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.
Conversion
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 the standard deviation: For the average of a sample of size n from some population in which the mean is μ and the standard deviation is σ, the standard error is σ/√n:
The label for columns contains the second decimal place of Z.
The values within the table are the probabilities corresponding to the table type. These probabilities are calculations of the area under the normal curve from the starting point to Z.
Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table. Because the normal distribution curve is symmetrical, probabilities for only positive values of Z are typically given. The user has to use a complementary operation on the absolute value of Z, as in the example below.
Types of tables
Z tables use at least three different conventions: ;Cumulative from mean: gives a probability that a statistic is between 0 and Z. Example: Prob = 0.2549 ;Cumulative: gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z. Example: Prob = 0.7549. ;Complementary cumulative: gives a probability that a statistic is greater than Z. This equates to the area of the distribution above Z.
Table examples
Cumulative from mean (0 to Z)
This table gives a probability that a statistic is between 0 and Z. Note that for z = 1, 2, 3, one obtains the results f = 0.6827, 0.9545, 0.9974, characteristic of the 68–95–99.7 rule.
Cumulative
This table gives a probability that a statistic is less than Z. The values are calculated using the cumulative distribution function of a standard normal distribution with mean of zero and standard deviation of one, usually denoted with the capital Greek letter , is the integral is related to the error function, or erf.
Complementary cumulative
This table gives a probability that a statistic is greater than Z. This table gives a probability that a statistic is greater than Z, for large integer Z values.
z
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
0
5.00000 E -1
1.58655 E -1
2.27501 E -2
1.34990 E -3
3.16712 E -5
2.86652 E -7
9.86588 E -10
1.27981 E -12
6.22096 E -16
1.12859 E -19
10
7.61985 E -24
1.91066 E -28
1.77648 E -33
6.11716 E -39
7.79354 E -45
3.67097 E -51
6.38875 E -58
4.10600 E -65
9.74095 E -73
8.52722 E -81
20
2.75362 E -89
3.27928 E -98
1.43989 E -107
2.33064 E -117
1.39039 E -127
3.05670 E -138
2.47606 E -149
7.38948 E -161
8.12387 E -173
3.28979 E -185
30
4.90671 E -198
2.69525 E -211
5.45208 E -225
4.06119 E -239
1.11390 E -253
1.12491 E -268
4.18262 E -284
5.72557 E -300
2.88543 E -316
5.35312 E -333
40
3.65589 E -350
9.19086 E -368
8.50515 E -386
2.89707 E -404
3.63224 E -423
1.67618 E -442
2.84699 E -462
1.77976 E -482
4.09484 E -503
3.46743 E -524
50
1.08060 E -545
1.23937 E -567
5.23127 E -590
8.12606 E -613
4.64529 E -636
9.77237 E -660
7.56547 E -684
2.15534 E -708
2.25962 E -733
8.71741 E -759
60
1.23757 E -784
6.46517 E -811
1.24283 E -837
8.79146 E -865
2.28836 E -892
2.19180 E -920
7.72476 E -949
1.00178 E -977
4.78041 E -1007
8.39374 E -1037
70
5.42304 E -1067
1.28921 E -1097
1.12771 E -1128
3.62960 E -1160
4.29841 E -1192
1.87302 E -1224
3.00302 E -1257
1.77155 E -1290
3.84530 E -1324
3.07102 E -1358
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Examples of use
A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. Only a cumulative from mean table is available.
What is the probability that a student scores an 82 or less?
What is the probability that a student scores a 90 or more?
What is the probability that a student scores a 74 or less?
What is the probability that a student scores between 74 and 82?
What is the probability that an average of three scores is 82 or less?
