Standard normal table

From Wikipedia, the free encyclopedia - View original article

 
Jump to: navigation, search

A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, which are the values of the cumulative distribution function of the normal distribution. It is 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. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal and then use the standard normal table to find probabilities.[1]


Normal and standard normal distribution[edit]

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.

Conversion[edit]

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}

For the average of a sample from a population n in which the mean is μ and the standard deviation is S, the standard deviation is S/√n:

t = \frac{\overline{X} - \mu}{S / \sqrt{n}}

Reading a Z table[edit]

Formatting / layout[edit]

Z tables are composed as follows:

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 0.7549 from a cumulative table.

Types of tables[edit]

Z tables use at least three different conventions:

Cumulative
gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z. Example: Prob(Z ≤ 0.69) = 0.7549. This value is found directly from the table.
Complementary cumulative
gives a probability that a statistic is greater than Z. This equates to the area of the distribution above Z.

Example: Find Prob(Z ≥ 0.69). Since this is the portion of the area above Z, the proportion that is greater than Z is found by subtracting Z from 1. That is Prob(Z ≥ 0.69) = 1 - Prob(Z ≤ .69) or Prob(Z  ≥  0.69) = 1 - .7549 = .2451.

Cumulative from mean
gives a probability that a statistic is between 0 (mean) and Z. Example: Prob(0 ≤ Z ≤ 0.69) = 0.2549

Table examples[edit]

Cumulative table[edit]

This table gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z.
z0.000.010.020.030.040.050.060.070.080.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

Cumulative from mean (0 to Z)[edit]

This table gives a probability that a statistic is between 0 (mean) and Z.
z0.000.010.020.030.040.050.060.070.080.09
0.00.000000.003990.007980.011970.015950.019940.023920.027900.031880.03586
0.10.039830.043800.047760.051720.055670.059660.06360.067490.071420.07535
0.20.079260.083170.087060.090950.094830.098710.102570.106420.110260.11409
0.30.117910.121720.125520.129300.133070.136830.140580.144310.148030.15173
0.40.155420.159100.162760.166400.170030.173640.177240.180820.184390.18793
0.50.191460.194970.198470.201940.205400.208840.212260.215660.219040.22240
0.60.225750.229070.232370.235650.238910.242150.245370.248570.251750.25490
0.70.258040.261150.264240.267300.270350.273370.276370.279350.282300.28524
0.80.288140.291030.293890.296730.299550.302340.305110.307850.310570.31327
0.90.315940.318590.321210.323810.326390.328940.331470.333980.336460.33891
1.00.341340.343750.346140.348490.350830.353140.355430.357690.359930.36214
1.10.364330.366500.368640.370760.372860.374930.376980.379000.381000.38298
1.20.384930.386860.388770.390650.392510.394350.396170.397960.399730.40147
1.30.403200.404900.406580.408240.409880.411490.413080.414660.416210.41774
1.40.419240.420730.422200.423640.425070.426470.427850.429220.430560.43189
1.50.433190.434480.435740.436990.438220.439430.440620.441790.442950.44408
1.60.445200.446300.447380.448450.449500.450530.451540.452540.453520.45449
1.70.455430.456370.457280.458180.459070.459940.460800.461640.462460.46327
1.80.464070.464850.465620.466380.467120.467840.468560.469260.469950.47062
1.90.471280.471930.472570.473200.473810.474410.475000.475580.476150.47670
2.00.477250.477780.478310.478820.479320.479820.480300.480770.481240.48169
2.10.482140.482570.483000.483410.483820.484220.484610.485000.485370.48574
2.20.486100.486450.486790.487130.487450.487780.488090.488400.488700.48899
2.30.489280.489560.489830.490100.490360.490610.490860.491110.491340.49158
2.40.491800.492020.492240.492450.492660.492860.493050.493240.493430.49361
2.50.493790.493960.494130.494300.494460.494610.494770.494920.495060.49520
2.60.495340.495470.495600.495730.495850.495980.496090.496210.496320.49643
2.70.496530.496640.496740.496830.496930.497020.497110.497200.497280.49736
2.80.497440.497520.497600.497670.497740.497810.497880.497950.498010.49807
2.90.498130.498190.498250.498310.498360.498410.498460.498510.498560.49861
3.00.498650.498690.498740.498780.498820.498860.498890.498930.498960.49900

[2]

Examples of use[edit]

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

P(X \le 82) = P\left(Z \le \frac{82 - 80}{5}\right) = P(Z \le 0.40) =0.15542 + 0.5000 = 0.65542
P(X \ge 90) = P\left(Z \ge \frac{90 - 80}{5}\right) = P(Z \ge 2.00) = 1 - P(Z \le 2.00) = 1 - 0.9772 = 0.0228
P(X \le 74) = P\left(Z \le \frac{74 - 80}{5}\right) = P(Z \le - 1.20) = 0.1151

If the table does not have negatives, one uses:

P(Z \le -1.20) = P(Z \ge 1.20) = 1 - 0.8849 = 0.1151
P(78 \le X \le 88) = P\left(\frac{78 - 80}{5} \le Z \le \frac{88 - 80}{5}\right)  = P(-0.40 \le Z \le 1.60) = P(Z \le 1.60) - P(Z \le - 0.40) = 0.9452 - 0.3446 = 0.6006
P(X \le 82) = P\left(Z \le \frac{82 - 80}{5/\sqrt{3}}\right) = P(Z \le 0.69) = 0.7549

References[edit]

  1. ^ Larson, Ron; Farber, Elizabeth (2004). Elementary Statistics: Picturing the World. 清华大学出版社. p. 214. ISBN 7-302-09723-2. 
  2. ^ "Cumulative Distribution Function of the Standard Normal Distribution". NIST. Retrieved 5 May 2012.