate a z score?

Three numbers: the raw value, the mean of its distribution, and the standard deviation. Without the spread you cannot say how unusual a deviation is — a 15-point gap is huge in one class and trivial in another.

Q: Can I calculate a z score for data that is not normal?

The arithmetic always works, and z remains an honest "distance in standard deviations". What you lose is the table interpretation: percentile readings from the normal table are only valid when the distribution is approximately normal.

Q: How do I find the raw score from a z score?

Rearrange: X = μ + zσ. Multiply z by σ first, then add the mean. For μ = 500, σ = 100, z = 1.28: X = 628. Example 5 above works two of these in full.

To calculate a z score: subtract the mean from your value, then divide by the standard deviation. z = (X − μ) / σ. Three numbers in, one number out — this page walks the steps slowly, with six worked examples including the two variations students actually get wrong: negative scores, and running the calculation backwards to find the raw value.

Type a z score or a probability

Area to the left—

Area to the right—

Between mean and z—

Percentile—

Shade:

Area between two z scores:

Positive z score table: cumulative probabilities P(Z ≤ z) from 0.00 to 3.99
z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998
3.5	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998
3.6	0.9998	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.7	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.8	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.9	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Once you have a z score, this is where it goes: the finder above converts it to areas and percentiles instantly. Work the examples below by hand first, then check yourself here.

The steps

Write down the three ingredients. The raw value X, the mean μ (or sample mean x̄), and the standard deviation σ (or s). Most calculation errors are really ingredient errors — a variance (σ²) used where σ belongs is the classic. If you are given variance, take its square root first.
Subtract: X − μ. This is the deviation from the mean, in the original units, with a sign. Positive means above average; negative means below. Keep the sign — it is half the information.
Divide by σ. The result is unit-free: "standard deviations from the mean". Round the final z to two decimal places if a table lookup follows.
Interpret before you move on. Say the sentence: "This value is ___ standard deviations above/below the mean." If the sentence surprises you (a tall person with a negative z), an ingredient is wrong — catch it here, not after the table.
Look it up. The z score table converts z into the proportion below, above or between — the part most questions actually ask for.

Six worked examples

Example 1 — the standard case

X = 85, μ = 70, σ = 10. z = (85 − 70) / 10 = 1.50. Table: 0.9332 — the 93.32nd percentile.

Example 2 — a negative z score

X = 54, μ = 60, σ = 8. z = (54 − 60) / 8 = −6 / 8 = −0.75. Table (negative): 0.2266 — the 22.66th percentile. The method does not change for negative results; only the table page does.

Example 3 — decimals in the ingredients

A component measures X = 25.4 mm against x̄ = 24.6 mm, s = 0.4 mm. z = (25.4 − 24.6) / 0.4 = 0.8 / 0.4 = 2.00. Only 2.28% of components sit further above the mean — worth a second look on a production line.

Example 4 — given the variance instead

A distribution has mean 100 and variance 225. The standard deviation is √225 = 15 — take the root before anything else. For X = 130: z = (130 − 100) / 15 = 2.00, the 97.72nd percentile. (Divide by 225 instead and you would get 0.13 — the single most common ingredient error in coursework.)

Example 5 — running it backwards: find the raw score

What mark sits at z = 1.28 in a class with μ = 70, σ = 10? Rearrange the formula: X = μ + zσ = 70 + 1.28 × 10 = 82.8. This direction — z first, X second — is how grade boundaries, admission cutoffs and screening thresholds are set. A second one: z = −0.75 with μ = 60, σ = 8 gives X = 60 + (−0.75 × 8) = 54, which is Example 2's score coming home.

Example 6 — comparing two distributions

Which is the stronger result: 1250 on a test scaled μ = 1050, σ = 200, or 31 on one scaled μ = 25, σ = 4? First: z = (1250 − 1050) / 200 = 1.00 (84.13th percentile). Second: z = (31 − 25) / 4 = 1.50 (93.32nd percentile). The second result is stronger relative to its group, by half a standard deviation — a judgement the raw scores (1250 vs 31) could never have supported.

Calculating from a sample vs a population

If the mean and standard deviation describe an entire population, use μ and σ. If you computed them from a sample, use x̄ and s — and know that the sample standard deviation divides by n − 1 rather than n (Bessel's correction), which slightly widens s to compensate for sampling. Spreadsheet functions encode the choice: STDEV.P for a population, STDEV.S for a sample in Excel; STANDARDIZE(x, mean, sd) then applies the formula with whichever spread you hand it. Using the population function on sample data understates s, inflates z, and exaggerates how extreme values look — a quiet way to manufacture "significant" results, and one reason methods sections state which was used.

Errors that survive into exam scripts

Dividing before subtracting. The mean subtraction comes first; X / σ − μ is a different, meaningless number.
Using σ². If the problem gives variance, square-root it (Example 4).
Dropping the sign. A z of −0.75 is not "0.75, but low". The table lookup differs by sign: 0.2266 versus 0.7734.
Rounding mid-calculation. Keep full precision until the final z; round once, to two decimals, for the table.
Answering z when the question asked for a probability. z is the middle of the problem, rarely the end. If the question says "what proportion…", the table step is still owed.

Checking your answer before you trust it

Thirty seconds of sanity-checking catches most wrong z scores before they cost anything. Sign check: a value above the mean must give a positive z, below must give negative — if not, the subtraction happened in the wrong order. Size check: the deviation divided by the spread should feel right: a gap smaller than one standard deviation must give |z| < 1, and a gap of exactly 2σ must give exactly 2 — if your arithmetic says a half-σ gap produced z = 4, a decimal point has wandered. Table check: the percentile your z produces should match the story — a score described as "top of the class" should not convert to the 40th percentile. And ingredient check: confirm the "standard deviation" you used was not a variance (square-root it) and not a range or an interquartile range wearing a similar name. These four checks are mechanical, fast, and — unlike re-reading your working — they actually catch errors, because each one approaches the number from a different direction.

Frequently asked questions

How do I find a z score without a calculator?

The arithmetic is one subtraction and one division, chosen by textbook writers to come out clean (deviations like 15 with σ = 10). Do the subtraction, then ask "how many σ fit into this gap?" — that count is z. Awkward divisions (like 12 / 7 = 1.71) are usually a signal the question intends a table value after rounding to two decimals.

How do I find the z score from a percentile?

That is a reverse lookup, not a formula calculation: turn the percentile into a decimal (95th → 0.95), find it in the body of the table, and read off the row and column — z = 1.645. The finder's probability box on this page automates it.

What do I need to calculate a z score?

Three numbers: the raw value, the mean of its distribution, and the standard deviation. Without the spread you cannot say how unusual a deviation is — a 15-point gap is huge in one class and trivial in another.

Can I calculate a z score for data that is not normal?

The arithmetic always works, and z remains an honest "distance in standard deviations". What you lose is the table interpretation: percentile readings from the normal table are only valid when the distribution is approximately normal.

How do I find the raw score from a z score?

Rearrange: X = μ + zσ. Multiply z by σ first, then add the mean. For μ = 500, σ = 100, z = 1.28: X = 628. Example 5 above works two of these in full.

More reps: the practice problems are half calculation questions by design, and the formula page explains why the formula is built the way it is.