Standard normal table

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, 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.