[Notes on a series of lectures given at the Glasgow University, January to March 1913].

  • Brownlee, John, 1868-1927.
Date:
[1913?]
    Examples of Distributions. {a) CONTINUOUS VARIATION. Symmetrical Distribution. Slightly Asymmetrical. Markedly Asymm etrical. Age Distribution. J Stature in Man.* Weight in Man. f Cases of Scarlet Fever. Rental of Houses. § Sta’ture in Nos. in each Weight in Nos. in No. of Rent No. of Houses Inches. Class. Lbs. each Class. Age. Cases. under in Thousands. 51-58 2] 90-100 2 0-1 246 £10 3175 58-59 4 22 100-110 34 1-2 773 £10-20 1451 59-GO I4J 110-120 152 2-3 1399 GO-61 411 120-130 390 3-4 1874 £20-30 442 61-62 83 - 294 130-140 867 4-5 2009 62-63 169^ 140-150 1623 5-6 1931 £30-40 260 63-64 394' 150-160 1559 6-7 1704 £40-50 151 64-65 669 2053 160-170 1326 7-8 1533 £50-60 90 65-66 990J 170-180 787 8-9 1236 66-67 12231 180-190 476 9-10 1014 67-68 1329 -3782 190-200 263 10-15 2921 £60-80 104 68-69 1230 200-210 107 15-20 921 69-70 10631 210-220 85 20-25 417 £80-100 47 70-71 646 -2101 220-230 41 25-35 327 71-72 392 230-240 16 35-45 85 Above 72-73 2021 240-250 11 45- 32 £100 110 73-74 79 r 313 250-260 8 74-75 32 260-270 1 75-76 16 270-280 ... 76-77 6 - 23 280-290 1 77-78 2J Totals, 8388 7749 * Report B.A, 1883, p. 256. f Report B.A. 1883. } Manehester Health Reports. § Goschen, quoted by Pearson, Phil. Trans. Roy. Soc. vol. 186 A, pp. 343-414.
    (b) DISCRETE VARIATION. Number ok Sepals in Flowers of Anemone nemorosa* Number of Petals in Ranunculus Bulbosum. f No. of Sepals. No. of Instances. Example (a). No. of Instances. Example (/^). No. of Petals. No. of Instances. 4 3 5 133 5 7 31 6 55 6 515 657 7 23 7 419 271 8 7 8 49 35 9 2 9 13 2 10 2 10 1 1 U ... 11 1 . . . 12 ... ... Yule, Biometrika, vol. i. p. 308. t De Vries, quoted by Pearson, loc. cit.
    Constants of Distributions. (1) Mean. If there be a number of quantities of definite measurement, then the ' term mean is used to denote the sum of these measurements divided by the total number of the quantities, or if ... be the measurements, n in number, and M the mean. The term mean as used in .statistics is equivalent to the term arithmetical mean in algebra. (2) Median. 'Phe median is the central value of the group when the measurements are arranged in order of magnitude, so that the number of instances above the median is equal to that below the median. If the groups are at all numerous, it is most easily calculated by simple proportion. Thus, taking the weights of British adults, we find 7749 instances. Of these 3068 are under 1.50 pounds, a defect of 806'5 below the median, and 3122 above 160 pounds, so that the median will be very approximately given by Median = 1504-^^x10 =155-2 lbs. 15o9 The first number is the weight at which the group begins: the multiplier 10 is the value of the group difference, and the fraction the proportional number of the group 1559 to be expected. (3) Mode. This is the mo.st frequent group in asymmetrical distributions, and in symmetrical distributions coincides with the mean and the median. The group in which the mode is situated can usually be easily seen, and if the middle point of this be denoted by zero, the distance of the mode from this can be calculated approximately by the formula m j m. — m, Mode = jT7 L^ r, 2 (m^ — 2m2 4- where n\, m.^, m^ are the numbers included in the successive groups, m^ being that of the group in which the mode is expected. More accurately, the median in general lies between the mode and the mean, so that its distance from the former is twice that from the latter, that is 2(Mean— Median) = Median —Mode, or Mode = 3 x Median — 2 x Mean = 150-8 lbs. in the case previously considered. — 792 By the formula given. Mode = 145 4 ^™ x 10 = 149-2 lbs., Z X or one per cent, of difference.