Standard normal table

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A standard normal table also called the "Unit Normal Table" is a mathematical table for the values of Φ, the cumulative distribution function of the normal distribution.

They are used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution.

Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The standard normal distribution, represented by the letter Z, is the normal distribution having a mean of 0 and a standard deviation of 1. Since probability tables cannot be printed for every normal distribution, (there are infinite), it is common practice to convert a normal to a standard normal, and use a Z table to find probabilities.

Reading the table

Tables use at least 3 different conventions, depending on the interpretation of the meaning of an entry such as 1.57:

Cumulative

This is most common, and gives Prob(Z ≤ 1.57) = 0.9418.

Complementary cumulative

The complement (1–x) of above: Prob(Z ≥ 1.57) = .0582.

Cumulative from zero

The cumulative probability, starting from 0: Prob (0 ≤ Z ≤ 1.57) = .4418

These can easily be checked by inspecting a number like 2.99:

• if this is approximately 1 (or rather 0.99..), then it displays cumulative probabilities;
• if this is approximately 0 (or rather 0.00..), then it displays complementary probabilities;
• if this is approximately 0.5 (or rather 0.49..), then it displays cumulative from 0 probabilities.

Printed tables usually give cumulative probabilities^{[citation needed]}, the chance that a statistic takes a value less than or equal to a number, from at least 0.00 to 2.99 by 1/100. To read the value 1.57 on a typical table, go to 1.5 down and 0.07 across. The probability of Z ≤ 1.57 = 0.9418.

If your table does not have negative values, use symmetry to find the answer. Remember that 50% falls below and above 0.

Converting from normal to standard normal

If X is a random variable from a normal distribution with mean μ and standard deviation σ, its Z-score may be calculated from X by subtracting μ and dividing by σ.

$Z = \frac{X - \mu}{\sigma} \!$

If you are using an average, divide the standard deviation by the square root of the sample size.

$Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \!$

Examples

A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5.

• What is the probability that a student scores an 82 or less?

Prob(X ≤ 82) = Prob(Z ≤ (82-80)/5) = Prob(Z ≤ .40) = .6554

• What is the probability that a student scores a 90 or more?

Prob(X ≥ 90) = Prob(Z ≥ (90-80)/5) = Prob(Z ≥ 2.00) = 1 - Prob(Z ≤ 2.00) = 1 - .9772 = .0228

• What is the probability that a student scores a 74 or less?

Prob(X ≤ 74) = Prob(Z ≤ (74-80)/5) = Prob(Z ≤ -1.20) = .1151

If your table does not have negatives, use Prob(Z ≤ -1.20) = Prob(Z ≥ 1.20) = 1 - .8849 = .1151

• What is the probability that a student scores between 78 and 88?

Prob(78 ≤ X ≤ 88) = Prob((78-80)/5 ≤ Z ≤ (88-80)/5) = Prob(-0.40 ≤ Z ≤ 1.60) = Prob(Z ≤ 1.60) - Prob(Z ≤ -0.40) = .9452 - .3446 = .6006

• What is the probability that an average of three scores is 82 or less?

Prob(X ≤ 82) = Prob(Z ≤ (82-80)/(5/√3)) = Prob(Z ≤ .69) = .7549

Partial Table

The below table read by using the rows to find the first digit, and the columns to find the second digit of a Z-score. To find 0.69, first look down the rows to find 0.6 and then across the columns to 0.09 and 0.7549 will be the result.

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

References

• (2004) Elementary Statistics: Picturing the World, 清华大学出版社.