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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.

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}} \!

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 0.000.010.020.030.040.050.060.070.080.09
0.0 0.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359
0.1 0.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753
0.2 0.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
0.3 0.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
0.4 0.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
0.5 0.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
0.6 0.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
0.7 0.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852
0.8 0.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133
0.9 0.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389
1.0 0.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621
1.1 0.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830
1.2 0.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015
1.3 0.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
1.4 0.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
1.5 0.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
1.6 0.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
1.7 0.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
1.8 0.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
1.9 0.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
2.0 0.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817
2.1 0.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
2.2 0.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.3 0.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.4 0.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.5 0.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.6 0.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.7 0.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.8 0.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.9 0.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.0 0.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

ReferencesEdit

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