The Sample Variance Formula: A Detailed Study of an Old Controversy


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1 The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Abstract The two biased ad ubiased formulae for the sample variace of a radom variable are uder scrutiy. New research result proves that the formula with a smaller divisor is ubiased ad the formula with a larger divisor is biased. This fact agrees with the curret belief i statistics literature. May mathematical proofs for both formulae cotai errors: This is the reaso for the cotroversy ad this ew research. The fial verdict comes from simulatio results. Keywords: Estimator epectatio operator Helmert s trasformatio sample variace. Itroductio Statisticias kow for a log time that there are two formulae for the sample variace of a radom variable differet by a divisor. For decades a umber of statisticias seem to be happy with a formula called the ubiased formula with a smaller divisor. This formula is correct but the proofs for its ubiasedess by a umber of authors are icorrect; for eample see P.G. Hoel 97) ad I. Guttma S.S. Wilks ad J.S. Huter 97). The other formula with a larger divisor called the biased formula is believed wrog but supported by a smaller umber of statisticias; for eample see R.V. Hogg ad A.T. Craig 978). I this paper we will give a detailed study of the formulae with their supported argumets. The paper is orgaized as follows. Sectio oe is the itroductio sectio. I sectio two we preset the formulae for the sample variace. I sectio three we preset three icorrect proofs collected from tetbooks for the ubiased formula. I sectio four we preset some argumets for the biased formula. I sectio five we discuss the Helmert s trasformatio ad its applicatio to the sample variace formula. I sectio si we report the simulatio results which give the verdict as to which formula is the right oe to use. Fially sectio seve cocludes the discussio of the paper. momet kow as the mea is give as t. ) There is o cotroversy about this formula. There is however a cotroversy for the estimator of the secod momet about the mea kow as the variace. There are two estimators kow by two formulae: ad ˆσ ubiased ˆσ biased t ) ) t ) 3) for this statistic called the sample variace. By developig the square i the last equatio we ca write ˆσ biased t ) t t + ) [ t + [ t [ t t ) t t ). From the last equatio statisticias call the variace the secod momet about the mea. The Formulae for the Sample Variace Suppose that we have a radom variable X has a distributio with mea µ ad variace σ which gives values t s t ) i a sample. The estimated value for the first 3 Argumets for the Ubiased Formula I this sectio we will cite three proofs from propoets for Equatio ). I oe old tetbook which the author of this paper ca o loger fid there is a argumet as follows. If we take the epectatio of both sides of the last
2 equatio i the last sectio we have E{ˆσ biased} E{ E{ t ) t } t } E{ ) t } E{ t } ) E{ t }.4) If t s are idepedet observatios the epected values of the crossproducts of the secod term o the right had side of the above equatio are zeros ad we obtai E{ˆσ biased} σ σ σ σ σ. The secod proof is take from P.G. Hoel 97) pages 99. From the properties of E ad the defiitio of σ it follows that [ E{ˆσ biased} E [ E [ E t ) [ t µ) µ)] t µ) µ) E t µ) E µ) σ σ σ σ σ. The first proof does ot use the mea ad its estimator but the secod proof uses both. The epected value of the estimator or sample variace ˆσ biased is ot the same as the populatio parameter σ so the proofs claim that the estimator ˆσ biased is biased hece the reaso for its ame. The followig argumet i aother tetbook I. Guttma S.S. Wilks ad J.S. Huter 97) pages 88 is a more laborious argumet ad stroger proof for the formula give by Equatio ). Suppose that ) is a radom sample of a radom variable X havig mea µ ad variace σ. The the radom variable L c + + c where the c i s are costats has the epectatio µ ad variace σ i c i. I the special case cosider we obtai + + E{ } µ V ar ) σ. The last equatio states that E{ µ) } σ. Now cosider the algebraic idetity i µ) i ) + µ). i i i c i Usig properties of the epectatio operator we have E i µ) E{ i ) } + E µ). 5) i i Ad makig use of a previous result we fid σ E{ i ) } + σ i i ad that meas E{ i ) } )σ. Fially we have i E{ i ) } σ. 6) i The first ad secod proofs try to prove that the estimator ˆσ biased is biased; the third proof the estimator ˆσ ubiased is ubiased. Ufortuately all the proofs are wrog as the discussio i the et sectio will reveal. 4 Argumets for the Biased Formula I this sectio we will give some argumets for the formula give by Equatio 3). These argumets are listed below. 4. Commo Sese Oce i a while a mathematicia gives a proof for his thikig o some problem that is so cotrary to commo sese thikig ad he believes i his proof. But ufortuately his proof fails him ad does ot solve the problem. The story of the Greek mathematicia Zeo of Elea BC) with the solved parado of Achilles ad the Tortoise is oe eample. The story of the sample variace formula might be aother. From commo sese thikig we kow that to obtai the variace we must divide the sum of squares of the values deviated from the mea by the total umber of values.
3 4. The Probability I R.V. Hogg ad A.T. Craig 978) pages 45 the authors of the referece advocated for the variace formula give by Equatio 3). These authors eplaied that the ratio / is the probability of each evet for the radom variable X to have the value t. With such a isight it is too clear to see that the correct sample variace formula must have the divisor ot ). This is because the formula is a reflectio of a epectatio operator E which is a probabilityweighted average. However these authors could either prove their preferred formula or discredit the other oe. Usig the probability cocept K. Vu 007) claimed that the epectatio operator E ca be represeted as E{) t } lim ) t where ) t is a fuctio of a radom variable with the ide t. Let us apply this techique to fid the ubiased estimator for the mea µ of a distributio. We have lim t lim t. Now assume that i the sample of values t s there is a value ad k is the umber of times t s have this value. The we ca write the last equatio as below lim t lim lim lim E{X} µ. t k P r{x } Whe the value approaches ifiity the rage of epads to cover the rage of the radom variable X ad at the value of ifiity the ratio k / simply becomes the probability desity for the radom variable X to have the value. Ad depedig o the ature of discrete or cotiuous type of the radom variable X the sum o the right had side of the last equatio will remai a sum or chage to a itegral. I either case the right had side simply becomes the epected value of t. For this to be true however the ratio k / must be a probability value. Usig this approach we ca fid the ubiased estimator for the variace easily. We are ow ready to preset ad prove our research result i a theorem. Theorem 4. The ubiased estimator for the variace of some distributio of a radom variable X with probability desity fuctio p) mea µ variace σ ad sample values t s t ) is ˆσ t ) where is the ubiased mea estimator give by Equatio ). Proof. From the equatio of the estimator we write ˆσ t ) t ). 7) Now assume that i the sample of values t s we have some t s with the value ad the umber of these t s is k times. Sice there is oly a fiite umber of values of we ca write ˆσ k ) ) {P r X } ) p. By takig the limit whe approaches ifiity the last equatio becomes σ µ) p) if X is discrete µ) p)d if X is cotiuous E{X EX)) }. At the limit of ifiity the sample becomes the populatio. Sice at the limit the estimator gives the defiitio of the populatio variace the estimator must be a ubiased estimator. This fact proves the theorem. QED Havig prove that the formula with the divisor is ubiased we must ow prove that the formula with the divisor ) is biased or the proofs i the last sectio prove othig. This is what we are goig to do et. 4.3 The Errors of the Icorrect Proofs Now we will ivestigate the proofs i the last sectio. The first proof does ot use the mea µ ad its estimator as the secod ad third proofs so it does ot make the mistake caused by the estimator but it has a flaw that is easy to fid. The flaw is about the defiitio of the variace. The variace is defied as the secod momet about the mea ot about zero. This meas that σ E{ t }. The argumet has E{ t } σ. This ca be oly true if the values t s
4 i the argumet have a zero mea. However if the values t s have a zero mea the the secod term i Equatio 4) will be zero. This will result i the correct epected value σ for the right had side of Equatio 4). The author of the argumet set out to prove somethig that is correct but obtaied a wrog result. While the secod ad third proofs appear to be mathematically flawless they actually prove othig. Their faults are i usig the value. The errors i the secod ad third proofs is i the probability desity distributios of X ad X. They are differet. Both have mea µ; but the variaces are σ ad σ / respectively. Assumig that the probability desity distributio of X is p ); the to obtai the variace about the mea of this variable we must write E{ µ) } µ) p )d σ. We oly ivestigate the case whe X is of cotiuous type; because it is the usual case for to be the average of values t s. But the argumet applies to the case whe X is of discrete type as well. Now if we apply the same procedure takig epectatio to the quatity t µ) we get E{ t µ) } t µ) p )d σ. This is because t has a wrog probability desity value. The result will lead to E{ t ) } σ which is cotrary to what Equatio 6) claims. 4.4 The Number Degrees of Freedom Aother argumet for the ubiased formula comes from the umber of degrees of freedom. Propoets for this formula claim that the sum of squares t ) has oly ) degrees of freedom because oe is lost with the calculatio of the average. I values t ) t there is oe that does ot have a free value; it ca be calculated from the other ) values. This argumet appears to be correct first ad it has may statisticias icludig the author of this paper uder its spell for some time. But it has a error like the formula it supports. The questio here is: How may degrees of freedom are there i the sum of squares t )? To aswer this questio we write the sum of squares as t ) t t ) ad cout the umber of degrees of freedom of the quatity o the right had side of the last equatio. The umber of degrees of freedom must be. Lookig at the sum o the left had side we accept the fact that it has oly ) free quatities t ). However each of these quatities has two degrees of freedom: Oe is give by t ; the other. Therefore the sum should have ) degrees of freedom. However we caot cout oe degree of freedom more tha oce. The result is: We have a total of degrees of freedom. This is why whe we remove the sum shows that it has degrees of freedom as the right had side tells us. 5 The Helmert Trasformatio A brilliat trasformatio that ca be used to solve the variace formula problem is the Helmert s trasformatio. It is a attempt to show the correct umber of degrees of freedom i the sum of squares of a sample variace formula. If we apply the Helmert s trasformatio to the variables ts t ) we obtai a ew set of variables as follows: y y y ) ) y The ew variables y i s i ) have a zero mea ad the same variace σ of the variable i. The Helmert s trasformatio is a orthogoal trasformatio; therefore the Jacobia of the trasformatio is oe ad we have t yt. Therefore we ca write yt yt y t t ) S. Sice the sum S has ) idepedet variables y i s i ) each with variace σ we ca obtai the variace for the radom variable by takig S divided by ). This meas that this is aother argumet for the ubiased formula ).
5 6 Simulatio Results As there are i s i ) i the sum of squares S we ca still argue that the sum has degrees of freedom a earlier argumet). If the variace ca be cosidered as a sum of squares divided by its umber of degrees of freedom the divisor for the sample variace formula should be. This meas that this is aother argumet for the biased formula 3). The cotroversy starts agai with the Helmert s trasformatio. The cotroversy of the variace formula becomes the cotroversy of the umber of degrees of freedom: Is it because of the variable i or is it ) because of the variable y i? As there are o criteria to choose the right umber of degrees of freedom the author of this paper decided to take the proofofthepuddig approach to solve this cotroversy with software simulatio. I this simulatio the author of this paper wrote a small software program usig the MATLAB laguage. Te thousad observatios of a stadardized ormally distributed radom variable were created. The observatios were multiplied by a costat to give the radom variable a variace of value 0 the added by aother costat to give it a mea of value 0. Te thousad observatios are a big umber big eough to be cosidered as a populatio. The sample size take from this populatio has various values from to 8. The the sample variaces were calculated from these samples by the two formulae ) ad 3). The variaces were calculated repeatedly te thousad times each time with a ew populatio of a stadardized ormally distributed radom variable to create populatios of the variaces. The the meas of these variace populatios were calculated ad reported. Table is the result of these simulatio rus. While the umber of degrees of freedom i the sum of squares S ca hardly be agreed upo the simulatio results support the ubiased formula ie. Equatio ). This result comes as a surprise to the author of this paper because of his support for the biased formula ad his fidig of the icorrect proofs for the ubiased formula. The result also tells us that the umber of degrees of freedom i the sum of squares S is ) ot. With the Helmert s trasformatio the proof for the ubiasedess of formula ) ca be established with the trasformed variable y i ad the approach used i the theorem i sectio four. Sice the simulatio supports the ubiased formula we must fid the error i the proof for the biased formula. This error ca be foud i Equatio 7). Oe etry i the mea will joi with the variable t to create a ratio value that will ot give a probability value. The proof is perfect for the mea but it fails for the variace with the way it is defied. Table. Sample Variaces Sample Size Ru Variace ˆσ biased ˆσ ubiased Coclusio I this paper the sample variace formulae are studied agai to determie the right oe for computatio. The old belief is the formula with a smaller divisor is the ubiased oe. While the problem appears to be easy the verdict must be decided with simulatio results as there are icorrect proofs for both formulae i cotetio ad o criteria for determiig the umber of degrees of freedom. Simulatio results support the formula with a smaller divisor. Refereces Irwi Guttma Samuel S. Wilks ad J. Stuart Huter 97). Itroductory Egieerig Statistics. Joh Wiley & Sos Ic. New York NY USA ISBN Paul G. Hoel 97). Itroductio to Mathematical Statistics. Joh Wiley & Sos Ic. New York NY USA 4th Editio ISBN Robert V. Hogg ad Alle T. Craig 978). Itroductio to Mathematical Statistics. Macmilla Publishig Co. Ic. New York NY USA Fourth Editio ISBN Ky M. Vu 007). The ARIMA ad VARIMA Time Series: Their Modeligs Aalyses ad Applicatios. AuLac Techologies Ic. Ottawa ON Caada ISBN MATLAB is a trade mark of The MathWorks Ic.
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