;24 



NATURE 



[September i, 1923 



these into consideration, my analysis shows that : 

 (A) the statistical correction may easily become 

 negative ; that is, the true correlation may be 

 considerably lower than the observed correlation. 

 On the other hand, if " errors " are independent 

 (or as my analysis shows, for particular values of 

 correlation between errors), then (B) the correlation 

 may be positive as found by Chapman, and the true 

 correlation higher than the observed. The question 

 is : under which category (A) or (B) above does the 

 work of Dines fall ? 



In the case of a balloon meteograph, all measure- 

 ments are made on one and the same trace,* and the 

 heights are calculated with the help of Laplace's 

 formula.^ This formula involves both pressure and 

 temperature, and a detailed examination shows that 

 it serves to introduce, through " interpolation," 

 correlation between errors of measurement in pressure 

 and temperature. Besides this " interpolation " 

 effect, correlation may also be introduced through 

 what Karl Pearson * calls the " atmosphere " of 

 measurement and through correlation of successive 

 judgments.* It is, therefore, not improbable that 

 Dines's work falls under (A) and gives values of cor- 

 relation coefficients higher than their true values. 

 My contention is this : (C) in the absence of definite 

 proof that Dines's work falls under (B), Chapman's 

 corrections cannot be accepted as real, and, to be 

 on the safe side, Dines's coefficients must be looked 

 upon as giving superior limits to the true correlation. 



Douglas ^^ found the values of correlation between 

 pressure and temperature at 10,000 feet to be 0-65, 

 which is considerably lower than Dines's figure 077 

 (and still more so than Chapman's corrected value). 

 I quoted Douglas's result, as I thought his work to 

 be free from the peculiar " interpolation " correlation 

 introduced by the use of Laplace's formula. On 

 this view, Douglas's work would probably come under 

 (B) and would give values of correlation lower than 

 true values. I now find stated in the note in Nature 

 that I have fallen into error in thinking " that 

 Douglas's coefficients are based on true heights." 

 (The fault, however, is scarcely mine, for Douglas 

 himself definitely stated ^^ that his observations 

 " refer to actual heights above mean sea-level, and 

 not to aneroid heights.") On the present view, 

 Douglas's work also would probably come under (A) 

 above, and even 0-65 would seem to be too high a 

 value for the true correlation. This corroborates 

 my contention (C) that Dines's coefficients are 

 probably too high. It is, therefore, clear that the 

 rectification of my error has further strengthened 

 my conclusion. I may note in passing that the low 

 values of the coefficients obtained by Douglas may 

 be easily explained in accordance with my analysis 

 if we assume that the magnitude of the correlations 

 between errors of measurement are lower in his case. 



In my other memoir " I pointed out certain 

 statistical discrepancies in the coefficients published 

 by Dines. It is stated in the note in Nature that 

 I seem " to have confused the T„ used by Dines, 

 namely, the mean temperature between i and 9 

 kilometres, with the mean temperature between o 

 and 9 kilometres," and that this supposed confusion 

 on my part " fully explains the discrepancies " noted 

 by me. I am unable to agree with this, as I do 

 not think I have made any confusion between the 

 two mean temperatures referred to above. On p. i 



• M.O. No. 210/, Geophys. Mem. 6, 1914. 



' M.O. No. 223, " Computer's Handbook," Section 2. 



• Phil. Trans. 198 A, 1902, " Errors of Judgment," etc. 



• Egon S. Pearson, Biometrika, xiv., 1922. 



" Quar. Jour. Met. Soc., xlvii., January 1921, p. 28, etc, 

 " Ibid. p. 25. 



" Mem. Ind. Met. Dept., vol. xxiv. Part I., " The Seat of Activity in the 

 Upper Air." 



NO. 2809, VOL. 112] 



and p. 3 of my memoir I have 

 that r, represents the mean ten 

 o and Z kilometres, and I li;t\r . ;• i / « 



distinct throughout. It is tru- i . • . ituted 

 dTt=dT„, but this is quite <liiJciciii lioin , itting 

 T,=T^, since dT, and rf7„ are lx)th si.-; 1 1 d 

 differences (which would ultimatelv l>e summcu I 

 averaged out) and not analytic diffcrentiaLs. I i 

 substitution is further discussed on p. of my meinuii . 

 Now if this substitution is justified, then it follows 

 from Laplace's equation that : (D) in the t as< of 

 the figures published by Dines it is actual! \ 

 to obtain higher values of the correlation co' ^ 



at levels considerably lower than 9 kilometres, in 

 view of the assumption involved it is, however, 

 necessary to test (D) by direct examination of the 

 data concerned. But in the absence of «»uch ex- 

 amination it is not sufficient to stat< 

 crepancies can be explained." 



To sum up, the main problem is to find ;(«] Hit- true 

 correlation, and (6) the region of the best correlation 

 in the case of upper air variables. It would seem 

 that in view of (A), (C), and (D) above, the work of 

 Dines and Chapman (which is flatly contradicted by 

 that of Douglas) cannot be accepted as final either 

 as regards (a) or as regards [b). F'urther advance is 

 not possible without a thorough statistical scrutiny 

 of the original data. 



May I, therefore, suggest that (i.) the original 

 material of Dines and l3ouglas (as well as other 

 fresh material, if available) be published with clear 

 statements about methods of measurement employed 

 and actual formulae (rigid or otherwise) used for 

 computation of heights, and that (ii.) such material 

 be submitted to some statistical expert like Prof. 

 Karl Pearson for examination and report. 



P. C. Mahalanobis. 



Presidency College, Calcutta, 

 June 20. 



The results of the British Registering Balloon 

 Ascents are pubUshed in full by the Meteorological 

 Office in the Annual Supplement to the Geophysical 

 Journal. A full description of the instruments, 

 methods, and formulae used have also been published 

 by the M.O., and will be found in the " Computer's 

 Handbook," M.O. 223, Section II., subsection ii. 

 They are open to anybody for use, and if Prof. 

 Mahalanobis will carry out the computation he 

 desires he will earn the thanks of meteorologists. 



It is difficult, however, to see how Prof. Maha- 

 lanobis can obtain a perfectly correct correlation 

 coefficient, in view of the fact that, with a coefficient 

 of 0-70 based on 400 observations, the causal standard 

 error is as high as 0-025. This fact suffices to explain 

 the differences between Dines's and Douglas's results, 

 which can scarcely be called a " flat contradiction." 



With reference to Prof. Mahalanobis' assumption, 

 that dTf =dT„, it may be pointed out that the result 

 of making this assumption is discussed in the papers 

 to which he referred, and also that no claim to 

 extreme accuracy in the correlation coefficient is 

 made by Dines. (See M.O. 2106, bottom of p. 43, 

 and p. 44, line 11 ; also Beitrdge zur Physik der 

 freien Atmosphdre, V. Band, Heft 4, pp. 222, 223, 

 and 225.) The Writer of the Note. 



Tubular Cavities in Sarsens. 



With regard to Mr. F. Chapman's letter on the 

 probable aeolian origin of sarsen rock (Nature, 

 August 18, p. 239), and his reference therein to my 

 previous note, may I say that I was not referring to 



