# THE PROBABLE ERROR OF A MEAN. Introduction

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3 Summig for all samples ad dividig by the umber of samples we get the moa value of s, which we will write s : s = µ µ = µ ( 1), where µ is the secod momet coefficiet i the origial ormal distributio of x: sice x 1, x, etc. are ot correlated ad the distributio is ormal, products ivolvig odd powers of x 1 vaish o summig, so that S(x1x) is equal to 0. If M R represet the Rth momet coefficiet of the distributio of s about the ed of the rage where s = 0, Agai M 1 ( 1) = µ. { ( )} S(x s 4 = 1 ) S(x1 ) ( ) S(x ( ) ( = 1 ) S(x 1) S(x1 ) S(x1 ) + = S(x4 1) + S(x 1x ) S(X4 1) 3 4S(x 1x ) 3 + S(x4 1) 4 + 6S(x 1x ) 4 +other terms ivolvig odd powers of x 1, etc. which will vaish o summatio. Now S(x 4 1) has terms, buts(x 1x ) has 1 ( 1), hece summig for all samples ad dividig by the umber of samples, we get M = µ 4 ( 1) +µ µ 4 ( 1) µ + µ 4 ( 1) 3 +3µ 3 ) 4 = µ 4 3 { +1}+ µ 3( 1){ +3}. Now sice the distributio of x is ormal, µ 4 = 3µ, hece M = µ ( 1) 3 { } = µ ( 1)(+1). I a similar tedious way I fid ad M 3 = µ 3 ( 1)(+1)(+3) 3 M 4 = µ 4 ( 1)(+1)(+3)(+5) 4. 3

4 The law of formatio of these momet coefficiets appears to be a simple oe, but I have ot see my way to a geeral proof. If ow M R be the Rth momet coefficiet of s about its mea, we have M = µ ( 1) { ( 1)(+1)(+3) M 3 = µ 3 3 {(+1) ( 1)} = µ 3 ( 1). 3( 1). (( 1) ( 1)3 3 = µ 3 ( 1) 3 { } = 8µ 3 ( 1) 3, M 4 = µ4 4 { ( 1)(+1)(+3)(+5) 3( 1) 1( 1) 3 ( 1) 4} = µ4 ( 1) 4 { } = 1µ4 ( 1)(+3) 4. Hece β 1 = M 3 M 3 = 8 1, β = M 4 M = 3(+3) 1), β 3β 1 6 = 1 {6(+3) 4 6( 1)} = 0. 1 Cosequetly a curve of Prof. Pearso s Type III may he expected to fit the distributio of s. The equatio referred to a origi at the zero ed of the curve will be where ad y = Cx p e γx, γ = M M 3 = 4µ ( 1) 3 8 µ ( 1) = µ p = 4 β 1 1 = 1 Cosequetly the equatio becomes y = Cx 3 e x µ, 1 = 3. which will give the distributio of s. The area of this curve is C x 3 e x µ dx = I (say). The first momet 0 coefficiet about the ed of the rage will therefore be C [ ] x 1 e x µ dx C µ x 1 e x x= µ 0 x=0 = + C 1 0 µ x 3 e x µ dx. I I I 4 }

5 The first part vaishes at each limit ad the secod is equal to 1 µ I I = 1 µ. ad we see that the higher momet coefficiets will he formed by multiplyig successively by +1 µ, +3 µ etc., just as appeared to he the law of formatio of M, M 3, M 4, etc. Hece it is probable that the curve foud represets the theoretical distributio of s ; so that although we have o actual proof we shall assume it to do so i what follows. The distributio of s may he foud from this, sice the frequecy of s is equal to that of s ad all that we must do is to compress the base lie suitably. Now if y 1 = φ(s ) be the frequecy curve of s ad y = ψ(s) be the frequecy curve of s, the y 1 d(s ) = y ds, y ds = y 1 sds, y = sy 1. Hece is the distributio of s. This reduces to y = Cs(s ) 3 e s µ. y = Cs e s σ. Hece y = Ax e s µ will give the frequecy distributio of stadard deviatios of samples of, take out of a populatio distributed ormally with stadard deviatio σ. The costat A may he foud by equatig the area of the curve as follows: The Area = A 0 I p = σ = σ x e x σ dx. ( Let I p represet ( e x σ )dx x p 1 d 0 dx ] x= [ x p 1 e x σ = σ (p 1)I p, x=0 sice the first part vaishes at both limits. + σ (p 1) 0 0 ) x p e x σ dx. x p e x σ dx 5

6 or By cotiuig this process we fid ( σ I = ( σ I = accordig is eve or odd. But I 0 is ad I 1 is Hece if be eve, while is be odd or 0 0 ) ) e x σ dx = xe x sigma dx = [ σ ( 3)( 5)...3.1I 0 ( 3)( 5)...4.I 1 ( π ) σ, ] e x x= σ x=0 = σ. Area A = (π )( ( 3)( 5) σ ), 1 A = Area ( 3)( 5)...4. ( σ ) 1 Hece the equatio may be writte ( ) N ( ) 1 y = ( 3)( 5) π σ x e x σ ( eve) y = N ( ) 1 ( 3)( 5)...4. σ x e x σ ( odd) where N as usual represets the total frequecy. Sectio II To show that there is o correlatio betwee (a) the distace of the mea of a sample from the mea of the populatio ad (b) the stadard deviatio of a sample with ormal distributio. (1) Clearly positive ad egative positios of the mea of the sample are equally likely, ad hece there caot be correlatio betwee the absolute value of the distace of the mea from the mea of the populatio ad the stadard. 6

8 Now let us suppose x measured i terms of s, i.e. let us fid the distributio of z = x/s. If we have y 1 = φ(x) ad y = ψ(z) as the equatios represetig the frequecy of x ad of z respectively, the y 1 dx = y dz = y 3 dx s, y = sy 1. Hece y = N ()s e s z σ (π)σ is the equatio represetig the distributio of z for samples of with stadard deviatio s. Now the chace that s lies betwee s ad s+ds is s+ds s 0 C σ 1 s e s σ ds C σ 1 s e s σ ds which represets the N i the above equatio. Hece the distributio of z due to values of s which lie betwee s ad s+ds is y = s+ds s C σ ( π 0 ) s 1 e s (1+z ) σ ds C σ 1 s e s σ ds = ( π ) s+ds s 0 C σ s 1 (1+z ) e s σ ds C σ s e s σ ds ad summig for all values of s we have as a equatio givig the distributio of z ( ) s+ds C π s σ s 1 (1+z ) e s y = σ ds. σ C σ s e s σ ds By what we have already proved this reduces to ad to 0 y = (1+z ) 1, if be odd y = (1+z ) 1, if be eve Sice this equatio is idepedet of σ it will give the distributio of the distace of the mea of a sample from the mea of the populatio expressed i terms of the stadard deviatio of the sample for ay ormal populatio. 8

9 Sectio IV. Some Properties of the Stadard Deviatio Frequecy Curve By a similar method to that adopted for fidig the costat we may fid the mea ad momets: thus the mea is at I 1 /I, which is equal to ( π ) σ, if be eve, or (π ) σ, if be odd. The secod momet about the ed of the rage is I = ( 1)σ. I The third momet about the ed of the rage is equal to I +1 I = I +1 I 1. I 1 I = σ the mea. The fourth momet about the ed of the rage is equal to I + = ( 1)(+1) I σ 4. If we write the distace of the mea from the ed of the rage Dσ/ ad the momets about the ed of the rage ν 1, ν, etc., the ν 1 = Dσ, ν = 1 σ, ν 3 = Dσ3, ν 4 = N 1 σ 4. From this we get the momets about the mea: µ = σ ( 1 D ), µ 3 = σ3 {D 3( 1)D +D } = σ3 D {D +3}, µ 4 = σ { 1 4D +6( 1)D 3D 4 } = σ4 { 1 D (3D +6)}. It is of iterest to fid out what these become whe is large. 9

10 is I order to do this we must fid out what is the value of D. Now Wallis s expressio for π derived from the ifiite product value of six π (+1) = () ( 1). If we assume a quatity θ ( = a 0 + a1 +etc.) which we may add to the +1 i order to make the expressio approximate more rapidly to the truth, it is easy to show that θ = ( π etc., ad we get ) = () ( 1). From this we fid that whether be eve or odd D approximates to whe is large. Substitutig this value of D we get ( µ = σ 1 1 ) (1, µ = σ3 3 + ) , µ 4 = 3σ ( ) 16. Cosequetly the ( value of the stadard ) deviatio of a stadard deviatio σ which we have foud becomes the same as that foud for () {1 (1/4)} the ormal curve by Prof. Pearso {σ/()} whe is large eough to eglect the 1/4 i compariso with 1. Neglectig terms of lower order tha 1/, we fid β 1 = 3 (4 3), β) = 3 ( 1 1 )( 1+ 1 Cosequetly, as icreases, β very soo approaches the value 3 of the ormal curve, but β 1 vaishes more slowly, so that the curve remais slightly skew. Diagram I shows the theoretical distributio of the stadard deviatios foud from samples of 10. Sectio V. Some Properties of the Curve y = ( 4 3. π if be eve if be odd ) ). (1+z ) 1 Writig z = taθ the equatio becomes y = etc. cos θ, which affords a easy way of drawig the curve. Also dz = dθ/cos θ. This expressio will be foud to give a much closer approximatio to π tha Wallis s 10

11 Hece to fid the area of the curve betwee ay limits we must fid etc. cos θdθ = { [ 3 cos etc. cos 4 3 ]} θsiθ θdθ + = etc. cos 4 θdθ etc.[cos 3 θsiθ], ad by cotiuig the process the itegral may he evaluated. For example, if we wish to fid the area betwee 0 ad θ for = 8 we have Area = π = 4 3. π θ 0 θ 0 cos 6 θdθ cos 4 θdθ π cos5 θsiθ = θ π + 1 π cosθsiθ π cos3 θsiθ π cos5 θsiθ ad it will be oticed that for = 10 we shall merely have to add to this same expressio the term π cos7 θsiθ. 11

12 The tables at the ed of the paper give the area betwee ad z ( or θ = π ) ad θ = ta 1 z. This is the same as 0.5+the area betwee θ = 0, ad θ = ta 1 z, ad as the whole area of the curve is equal to 1, the tables give the probability that the mea of the sample does ot differ by more tha z times the stadard deviatio of the sample from the mea of the populatio. The whole area of the curve is equal to etc. π cos θdθ 1 π ad sice all the parts betwee the limits vaish at both limits this reduces to 1. Similarly, the secod momet coefficiet is equal to etc. π cos θta θdθ 1 π = etc. = 3 1 = π 1 π (cos 4 θ cos θ)dθ Hece the stadard deviatio of the curve is 1/ ( 3). The fourth momet coefficiet is equal to etc. π cos θta 4 θdθ 1 π = etc. + 1 π 1 π (cos 6 θ cos 4 θ +cos θ)dθ = ( ) 3 +1 = 3 ( 3)( 5). The odd momets are of course zero, a the curve is symmetrical, so β 1 = 0, β = 3( 3) 5 = Hece as it icreases the curve approaches the ormal curve whose stadard deviatio is 1/ ( 3). β, however, is always greater tha 3, idicatig that large deviatios are mere commo tha i the ormal curve. I have tabled the area for the ormal curve with stadard deviatio 1/ 7 so as to compare, with my curve for = It will be see that odds laid 3 See p. 9 1

17 Mea value of stadard deviatios: Calculated.186 ± 0.03 Observed.179 Differece = Stadard deviatio of stadard deviatios: Calculated 0.94 ± Observed Differece = Compariso of Fit. Theoretical Equatio: y = (π)σ x e x σ Scale i terms of stadard deviatios of populatio Calculated frequecy Observed frequecy Whece χ = 1.80, P = Value of stadard deviatio: Calculated 1(±0.017) Observed 0.98 Differece = Compariso of Fit. Theoretical Equatio: y = N π cos4 θ, z = taθ Scale of z Calculated frequecy Observed frequecy Differece Whece χ = 7.39, P = 0.9. A very close fit. We see the that if the distributio is approximately ormal our theory gives us a satisfactory measure of the certaity to be derived from a small sample i both the cases we have tested; but we have a idicatio that a fie groupig is of advatage. If the distributio is ot ormal, the mea ad the stadard deviatio of a sample will be positively correlated, so although both will have greater variability, yet they will ted to couteract oe aother, a mea derivig largely from the geeral mea tedig to be divided by a larger stadard deviatio. Cosequetly, I believe that the table give i Sectio VII below may be used i estimatig the degree of certaity arrived at by the mea of a few experimets, i the case of most laboratory or biological work where the distributios are as a rule of a cocked hat type ad so sufficietly early ormal 17

18 18

19 3. 4 Sectio VII. Tables of ( odd ) ta 1 z 1.1 π eve 1 π cos θdθ for values of from 4 to 10 iclusive Together with (π) 7 x e 7x dx for compariso whe = 10 z ( = x s) = 4 = 5 = 6 = 7 = 8 = 9 = 10 ( For compariso 7 ) x (π) e 7x dx Explaatio of Tables The tables give the probability that the value of the mea, measured from the mea of the populatio, i terms of the stadard deviatio of the sample, will lie betwee ad z. Thus, to take the table for samples of 6, the probability of the mea of the populatio lyig betwee ad oce the stadard deviatio of the sample is 0.96, the odds are about 4 to 1 that the mea of the populatio lies betwee these limits. 19

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