the proportions between the mean and these z values are 0.4732 and 0.2734 respectively . the proportion between z1 and z2 is therefore 0.4732 - 0.2734 = 0.1998 . this is the same as the proportion between z1 = - 1.93 and z2 = - 0.75 , since the curve is symmetrical . 4.16 . as well as occurring in the equation of the normal and other curves , the mean and variance parameters have another valuable property . this is the fact that they are additive . if we have two populations , with means m1 and m2 , and we add the variate values of these populations in pairs , we find that the mean of the sum ( m1+2 ) is the sum of the means ( &amp;formula; ) or &amp;formula; . the mean difference between pairs of population values is the difference of the means of the separate populations , i.e &amp;formula; . these simple properties are not , in general , possessed by medians , modes , or other position parameters . 4.17 . a similar property exists for variances , but in this case we must take account of the correlation between the two sets of data which are to be added or subtracted . the extent of correlation is expressed by the correlation coefficient r ( Greek letter rho , pronounced roe ) . this coefficient is positive when high values of one variate are paired with high values of the other , and similarly for low values ; it is negative when high values of one variate are paired with low values of the other , and it is zero when there is no systematic linear relationship between the variates . the coefficient r can take all fractional values between + 1.0 and - 1.0 ( for further discussion of r see chapter 9 , particularly 9.4 ) . we may now state that the variance of a sum ( s21+2 ) is &amp;formula; . a similar property holds for the variance of the differences between two correlated populations given as &amp;formula; . if it happens that our two populations are uncorrelated ( r = 0 ) , then the last terms in equations 4.10 and 4.11 vanish ( i.e 2rs1s2 = 0 ) and the sum or difference of the variates has a variance equal to the sum of the separate variances , or , &amp;formula; . these additive properties are not in general possessed by the other measures of dispersion that have been discussed . 4.18 . the data already used in table 4 . A are written out in full in table 4 . C , which illustrates how the above five formulae work . here , the individual values of X1 and X2 are put opposite one another so that r = 3/4 . the values of X3 and X4 are put together so that r = 0 . the actual means and variances of the sums and differences of X1 with X2 and X3 with X4 , may be compared with the results of using the above formulae . &amp;formula; . these results agree with those calculated in table 4 . C . the reader should notice that in this table , &amp;formula; . 4.19 . measuring scales and parameters . all the parameters we have discussed may be justifiably used with measurements on a ratio or interval scale . nominal scales , by definition , do not justify the calculation of any position or dispersion parameters , since in such scales there is no dimension or singleness of direction involved . in nominal scales , events are numbered to show they are the same or different from other events , i.e the numbers reflect qualitative , not quantitative characteristics in the data . an ordinal scale does reflect quantitative features of the material measured , i.e a dimension or singleness of direction , but it does so by inconstant units of unknown size . the numbers which constitute an ordinal scale may vary by fixed and known amounts ( such as in ranking ) , but this in no way implies that the objects measured by these numbers also change by fixed amounts . the lack of isomorphism between number intervals and object intervals in ordinal scales of all types , makes the addition and subtraction of ordinal measurements illegitimate . addition and subtraction of numbers signifies an imaginary movement over certain intervals . if these numerical intervals do not correspond to object intervals , addition or subtraction of the numbers may lead to false conclusions about the objects they are supposed to represent . since addition and subtraction of ordinal measurements are not legitimate , the calculation of means is not justified , and the use of medians , which do not require the addition of X values , is more permissible . 4.20 . an illustration of the type of error which means of ordinal scales may engender , will clarify the above discussion and bring to light some further relevant considerations . imagine a set of objects A , B , C , &amp;hellip; which differ from one another by equal amounts of some variable . let the true interval scale , measuring these objects , be represented by the italic numbers 1 , 2 , 3 , &amp;hellip; if all knowledge of the interval sizes is denied us , we may construct a standard ordinal scale , which may be represented by normal numbers , 1 , 2 , 3 , &amp;hellip; the relation between the true and the ordinal numbers might be - relative to the interval scale , this ordinal scale is stretched at B , F , H , I , J , M , and N , and compressed between O and P . if we measure the objects ACK and CDE on our ordinal scale , the means of these two groups of objects are each equal to 3 , i.e the mean object is D for both sets . yet the positions of the two sets of objects are different when measured on the true interval scale , which yields means of 5 and 4 respectively , i.e objects E and D . the point being made here is not that the numerical values of the means differ from one scale to another , but that the two scales yield different conclusions about the similarity between the two groups of three objects . the mean centigrade temperature of a set of objects will be numerically different from the mean fahrenheit temperature , yet both means will refer to the same object , because these scales are interval scales . the ordinal scale means of objects deo and AGP are 5 and 6 , while the interval scale means agree at the value 8 . this illustrates the error converse to that already given , the ordinal scale producing a difference where none exists . 4.21 . means and medians . the medians of the ordinal measurements of the first two groups given above are 2(ACK) and 3(CDE) . this observation shows that means and medians do not necessarily agree in the conclusions they yield . the interval scale means show that CDE sits to the left of ACK , ordinal scale means make both groups equal in position , and now , ordinal scale medians place CDE to the right of ACK . which of these conclusions is correct ? the truth is that the first and last are both correct , though they disagree ! this apparent paradox is resolved when we note that means refer to the interval properties of objects and medians to their ordinal properties . if only order is known , medians will yield conclusions which are correct so far as order is concerned . if intervals are known , these supersede simple order , and means will yield conclusions which are correct relative to this improved knowledge . note that the medians of both the interval and ordinal measurements of ACK and CDE agree in selecting objects C and D . we may say that a mean is a strong parameter which requires known intervals and if applied to a weak scale ( ordinal ) may yield false conclusions . a median is a weak parameter and if applied to a strong scale ( interval or ratio ) will yield a result comparable to that obtainable from any weak equivalent of this scale . finally , we should note that the numerical size of a difference between means of interval or ratio scale data is an indication of the extent to which the data differ in position , but the numerical size of a difference between medians of any data is not an indication of the extent of difference . 4.22 . variances and semi-interquartile ranges . the argument against ordinal scale means can be extended to the use of variances on ordinal scale data . is there any dispersion parameter which may be legitimately used on ordinal measurements ? the obvious candidate for this role is the semi-interquartile range , but although this is a parameter concerned chiefly with order , it is unsatisfactory . the semi-interquartile ranges of two sets of ordinal results might show them to be similar ( or different ) in dispersion , but the use of some other order parameter ( e.g half the distance between the top tenth and the bottom tenth of the data ) might show them to be different ( or similar ) , and we have no reason for choosing one kind of order parameter rather than another . we shall not pursue this argument further , except to say that dispersion is almost synonymous with distance and the distance between objects is something about which ordinal scales tell us very little . to seek a dispersion parameter for ordinal scale data is to ask from the scale more than it is able to tell us . 4.23 . a mechanical analogy . we may imagine a variate X to be represented by a horizontal uniform rod of negligible mass which is marked off in the units of X . each individual in the population can be represented by a small weight . we can now attach these weights to the uniform rod at the points which represent their variate value . the resulting assembly will resemble a histogram turned upside down . an illustration is given in fig 4 . A . in this illustration , each individual f is represented by a weight hung from its value of X . if we try to find that point on the rod which will balance the whole assembly , we discover it as m . in other words , the mean of a distribution is its centre of gravity . when the apparatus is hung from its centre of gravity , we may give one end of it a little push . this will set it spinning or rotating about the point of suspension . the amount of spinning it does depends on how spread out the weights are along the rod . if the weights are clustered closely around the centre of gravity , it will be highly stable and swing very little . if they are spread out along the length of the rod , it will be unstable and swing a great deal . the stability of the apparatus is given by s2 . in other words , the variance of a distribution is its moment of inertia . 4.24 . short cuts in calculating . we have already learned that frequency distributions provide easier arithmetic than a set of disorganised measurements ( 4.5 ) . there are techniques which make calculation still less laborious , and these may well be discussed here . in calculating the mean of a set of data , we must add all the values of the variate and divide the total so obtained by N . when the variate values are large numbers ( such as age in months ranging from 120 to 145 months ) , addition is laborious and , consequently , liable to error . a short cut which reduces the size of the values to be added is to accept a central value arbitrarily ( A ) before we begin the calculation and write all variate values ( X ) as deviations ( x&amp;prime; ) from this . the mean of the data can then be found from &amp;formula; . this formula derives from the fact that the sum of the deviations of a set of numbers from their mean ( Sx ) is zero ( 4.8 ) . it follows that if Sfx&amp;prime; = 0 then A = m , and we have chosen the mean as our central value by accident . if Sfx&amp;prime; is positive , then the A chosen must have been smaller than m . if Sfx&amp;prime; is negative , then the A chosen was larger than m . 4.25 . the major difficulty encountered in calculating the variance or standard deviation of data , is that if m is , say , 74.98 , then all deviations from this value must involve two places of decimals . squaring numbers containing two places of decimals is a tedious matter . this difficulty can be circumvented by using the deviations from A mentioned above . the formula for the variance then becomes - &amp;formula; and the standard deviation is &amp;formula; . the reason we subtract the correction term &amp;formula; is that the sum of squares of deviations from a mean , is smaller than squares about any other point . 