General average

“ Again, General average is the apportionment of loss caused by measures taken for the ship's safety, as cutting away the masts, throwing overboard cargo, accepting towage, or the like.—adj. containing a mean value: ordinary.—v.t. to fix an average.—v.i. to exist in, or form, a mean quantity. ”

“ The general average, for France, of the number of openings per inhabitant, is one and a half. In one hundred departments, in which the number is less than the mean, fifty-five show a higher mortality, and forty-five a mortality equal to the average ”

“ 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. ”

“ 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. ”

“ By reducing the facts to figures comparisons can easily be made and the proportion of one thing to another is more accurately presented. By divesting facts from their aspect in daily life and the caprices of temperament, a truer estimate of tendencies can be made. Because statistics generally deal with large [Pg 582] masses or great numbers, standards, ratios, and types can easily be established. The mortality statistics of insurance companies afford a good example of general averages or ratios that hold true when applied to such an uncertain proposition as the length of human life. ”

“ In height he gained from the 25 percentile to the 40 percentile. His total gain in all tests is 30 percentile out of a possible 415, and the average gain is ·05. The reader may see by scanning the table that the gains in the test group are practically equaled by those in the control group. There seems to be no consistent relationship between a low score in the first test and a large gain. ”

“ It is true that wealth has been greatly increased, and that the average of comfort, leisure, and refinement has been raised; but these gains are not general. In them the lowest class do not share. [1] I do not mean that the9 condition of the lowest class has nowhere nor in anything been improved; but that there is nowhere any improvement which can be credited to increased productive power. I mean that the tendency of what we call material progress is in nowise to improve the condition of the lowest class in the essentials of healthy, happy human life. ”

“ We saw that a resort to the average had the effect of ‘humping up’ our curve more towards the centre, expressive of the fact that the errors of averages are of a better, i.e. smaller kind. But it must be noticed that exactly the same characteristics will follow, as a general rule, from any other such mode of dealing with the individual errors. No strict proof of this fact can be given here, but a reference to one of the familiar results of taking combinations of things will show whence this tendency arises. Extreme results, as yielded by an average of any kind, can only be got in one way, viz. ”

“ Many serious difficulties have to be overcome, and the final result must be regarded as merely approximate. For example, some observers have studied the entire population of districts; others draw their figures from the army alone, from which, of course, all the abnormally short men have been eliminated. Some give averages alone; others work by percentile grades. To be sure, these two methods give parallel results; but how discover the average from them? ”

“ 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. ”

“ The percentage is, of course, rarely or never precisely realised in any one year, but the longer the period of years we take for examination, the more exactly do the deviations from the average in individual years compensate one another. ”

“ 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. ”

“ Now when we are referring to averages it must be remembered that the minimum number of different occurrences necessary in order to justify the average is that which enables each of them to present itself once. ”

“ 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. ”

“ Even outside the field of finance, Americans are apt to be unduly interested in discovering what average opinion believes average opinion to be; and this national weakness finds its nemesis in the stock market. ”

“ 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. ”

“ Thus if in the first of the averages above instanced (the arithmetic mean) the constituent xr, occurs l times, the expression is to be modified by putting lxr for xr in the numerator, and in the denominator, for n, n+r−1. The definition of an average covers a still wider held. The process employed need not be a function. [156] One of the most important averages is formed by arranging the constituents in the order of magnitude and taking for the average a value which has as many constituents above it as below it, the median. ”

“ We being the first troops who had arrived, received a very large proportion of the attention of General Pershing and his staff. [77] The General once came out to look over the Twenty-sixth Infantry, and stopped in front of the redoubtable Sergeant Murphy and his platoon. Now, Sergeant Murphy could stand with equanimity as high an officer as a colonel, but a general was one too many. He was not afraid of a machine gun or a cannon, but a star on a man's shoulder petrified him. After the General had watched for a minute, the good sergeant had his platoon tied up in thirteen different ways. ”

“ One is that the result formulated in the statistical law is always one to which the actual course of events may reasonably be expected to conform within certain limits of deviation, never one to which we have a right to expect absolute conformity. Not only is the actual number of marriages, e.g., in any one year, usually slightly above or below the average percentage computed, e.g., for a ten years’ period, but as we compare one longer period with others, the average percentage for the longer period itself fluctuates. ”

“ 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. ”

“ 'Winwood Reade is good upon the subject,' said Holmes. 'He remarks that, while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician. ”

“ We see then that if the distribution is approximately normal our theory gives us a satisfactory measure of the certainty to be derived from a small sample in both the cases we have tested; but we have an indication that a fine grouping is of advantage. If the distribution is not normal, the mean and the standard deviation of a sample will be positively correlated, so that although both will have greater variability, yet they will tend to counteract each other, a mean deviating largely from the general mean tending to be divided by a larger standard deviation. ”

“ 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. ”

“ It would seem, therefore, that we might find a sort of measure of the deviation of an ensemble from statistical equilibrium in the excess of the average index above the minimum which is consistent with the condition of the invariability of the distribution with respect to the constant functions of phase. But we have seen that the index of probability is constant in time for each system of the ensemble. The average index is therefore constant, and we find by this method no approach toward statistical equilibrium in the course of time. ”

“ 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 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. ”

“ Before closing this short survey of the very important subject of averages or means, it is needful to discuss briefly the nature of the phenomena which they may safely be regarded as indicating, when they have been properly obtained. Given a weighted mean of a series of numbers referring to no matter what phenomenon, it is obvious that the value of the mean as a type of the whole series will depend entirely on the extent of divergence from it of the members of the series as a body. If we are told that there are in a certain district moo men, and that their average height is 5 ft. ”

“ The average is, however, an entirely ideal quantity which need not agree with the measurement of any individual; and it is a little curious that it is so universally used in statistics. Of a series of measurements made on one and the same dimension, the average is demonstrably nearest the true value, and consequently engineers, physicists, astronomers and others who aim at the greatest possible precision in the measurement of the individual phenomenon must make use of it. ”

“ In these works he shows how the numbers representing the individual qualities of man are grouped round the numbers referring to the “average man” in a manner exactly corresponding to that in which single results of observation are grouped round the mean result, so that the principles of the theory of probabilities may be applied to statistical researches on the subjects. ”

“ You've seen an unemployment rate that was well below the national average 12 months ago now being well, well above -- almost twice what the national average is -- is twice what the national average is. ”

“ It is evident, therefore, that in close-breeding, and in the crossing of varieties of different degrees of divergence, there is, first, a less than average result, then an average, then better than an average, then this better result quickly reaches a maximum and again declines, crosses the line of average and becomes bad, and finally infinitely bad, or dies out. ”

“ If we have reason to suppose that the possible calibres partake of the nature of a continuous magnitude,—i.e. that all values, with certain limits, are to be considered as admissible, (an assumption which we always make in our ordinary inverse step from an observation or magnitude to the thing observed or measured) —then we should be justified in selecting the average as the likelier value. ”

“ I should put down their answers as to what 6 times 9 amounts to, I should work out the average of their answers to six places of decimals, and should then decide that, at the present stage of human development, this average is the value of 6 times 9. ”

“ The Upper Surface shows the Theoretical Distribution of Grades, the Lower that must convenient in Practise. and that ability is distributed somewhat like wages which are roughly proportional to it. If the average earnings of men in this country are $600 a year, it is clear that the positive deviations from the average are many times the negative deviations. ”