Percentage

“ 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. ”

“ A standard basis for the estimate is not possible because in some businesses credit is extended much more carefully than in other businesses and in some collections are followed up more vigorously than in others. In making the estimate two methods are used, one being a certain percentage of the outstanding accounts, the other being a certain percentage of the sales made during the period. ”

“ A test score in and of itself is usually difficult to interpret. For example, if you learned that your score on a measure of shyness was 35 out of a possible 50, you would have little idea how shy you are compared to other people. More relevant is the percentage of people with lower shyness scores than yours. This percentage is called a percentile. If 65% of the scores were below yours, then your score would be the 65th percentile. ”

“ Among the subjects which should be curtailed or omitted are compound proportion, cube root, abstract mensuration and the greater part of commercial arithmetic. Percentage should be reduced, and the needs of practical life—profit and loss, bank discount, compound interest, with such complications as result from fractional periods of time—are useless and undesirable. The metric system should be taught in application to actual measurements, and the weights and measures handled. ”

“ Other experts, testifying to an aggregate, without assigning a proportion to the contributory factors, give estimates so vague as to be little more than guesses, one of them, for illustration, holding to the opinion that ten would be a fair percentage, yet unable to give a reason why the amount should not be less or greater. ”

“ Better methods are (1) to deduct a percentage from the gross score of successful shooters, (2) to handicap by points, as in other pastimes, or (3) to rate a shooter according to the average of his last year’s performances, re-rating him monthly, or at convenient intervals, the system being to add his average of the current year to his average of last year, and divide the sum by two to form his new rating. ”

“ 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 comparison of the facts and the report can be made only when the latter has been analyzed into its single statements (Einzelangaben) and the percentage of right, wrong and indeterminate statements has been calculated; and, because of the varied character of the statements, it is necessary to calculate separately the proportions for particular categories (e. g., with reference to matters of color or of space relations) as well as to make a general calculation. ”

“ Applicable percentage quality increases.—
“ (1) IN GENERAL.—Subject to the succeeding paragraphs, in the case of a qualifying plan with respect to a year beginning with 2012, the applicable percentage under subsection (n) (2) (B) shall be increased on a plan or contract level, as determined by the Secretary—
“ (A) for 2012, by 1.5 percentage points;
“ (B) for 2013, by 3.0 percentage points; and
“ (C) for 2014 or a subsequent year, by 5.0 percentage points. ”

“ Household income or consumption by percentage share: Data on household income or consumption come from household surveys, the results adjusted for household size. Nations use different standards and procedures in collecting and adjusting the data. Surveys based on income will normally show a more unequal distribution than surveys based on consumption. ”

“ To make a more exact calculation, it would be necessary to compare the number of superior men who have arisen from some hundreds of the most distinguished families of a country with that of distinguished men who have arisen from some millions of the rest of the people, and then calculate the percentage. It is also necessary to take into account the means employed in the education of the individuals. ”

“ If there is this tendency to adopt 100 as a basis instead of 10, the teaching of decimals might sometimes be simplified by proceeding from percentages to percentages of percentages, i.e. by commencing with centesimals instead of with decimals.
22. Perception of Number.—In using material objects as a basis for developing the number-concept, it must be remembered that it is only when there are a few objects that their number can be perceived without either counting or the performance of some arithmetical process such as addition. ”

“ '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. ”

“ The modern method is to deal with fractions which have 100 as denominator; such fractions are called percentages. They only apply accurately to divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of fitting in with the denary scale of notation, and they can be extended to other divisions by using a mixed number as numerator. ”

“ 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. ”

“ 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. ”

“ In using percentiles it is to be remembered that equal differences between percentiles are not comparable in the same distribution except in the sense of the same extra proportions of the group to be met in competition. A change in the degree of ability from the lowest percentile to the lowest 2 percentile would be very different from the change in the degree represented by the 50 percentile to the next percentile above. ”

“ If the unit of the scale is some measure of the variability of a defined group, as it is in the majority of our present educational scales, this median point may well be calculated to the nearest tenth of a unit, or, if there are two hundred or more individual measurements in the distribution, it may be found interesting to calculate the median point to the nearest hundredth of a scale unit. Very seldom will anything be gained by carrying the calculation beyond the second decimal place. ”

“ If R is an integer, the Pth percentile is be the number with rank R. When R is not an integer, we compute the Pth percentile by interpolation as follows:
1. Define IR as the integer portion of R (the number to the left of the decimal point) . For this example, IR=2.
2. Define FR as the fractional portion of R. For this example, FR=0.25.
3. Find the scores with Rank IR and with Rank IR + 1. For this example, this means the score with Rank 2 and the score with Rank 3. The scores are 5 and 7. ”

“ 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. ”

“ By simple induction we know that the median of a large sample of observations is a probable value for the true median; how probable is determined as follows from a selection of our data. First suppose that all the observations are of the same weight. If x′ were the true median, the probability that as many as 1⁄2n + r of the observations should fall on either side of that point is given by the normal law for which the exponent is -2r2/n. ”

“ 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. ”

“ 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. ”

“ Let us suppose that we desire to find the degree of correlation in variation, or deviation from the mean, between an organ A, called subject, and a second organ B, called relative. We first take all the individuals of one (subject) class; that is, individuals whose subject organ deviates from the mean by a constant quantity, p. We next find for those individuals the average deviation-from-the-mean of the organ B, and call it q. We then find the ration q/p; this is the partial index of correlation. ”

“ 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. ”

“ The whole food of a country divided by the sum of its population, constitutes the share of each person. Here, the food is the numerator, and the population the denominator of a fraction. In order that this fraction shall diminish, it is not necessary that the numerator shall continue stationary while the denominator increases: it is sufficient that it shall not increase as fast; and this is the case with food, which, we know, cannot increase as rapidly as an unchecked population.
I have also stated that the channel of employment can receive as many labourers as can possibly be maintained. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Unless otherwise specified, when we refer to “percentile,” we will be referring to this third definition of percentiles. Let's begin with an example. Consider the 25th percentile for the 8 numbers in Table 1. Notice the numbers are given ranks ranging from 1 for the lowest number to 8 for the highest number. ”