The Chi Squared Distribution Page 1
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1 The Chi Square Distributio Page Cosier the istributio of the square of a score take from N(, The probability that z woul have a value less tha is give by z / g ( ( e z if > F π, if < z where ( e g e z π with g ( The pf of this istributio is give by the erivative π / e ( ( g ( if > f F π if < This is the chi square istributio with a sigle egree of freeom Degrees of freeom are ofte esigate by the Greek letter u Here The momet geeratig fuctio of chi-square,, is give by t t e e M ( t e π We restrict t to the omai, The itegral ca be evaluate by makig the substitutio ( t u a usig e the ormalizatio property (ie, u π of the pf ( t u e e M ( t ( t u ( t π π ( u Now for iepeet raom variables,, 3,, the momet geeratig fuctio of the sum Y α + α+ α αis give by the prouct of the separate momets t αkk k M t e f,, ( ( Y 3 3 t αkk k e f( f( f( 3 tα tα tα e f( e f( e f( M ( αt M ( αt M ( αt If each of the iepeet variables comes from the same populatio a the coefficiets are all oe this simplifies to ( ( MY t M t Al Lehe Maiso Area Techical College 8/4/3 (3 Now cosier the istributio of the sum of the squares of a scores take iepeetly from N(, Callig this sum from equatios ( a (3 it has the momet geeratig fuctio This fuctio has the followig properties ( ( M t t (4 (
2 The Chi Square Distributio Page t M t t t t ( ( ( + ( ( ( + M ( ( ( ( + ( ( ( ( M t t t Thus, without a etaile kowlege of the pf of we kow its mea a variace! + ( ( (5 ( ( μ M (6 Thus the sum of square z scores from a ormal istributio has a mea of a a staar eviatio of But i fact the momet geeratig fuctio (4 ca also give us the form of the pf for From equatio ( it woul be reasoable to guess that for some ormalizatio factor A( / p Ae ( if > f ( if < The momet geeratig fuctio of this pf is give below t t p ( t p M t e A e e A e ( ( ( Makig the substitutio ( t u, the itegral ca be epresse i terms of the gamma fuctio ( p p+ ( ( ( ( p+ u p p+ ( ( ( p+ ( + M t A t e u u A t p Comparig this result with Equatio (4 we coclue that istributio with has the followig pf f ( p + a A ( / e if > if < (7 + Thus, the chi-square From equatios (5 a (6 we alreay kow the mea a staar eviatio of this istributio The value of the moe ca be etermie by calculatig the first erivative / ( ( e if > f ( if < Moe for Graphs of the istributio for various egrees of freeom are show i Figure v Al Lehe Maiso Area Techical College 8/4/3
3 The Chi Square Distributio Page 3 Figure I the Appei it is show that for large the istributio for v approaches N (, Now cosier iepeet raom variables,, 3, all raw iepeetly from the same populatio From this sample we ca compute the sample mea a sample variace s i i ( i i ( i i For ay iepeet sample the mea of the sample variace is the populatio variace ( i ( ( μ ( μ s i Al Lehe Maiso Area Techical College 8/4/3
4 The Chi Square Distributio Page 4 If the paret populatio is N (, μ we ca be more precise Uer these circumstaces it ca be show that the sample mea a sample variace are iepeet raom variables Furthermore, the sample mea is ormally istribute like N, μ The sum ( i μ is istribute like chi square with egrees of freeom i ( ( ( ( ( ( ( i i i i + i i i ( μ ( μ ( ( i i + i i i So i ( μ ( μ ( i + + i μ μ μ μ But for ay sample the eviatios about the sample mea a up to zero ( Sice the sample variace a the sample mea are iepeet raom variables, the momet geeratig fuctio of ( μ + is the prouct of the momet geeratig fuctio of a ( μ But ( μ is the square of a z score from a ormal populatio a has the momet geeratio fuctio give by equatio( The momet geeratig fuctio of is give by equatio (4 Thus, we have that M t t M t t Solvig for M ( t we get ( ( ( M t t ( ( ((, which is the momet geeratig fuctio of a chi square istributio with egrees of freeom We therefore have the result that for a iepeet sample take from a ( ormal istributio that s is istribute like chi square with I this case we have ( s or s which we alreay kew to be true What s ew is that we ow 4 kow the variace of the sample variace! ( s s or Note that the sprea of the sample variace scales like the populatio variace (it must have square uits! a ecreases roughly like the iverse square root of the sample size Thus, for large eough samples the value of the sample variace coverges to the populatio variace Ufortuately, these results are ot robust for epartures of the paret populatio from ormality so that i practice the istributio of sample variaces ca look very ifferet tha a chi square istributio Al Lehe Maiso Area Techical College 8/4/3
5 The Chi Square Distributio Page 5 Appei : The Chi Square Distributio for Large Degrees of Freeom / e if > I the chi square istributio f (, let + z This i effect eamies the ature of the if < istributio at z staar eviatios from the mea For large the gamma fuctio is well approimate by Stirlig s formula ( u u u ( u e π ( u e Usig this approimatio i the chi-square istributio yiels the followig: + z / z e z ep + + ( l + z / e + ( π π e e z z ep + + ( l ( + z ep l z π π ep ( l( ( e z ep + + ( l + z ( l ( + ( ep ( A π π 3 l + + O, the fuctio A( Usig the Maclauri series for the atural logarithm: ( ( follows: So we have that ( z z 5 ( ( ( ca be epae as A + z + Ov + Ov + z 5 z 5 z + z + Ov ( + + Ov ( + Ov ( z f + z z e which meas that as the egrees of freeom icrease a chi square π istributio approaches N (,, a ormal istributio with the same mea a staar eviatio as the chi square istributio Al Lehe Maiso Area Techical College 8/4/3
6 The Chi Square Distributio Page 6 Appei : The Epecte Value of the Sample Staar Deviatio from A Normal Populatio It was show above that for ay iepeet sample the mea of the sample variace is the populatio variace, ( s, a that if the paret populatio is ormally istribute that s is istribute like chi square with Here we cosier the istributio of the sample staar eviatio of scores take from a ormal ( populatio Now, s s Y, wherey is a raom variable chose from a chi square istributio s with egrees of freeom The iverse trasformatio is Y Y s with, so the probability istributio of s s is give by So the epectatio of s is give by u with the iverse trasformatio s s ep s Y s s f ( ( s s f Y f s ( s s ep ( s ep u with u, u ( ( + + Makig the u substitutio, ( u s u e u + + ( + For a sample of size iepeet scores from a ormal istributio the epecte value of the sample staar eviatio is give by s Al Lehe Maiso Area Techical College 8/4/3
7 The Chi Square Distributio Page 7 A graph of versus is show i Figure Sice s the sample staar eviatio, eve for a ormal istributio, is ot a ubiase estimator of the populatio staar eviatio I fact the ratio is always less tha oe, so the sample staar eviatio, o the average, uerestimates the populatio staar eviatio Usig Stirlig s formula it s ot ifficult to show as the sample size icreases that I practice the sample staar eviatio is still the most commoly use estimator for the populatio staar eviatio Most tests o the variability of a populatio are formulate i terms of the variace rather tha the staar eviatio Figure Al Lehe Maiso Area Techical College 8/4/3
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