Z Score Practice Problems with Answers

Twenty graded problems — table reads first, then raw-score conversions, then the mixed shapes exams actually set. Attempt each one before opening the solution: every solution shows the move, the table reads, and the arithmetic, so a wrong attempt still teaches you where it turned. You will need the z score table (open it in another tab) or the finder below.

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:

Full z score table: cumulative probabilities P(Z ≤ z) from -3.99 to 3.99
z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
-3.9	0.0000	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
-3.8	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
-3.7	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0002
-3.6	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002
-3.5	0.0002	0.0002	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003
-3.4	0.0003	0.0003	0.0004	0.0004	0.0004	0.0004	0.0004	0.0004	0.0005	0.0005
-3.3	0.0005	0.0005	0.0005	0.0005	0.0006	0.0006	0.0006	0.0006	0.0006	0.0007
-3.2	0.0007	0.0007	0.0007	0.0008	0.0008	0.0008	0.0008	0.0009	0.0009	0.0009
-3.1	0.0010	0.0010	0.0010	0.0011	0.0011	0.0011	0.0012	0.0012	0.0013	0.0013
-3.0	0.0013	0.0014	0.0014	0.0015	0.0015	0.0016	0.0016	0.0017	0.0018	0.0018
-2.9	0.0019	0.0019	0.0020	0.0021	0.0021	0.0022	0.0023	0.0023	0.0024	0.0025
-2.8	0.0026	0.0026	0.0027	0.0028	0.0029	0.0030	0.0031	0.0032	0.0033	0.0034
-2.7	0.0035	0.0036	0.0037	0.0038	0.0039	0.0040	0.0041	0.0043	0.0044	0.0045
-2.6	0.0047	0.0048	0.0049	0.0051	0.0052	0.0054	0.0055	0.0057	0.0059	0.0060
-2.5	0.0062	0.0064	0.0066	0.0068	0.0069	0.0071	0.0073	0.0075	0.0078	0.0080
-2.4	0.0082	0.0084	0.0087	0.0089	0.0091	0.0094	0.0096	0.0099	0.0102	0.0104
-2.3	0.0107	0.0110	0.0113	0.0116	0.0119	0.0122	0.0125	0.0129	0.0132	0.0136
-2.2	0.0139	0.0143	0.0146	0.0150	0.0154	0.0158	0.0162	0.0166	0.0170	0.0174
-2.1	0.0179	0.0183	0.0188	0.0192	0.0197	0.0202	0.0207	0.0212	0.0217	0.0222
-2.0	0.0228	0.0233	0.0239	0.0244	0.0250	0.0256	0.0262	0.0268	0.0274	0.0281
-1.9	0.0287	0.0294	0.0301	0.0307	0.0314	0.0322	0.0329	0.0336	0.0344	0.0351
-1.8	0.0359	0.0367	0.0375	0.0384	0.0392	0.0401	0.0409	0.0418	0.0427	0.0436
-1.7	0.0446	0.0455	0.0465	0.0475	0.0485	0.0495	0.0505	0.0516	0.0526	0.0537
-1.6	0.0548	0.0559	0.0571	0.0582	0.0594	0.0606	0.0618	0.0630	0.0643	0.0655
-1.5	0.0668	0.0681	0.0694	0.0708	0.0721	0.0735	0.0749	0.0764	0.0778	0.0793
-1.4	0.0808	0.0823	0.0838	0.0853	0.0869	0.0885	0.0901	0.0918	0.0934	0.0951
-1.3	0.0968	0.0985	0.1003	0.1020	0.1038	0.1056	0.1075	0.1093	0.1112	0.1131
-1.2	0.1151	0.1170	0.1190	0.1210	0.1230	0.1251	0.1271	0.1292	0.1314	0.1335
-1.1	0.1357	0.1379	0.1401	0.1423	0.1446	0.1469	0.1492	0.1515	0.1539	0.1562
-1.0	0.1587	0.1611	0.1635	0.1660	0.1685	0.1711	0.1736	0.1762	0.1788	0.1814
-0.9	0.1841	0.1867	0.1894	0.1922	0.1949	0.1977	0.2005	0.2033	0.2061	0.2090
-0.8	0.2119	0.2148	0.2177	0.2206	0.2236	0.2266	0.2296	0.2327	0.2358	0.2389
-0.7	0.2420	0.2451	0.2483	0.2514	0.2546	0.2578	0.2611	0.2643	0.2676	0.2709
-0.6	0.2743	0.2776	0.2810	0.2843	0.2877	0.2912	0.2946	0.2981	0.3015	0.3050
-0.5	0.3085	0.3121	0.3156	0.3192	0.3228	0.3264	0.3300	0.3336	0.3372	0.3409
-0.4	0.3446	0.3483	0.3520	0.3557	0.3594	0.3632	0.3669	0.3707	0.3745	0.3783
-0.3	0.3821	0.3859	0.3897	0.3936	0.3974	0.4013	0.4052	0.4090	0.4129	0.4168
-0.2	0.4207	0.4247	0.4286	0.4325	0.4364	0.4404	0.4443	0.4483	0.4522	0.4562
-0.1	0.4602	0.4641	0.4681	0.4721	0.4761	0.4801	0.4840	0.4880	0.4920	0.4960
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

Set A — reading the table (problems 1–6)

1. Find P(Z ≤ 1.25).

Direct read: row 1.2, column 0.05 → 0.8944. 89.44% of values fall below z = 1.25.

2. Find P(Z > 0.92).

Right tail: Φ(0.92) = 0.8212, so 1 − 0.8212 = 0.1788.

3. Find P(Z ≤ −0.67).

Negative table, direct read: row −0.6, column 0.07 → 0.2514. Below the mean, so an answer under 0.5 — check passed.

4. Find P(−1.50 < Z < 1.50).

Between: Φ(1.50) − Φ(−1.50) = 0.9332 − 0.0668 = 0.8664. (Compare with the empirical rule's ±1 band of 0.6827 — widening to ±1.5 captures much more.)

5. Find P(Z > 2.45).

Right tail: 1 − Φ(2.45) = 1 − 0.9929 = 0.0071 — about 7 in 1,000.

6. Find the area between the mean and z = 1.13.

Mean-to-z shape: Φ(1.13) − 0.5000 = 0.8708 − 0.5000 = 0.3708.

Set B — raw scores in, answers out (problems 7–12)

7. A test has μ = 100, σ = 15. Amara scores 130. What percentage scored below her?

z = (130 − 100) / 15 = 2.00. Φ(2.00) = 0.9772 → 97.72% scored below her; she is in the top 2.28%.

8. A scale has μ = 50, σ = 5. Ben scores 43. What is his percentile?

z = (43 − 50) / 5 = −1.40. Φ(−1.40) = 0.0808 → the 8.08th percentile, roughly the 8th.

9. Women's heights average 165 cm with σ = 6 cm. What proportion are taller than 174 cm?

z = (174 − 165) / 6 = 1.50. Right tail: 1 − 0.9332 = 0.0668 — about 1 in 15.

10. Daily rainfall in a dry season averages 30 mm-equivalent units on a normalised index with σ = 4. On what share of days does the index exceed 38?

z = (38 − 30) / 4 = 2.00. Right tail: 1 − 0.9772 = 0.0228 — about 2.3% of days. (Real rainfall is skewed, so this is the normal model's answer, not a forecast.)

11. A distribution has μ = 200, σ = 25. What raw score corresponds to z = −0.50?

Run the formula backwards: X = μ + zσ = 200 + (−0.50 × 25) = 187.5.

12. Which is the stronger relative performance: 92 where μ = 80, σ = 8, or 45 where μ = 38, σ = 4?

First: z = (92 − 80) / 8 = 1.50, the 93.32nd percentile. Second: z = (45 − 38) / 4 = 1.75, and Φ(1.75) = 0.9599, the 95.99th percentile. The second performance is stronger, by a quarter of a standard deviation.

Set C — exam shapes (problems 13–20)

13. Find P(0.33 < Z < 1.44).

Two positives, subtract: Φ(1.44) − Φ(0.33) = 0.9251 − 0.6293 = 0.2958.

14. A two-tailed test produces z = 2.03. Find the p-value and state the verdict at α = 0.05.

One tail: 1 − Φ(2.03) = 1 − 0.9788 = 0.0212. Doubled: p = 0.0424, which is below 0.05 → reject the null hypothesis at the 5% level. (It would not survive α = 0.01.)

15. Find the z score that cuts off the bottom 5% of the distribution.

Reverse lookup for 0.0500 in the negative half: between −1.64 (0.0505) and −1.65 (0.0495); exact value z = −1.645.

16. Find P(Z ≤ 0.08), and explain why the answer is only slightly above 0.5.

Direct read: 0.5319. z = 0.08 is barely above the mean, and the curve is at its densest near the mean, but 0.08 of a standard deviation is a short walk — it collects only about 3.19 points of area past the halfway mark.

17. Find the z score for the 99th percentile.

Reverse lookup for 0.9900: the body of the table shows 0.9901 at z = 2.33; the exact value is z = 2.326 — also the one-tailed critical value at α = 0.01.

18. Find P(−1.96 < Z < 1.96) without using the words "confidence interval".

Φ(1.96) − Φ(−1.96) = 0.9750 − 0.0250 = 0.9500. (It is, of course, the middle 95% — the same number that makes 1.96 famous.)

19. Find P(|Z| > 2.50).

"Outside" shape: one tail beyond 2.50 is 1 − 0.9938 = 0.0062; double it → 0.0124, about 1.24% of values.

20. A teacher wants only the top 10% to earn an A. The class has μ = 70, σ = 10. What is the lowest A mark?

Top 10% = 90th percentile → z = 1.282 (reverse lookup). X = 70 + 1.282 × 10 = 82.8, so the lowest A is 83 (marks are whole numbers, and 82.8 rounds up to the first mark that clears the bar).

How to use this set well

Work on paper, with the printed table if you have it — the skill being trained is the lookup, not the answer. When a solution disagrees with your attempt, do not just note the right number: find the exact step where your path forked (sign, tail, subtraction order, ingredient). Nearly every z-table error belongs to one of those four families, and each family has a vaccine: sketch the curve (sign and tail), write larger-minus-smaller before touching numbers (subtraction order), and square-root any variance before dividing (ingredients). Finish the sets, then revisit the two problems that beat you in a week — spaced repetition on your own mistakes is worth more than twenty fresh problems.

For guided reps first, go back through the worked examples; for the method itself, how to read a z score table.