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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.<ref>{{cite book |title=Elementary Statistics: Picturing the World|first1=Ron
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|last1=Larson|first2=Elizabeth|last2=Farber|publisher=清华大学出版社|year=2004|isbn=7-302-09723-2|page=214}}</ref>
 
 
==Normal and standard 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.
 
===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 σ.
 
: <math>Z = \frac{X - \mu}{\sigma} </math>
 
For the average of a sample from a population n in which the mean is μ and the standard deviation is S, the standard error is S/&radic;''n'':
 
: <math>t = \frac{\overline{X} - \mu}{S / \sqrt{n}} </math>
 
==Reading a Z table==
===Formatting / layout===
Z tables are composed as follows:
* The label for rows contains the first two most significant digits of Z.
* The label for columns contains the least significant digits of Z.
* The values within the table are the probabilities.  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 '''0.7549''' from a ''cumulative'' table.
 
===Types of tables===
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&nbsp;≤&nbsp;0.69)&nbsp;=&nbsp;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''&nbsp;≥&nbsp;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''&nbsp;≥&nbsp;0.69) = 1 - Prob(Z ≤ .69) or Prob(''Z''&nbsp; ≥ &nbsp;0.69) = 1 - .7549 = .2451.
;Cumulative from mean: gives a probability that a statistic is between 0 (mean) and Z.  Example: Prob(0&nbsp;≤&nbsp;Z&nbsp;≤&nbsp;0.69)&nbsp;=&nbsp;0.2549
 
==Table examples==
===Cumulative table===
[[File:Z cumulative.svg|thumb|right|This table gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z.]]
{| border="1"
! ''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
|}
 
===Cumulative from mean (0 to Z)===
[[File:Z cumulative from mean.svg|thumb|right|This table gives a probability that a statistic is between 0 (mean) and Z.]]
{| border="1"
! ''z'' !!0.00!!0.01!!0.02!!0.03!!0.04!!0.05!!0.06!!0.07!!0.08!!0.09
|-
! 0.0
|0.00000||0.00399||0.00798||0.01197||0.01595||0.01994||0.02392||0.02790||0.03188||0.03586
|-
! 0.1
|0.03983||0.04380||0.04776||0.05172||0.05567||0.05966||0.0636||0.06749||0.07142||0.07535
|-
! 0.2
|0.07926||0.08317||0.08706||0.09095||0.09483||0.09871||0.10257||0.10642||0.11026||0.11409
|-
! 0.3
|0.11791||0.12172||0.12552||0.12930||0.13307||0.13683||0.14058||0.14431||0.14803||0.15173
|-
! 0.4
|0.15542||0.15910||0.16276||0.16640||0.17003||0.17364||0.17724||0.18082||0.18439||0.18793
|-
! 0.5
|0.19146||0.19497||0.19847||0.20194||0.20540||0.20884||0.21226||0.21566||0.21904||0.22240
|-
! 0.6
|0.22575||0.22907||0.23237||0.23565||0.23891||0.24215||0.24537||0.24857||0.25175||0.25490
|-
! 0.7
|0.25804||0.26115||0.26424||0.26730||0.27035||0.27337||0.27637||0.27935||0.28230||0.28524
|-
! 0.8
|0.28814||0.29103||0.29389||0.29673||0.29955||0.30234||0.30511||0.30785||0.31057||0.31327
|-
! 0.9
|0.31594||0.31859||0.32121||0.32381||0.32639||0.32894||0.33147||0.33398||0.33646||0.33891
|-
! 1.0
|0.34134||0.34375||0.34614||0.34849||0.35083||0.35314||0.35543||0.35769||0.35993||0.36214
|-
! 1.1
|0.36433||0.36650||0.36864||0.37076||0.37286||0.37493||0.37698||0.37900||0.38100||0.38298
|-
! 1.2
|0.38493||0.38686||0.38877||0.39065||0.39251||0.39435||0.39617||0.39796||0.39973||0.40147
|-
! 1.3
|0.40320||0.40490||0.40658||0.40824||0.40988||0.41149||0.41308||0.41466||0.41621||0.41774
|-
! 1.4
|0.41924||0.42073||0.42220||0.42364||0.42507||0.42647||0.42785||0.42922||0.43056||0.43189
|-
! 1.5
|0.43319||0.43448||0.43574||0.43699||0.43822||0.43943||0.44062||0.44179||0.44295||0.44408
|-
! 1.6
|0.44520||0.44630||0.44738||0.44845||0.44950||0.45053||0.45154||0.45254||0.45352||0.45449
|-
! 1.7
|0.45543||0.45637||0.45728||0.45818||0.45907||0.45994||0.46080||0.46164||0.46246||0.46327
|-
! 1.8
|0.46407||0.46485||0.46562||0.46638||0.46712||0.46784||0.46856||0.46926||0.46995||0.47062
|-
! 1.9
|0.47128||0.47193||0.47257||0.47320||0.47381||0.47441||0.47500||0.47558||0.47615||0.47670
|-
! 2.0
|0.47725||0.47778||0.47831||0.47882||0.47932||0.47982||0.48030||0.48077||0.48124||0.48169
|-
! 2.1
|0.48214||0.48257||0.48300||0.48341||0.48382||0.48422||0.48461||0.48500||0.48537||0.48574
|-
! 2.2
|0.48610||0.48645||0.48679||0.48713||0.48745||0.48778||0.48809||0.48840||0.48870||0.48899
|-
! 2.3
|0.48928||0.48956||0.48983||0.49010||0.49036||0.49061||0.49086||0.49111||0.49134||0.49158
|-
! 2.4
|0.49180||0.49202||0.49224||0.49245||0.49266||0.49286||0.49305||0.49324||0.49343||0.49361
|-
! 2.5
|0.49379||0.49396||0.49413||0.49430||0.49446||0.49461||0.49477||0.49492||0.49506||0.49520
|-
! 2.6
|0.49534||0.49547||0.49560||0.49573||0.49585||0.49598||0.49609||0.49621||0.49632||0.49643
|-
! 2.7
|0.49653||0.49664||0.49674||0.49683||0.49693||0.49702||0.49711||0.49720||0.49728||0.49736
|-
! 2.8
|0.49744||0.49752||0.49760||0.49767||0.49774||0.49781||0.49788||0.49795||0.49801||0.49807
|-
! 2.9
|0.49813||0.49819||0.49825||0.49831||0.49836||0.49841||0.49846||0.49851||0.49856||0.49861
|-
! 3.0
|0.49865||0.49869||0.49874||0.49878||0.49882||0.49886||0.49889||0.49893||0.49896||0.49900
|}
<ref>{{cite web|title=Cumulative Distribution Function of the Standard Normal Distribution|publisher=NIST|url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm|accessdate=5 May 2012}}</ref>
 
==Examples of use==
 
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?
 
:: <math>P(X \le 82) = P\left(Z \le \frac{82 - 80}{5}\right) = P(Z \le 0.40) =0.15542 + 0.5000 = 0.65542</math>
 
* What is the probability that a student scores a 90 or more?
 
:: <math>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</math>
 
* What is the probability that a student scores a 74 or less?
 
:: <math>P(X \le 74) = P\left(Z \le \frac{74 - 80}{5}\right) = P(Z \le - 1.20) = 0.1151</math>
 
If the table does not have negatives, one uses:
 
:: <math>P(Z \le -1.20) = P(Z \ge 1.20) = 1 - 0.8849 = 0.1151</math>
 
* What is the probability that a student scores between 78 and 88?
 
:: <math>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</math>
 
* What is the probability that an average of three scores is 82 or less?
 
:: <math>P(X \le 82) = P\left(Z \le \frac{82 - 80}{5/\sqrt{3}}\right) = P(Z \le 0.69) = 0.7549</math>
 
==References==
{{reflist}}
 
[[Category:Continuous distributions]]
[[Category:Mathematical tables]]

Latest revision as of 18:02, 13 March 2014

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