# 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 tableEdit

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 normalEdit

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.

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

### ExamplesEdit

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 TableEdit

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 |

## ReferencesEdit

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

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