Median

“ The median point of any distribution of measures is that point on the scale which divides the distribution into two exactly equal parts, one half of the measures being greater than this point on the scale, and the other half being smaller. When the scales are very crude, or when small numbers of measurements are being considered, it is not worth while to locate this median point any more accurately than by indicating on what step of the scale it falls. ”

“ The median step of a distribution is the step which contains within it the median point. Similarly, the median measure in any distribution is the measure which contains the median point. In a distribution containing 25 measures, the 13th measure is the median measure, because 12 measures are greater and 12 are less than the 13th, while the 13th measure is itself divided into halves by the median point. Where a distribution contains an even number of measures, there is in reality no median measure but only a median point between the two halves of the distribution. ”

“ Draw a scheme of variates on the blackboard, and show that it consists of two parts; the median which represents a constant, and the curve which represents the variations from it. Draw a horizontal line from limit to limit, through the top of the median, to serve as axis to the curve. Divide the axis centesimally, and wipe out everything except curve, axis and limits. This forms a scheme of distribution of deviates. Draw ordinates from the axis to the curve at the twenty-fifth and seventy-fifth divisions. These are the "quartile" deviates. ”

“ The median is often preferred to the average in psychological work, not only because it is more easily computed, but because it is less affected by the eccentric or unusual performances of a few individuals, and therefore more fairly represents the whole population. ”

“ This probability that the observed median will differ from the true one by a certain number of observations is connected with the probability that they will differ by a certain extent of the abscissa, by the proposition that the number of observations contained between the true and apparent median is equal to the small difference between them multiplied by the density of observations at the median—in the case of normal and generally symmetrical curves the greatest ordinate. ”

“ It will be found much more advantageous in dealing with educational statistics to consider only the median point, and to use only the n/2 formula given in a previous paragraph, for practically all educational scales are or may be thought of as continuous scales rather than scales composed of discrete steps. ”

“ Then we should have two groups each containing 25 million, one falling below and one rising above the average. These are the ordinary men who depart from the median by an amount less than the probable error. Then at the upper end we have the group of 25 million individuals who through some special trait or through a combination of traits rise above the others. At the extreme end of this group are the thousand preeminent men of our list. ”

“ In asymmetrical curves of error the arithmetic average must he distinguished from, 5. (3) the Maximum Ordinate average, 6. (4) and the Median. 7. Diagram in illustration. 8—10. Average departure from the average, considered under the above heads, and under that of 11. (5) The (average of) Mean Square of Error. 12—14. The objects of taking averages. 15. Mr Galton's practical method of determining the average, 16, 17. No distinction between the average and the mean. 18—20. Distinction between what is necessary and what is experimental here. ”

“ A method of rejecting data analogous to the use of percentiles in one dimension is practised when, given the frequency of observations for each increment of area, e.g. each ∆x ∆y, we utilize only the frequency for integral areas. Mr Sheppard has given an elegant solution of the problem: to find the correlation between two attributes, given the medians L, and M, of a normal group for each attribute and the distribution of the total group, as thus. [261] ”

“ And so if i, instead of being a particular sort of instance, is an instance known to be under the influence of a particular set of circumstances, the presumption drawn from the numerical proportions in the whole genus would probably, in such a case, only mislead. A general average should only be applied to cases which are neither known, nor can be presumed, to be other than average cases. Such averages, therefore, are commonly of little use for the practical guidance of any affairs but those which concern large numbers. ”

“ A mere average, therefore, might lead to conclusions only the more misleading, for their deceptive semblance of accuracy. There would be less danger of error in taking the average of only a small number of years, and correcting it by a conjectural allowance for the character of the seasons, than in trusting to a longer average without any such correction. ”

“ Of course, on this supposition, the percentages must all be multiplied together. This she has done correctly, but I can give her no honours, as I do not think the question will fairly bear her interpretation. Three Score and Ten makes it "19 and 3⁄8ths." Her solution has given me—I will not say "many anxious days and sleepless nights," for I wish to be strictly truthful, but—some trouble in making any sense at all of it. She makes the number of "pensioners wounded once" to be 310 ("per cent.," I suppose!) : dividing by 4, she gets 77 and a half as "average percentage:" again dividing by 4. ”

“ The process by which an average is obtained, it may be urged, insures this tendency to equalization amongst the magnitudes with which it deals. For instance, let there be a party of ten men, of whom four are tall and four are short, and take the average of any five of them. Since this number cannot be made up of tall men only, or of short men only, it stands to reason that the averages cannot differ so much amongst themselves as the single measures can. ”

“ The average of the grades assigned in these courses does not vary considerably, but the distribution is different. In the courses in English the distribution tends to follow the normal curve of error, with the failures as a separate group or species. In the courses in mathematics and history the groups are more nearly equal in size, except in the case of 'excellent.' Here the range of ability is presumably greater in D and F than in B and C. The distribution in economics is intermediate. ”

“ In the study of the data of 1880 I was led to the conclusion that the average per capita expenditure (and therefore the annual product in its final stage) of the people of this country for the necessaries, comforts and luxuries of life could not be less than what $200 per head could buy at the prices of the census year. The proportion of the population occupied for gain varied but a fraction from one in three, leading to the conclusion that the average product of each person occupied for gain could not be less than $600 worth a year. ”

“ In all calculations concerning the average rate of progress of a community, or any of its operations, the geometric mean should be employed. For if a quantity increases 100 per cent. in 100 years, it would not on the average increase 10 per cent. ”

“ Theorem V. If an ensemble of systems is so distributed in phase that the index of probability is a linear function of , , etc., (these letters denoting functions of the phase,) the average value of the index is less than for any other distribution in which the functions , , etc. have the same average values.
Theorem VI. The average value in an ensemble of systems of (where denotes as usual the index of probability and any function of the phase) is less when the ensemble is so distributed that is constant than for any other distribution whatever. ”

“ When in an imperfect balance, we reverse the weights to eliminate error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of a price is not 50 per cent. because the ratio 150 : 200 is not the same as 100 : 150. The mean ratio is as unity to √1·00 × 2·00 or 1 to 1·41. ”

“ An average is found by adding together several measurements of the same kind and dividing by the number of measurements. In calculating an average it should be observed that the addition of any numerical quantity (positive or negative) to each of the measurements produces the addition of the same quantity to the average, so that the calculation may often be simplified by taking some particular measurement as a new zero from which to measure. ”

“ Theorem VII. If a system which in its different phases constitutes an ensemble consists of two parts, and we consider the average index of probability for the whole system, and also the average indices for each of the parts taken separately, the sum of the average indices for the parts will be either less than the average index for the whole system, or equal to it, but cannot be greater. The limiting case of equality occurs when the distribution in phase of each part is independent of that of the other, and only in this case. ”

“ If the income distribution remains the same (all incomes grow with the same rate) then also the average income, y = W grows at the same rate, and then also the net income distribution remains the same, and then the ratio of net average to subsistence remains the same too. ”

“ We have also shown that the average square of the anomalies of , that is, of the deviations of the individual values from the average, is in general of the same order of magnitude as the reciprocal of the number of degrees of freedom, and therefore to human observation the individual values are indistinguishable from the average values when the number of degrees of freedom is very great. ”

“ TEST OF POPULATION. “The first and most important is the relation of expenditure to the population. In some ratio corresponding to the increase of population it may be reasonable to increase the expenditures of a government. This is the test usually applied in Europe. In an official table I have before me the expenditures of the British government for the last fifteen years, I find the statement made over against the annual average of each year of the expenditure per capita of the population. ”

“ X.12 We choose, then, as one of the best index numbers of prices, the average price of the goods sold, those goods being measured in units worth a dollar in the base year; in other words, the ratio of the value of sales at actual prices to the value of the same sales if made at base prices; in still other words, the weighted arithmetical average of all price ratios, each ratio being weighted according to the values sold, reckoned at base prices. ”

“ The full statement of these facts, made in graphic form, shows an ascending line of density of population, as we proceed from left to right, an ascending line of price for raw produce, a descending line of price for highly wrought merchandise, and descending lines for wages and interest. All these lines represent the facts in a broadly general way. They deal with averages and not with particular rates. The labor whose earning power descends along the line numbered 5 is of many kinds, and the produce of which the average values vary along the lines numbered 2 and 4 is of many varieties. ”

“ Where the average for a particular state for the first year is less than the general average, the per capita sum is to be increased by equal gradations, until, at the end of four years, it equals the general average. ”

“ For, if a 1 is equally likely to become a 1, 2, 3, 4, 5, 6 or 7, and so also of a 2, a 3, a 4, etc., the average result will be 16. [2]
Any quantity below 16 as the sum of a column then means some permanence of interests in the individual in question, and the degree of permanence is measured by the divergence from 16 toward 0.
For the permanence from the elementary-school period to the junior year of college or professional school in my hundred individuals this figure is, on the average 9, three fifths of the individuals showing sums of from 6 to 12 for column 2 of Table 3. ”

“ It gives, in short, the average tint of every unit of area in the picture, measured from the fiducial lines as co-ordinates. Now every statistician knows, by experience, that numerical averages usually begin to agree pretty fairly when we deal with even twenty or thirty cases. Therefore we should expect to find that any groups of twenty or thirty men of the same class would yield composites bearing a considerable likeness to one another. ”

“ Provided only that the conditions of production remain stable, averages of large numbers will always practically present much the same general characteristics. The far more important distinction lies between the average of a few, with its fluctuating values and very imperfect and occasional attainment of its ultimate goal, and the average of many and its gradually close approximation to its ultimate value: i.e. to its objective point of aim if there happen to be such. ”

“ Consider for instance the tables of human stature. These are often very extensive, including tens or hundreds of thousands. In such cases the mean or central value is determinable with just as 472 great certainty as by any à priori rule. That is, if we took another hundred thousand measurements from the same class of population, we should feel secure that the average would not be altered by any magnitude which our measuring instruments could practically appreciate. ”