The proportions between the mean and these z values are 0.4732 and 0.2734 respectively .sx The proportion between z;1 ; and z;2 ; is therefore 0.4732 - 0.2734 = 0.1998. This is the same as the proportion between z;1 ; = - 1.93 and z;2 ; = - 0.75 , since the curve is symmetrical .sx 4.16 As well as occurring in the equation of the normal and other curves , the mean and variance parameters have another valuable property .sx This is the fact that they are additive .sx If we have two populations , with means 15m;1 ; and 15m;2 ; , and we add the variate values of these populations in pairs , we find that the mean of the sum ( 15m;1+2; ) is the sum of the means ( 15m;1 ; + 15m;2; ) or The mean difference between pairs of population values is the difference of the means of the separate populations , i.e. These simple properties are not , in general , possessed by medians , modes , or other position parameters .sx 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 .sx The extent of correlation is expressed by the correlation coefficient 5r ( Greek letter rho , pronounced " roe") .sx 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 .sx The coefficient 5r can take all fractional values between + 1.0 and - 1.0 ( for further discussion of 5r see Chapter 9 , particularly ) .sx We may now state that the variance of a sum ( 15s :sx 2:;1+2; ) is A similar property holds for the variance of the differences between two correlated populations given as If it happens that our two populations are uncorrelated ( 5r = 0 ) , then the last terms in equations 4.10 and 4.11 vanish ( i.e. 215rs;1;15s;2 ; = 0 ) and the sum or difference of the variates has a variance equal to the sum of the separate variances , or , These additive properties are not in general possessed by the other measures of dispersion that have been discussed .sx 4.18 The data already used in Table 4 .sx A are written out in full in Table 4 .sx C , which illustrates how the above five formulae work .sx Here , the individual values of X;1 ; and X;2 ; are put opposite one another so that 5r = 3/4 .sx The values of X;3 ; and X;4 ; are put together so that 5r = 0 .sx The actual means and variances of the sums and differences of X;1 ; with X;2 ; and X;3 ; with X;4 ; , may be compared with the results of using the above formulae .sx These results agree with those calculated in Table 4 .sx C. The reader should notice that in this table , [FORMULA] .sx 4.19 Measuring Scales and Parameters .sx All the parameters we have discussed may be justifiably used with measurements on a ratio or interval scale .sx 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 .sx 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 .sx 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 .sx 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 .sx 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 .sx Addition and subtraction of numbers signifies an imaginary movement over certain intervals .sx 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 .sx 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 .sx 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 .sx Imagine a set of objects A , B , C, .sx . which differ from one another by equal amounts of some variable .sx Let the " true " interval scale , measuring these objects , be represented by the italic numbers 1 , 2 , 3, .sx . 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, .sx . 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 .sx 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 .sx 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 .sx The ordinal scale means of objects DEO and AGP are 5 and 6 , while the interval scale means agree at the value 8 .sx This illustrates the error converse to that already given , the ordinal scale producing a difference where none exists .sx 4.21 Means and Medians .sx The medians of the ordinal measurements of the first two groups given above are 2(ACK ) and 3(CDE) .sx This observation shows that means and medians do not necessarily agree in the conclusions they yield .sx 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 .sx Which of these conclusions is correct ?sx The truth is that the first and last are both correct , though they disagree !sx This apparent paradox is resolved when we note that means refer to the interval properties of objects and medians to their ordinal properties .sx If only order is known , medians will yield conclusions which are correct so far as order is concerned .sx If intervals are known , these supersede simple order , and means will yield conclusions which are correct relative to this improved knowledge .sx 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 .sx 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 .sx 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 .sx 4.22 Variances and Semi-interquartile Ranges .sx The argument against ordinal scale means can be extended to the use of variances on ordinal scale data .sx Is there any dispersion parameter which may be legitimately used on ordinal measurements ?sx The obvious candidate for this role is the semi-interquartile range , but although this is a parameter concerned chiefly with order , it is unsatisfactory .sx 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 .sx 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 .sx To seek a dispersion parameter for ordinal scale data is to ask from the scale more than it is able to tell us .sx 4.23 A Mechanical Analogy .sx 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 .sx We can now attach these weights to the uniform rod at the points which represent their variate value .sx The resulting assembly will resemble a histogram turned upside down .sx An illustration is given in Fig. 4 .sx 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 5m .sx In other words , the mean of a distribution is its centre of gravity .sx When the apparatus is hung from its centre of gravity , we may give one end of it a little push .sx This will set it spinning or rotating about the point of suspension .sx The amount of spinning it does depends on how spread out the weights are along the rod .sx If the weights are clustered closely around the centre of gravity , it will be highly stable and swing very little .sx If they are spread out along the length of the rod , it will be unstable and swing a great deal .sx The stability of the apparatus is given by 15s :sx 2:. In other words , the variance of a distribution is its moment of inertia .sx 4.24 Short Cuts in Calculating .sx We have already learned that frequency distributions provide easier arithmetic than a set of disorganised measurements ( ) .sx There are techniques which make calculation still less laborious , and these may well be discussed here .sx 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 .sx 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@7 ) from this .sx The mean of the data can then be found from This formula derives from the fact that the sum of the deviations of a set of numbers from their mean ( 15Sx ) is zero ( ) .sx It follows that if 15Sfx@7 = 0 then A = 5m , and we have chosen the mean as our central value by accident .sx If 15Sfx@7 is positive , then the A chosen must have been smaller than 5m .sx If 15Sfx@7 is negative , then the A chosen was larger than 5m .sx 4.25 The major difficulty encountered in calculating the variance or standard deviation of data , is that if 5m is , say , 74.98 , then all deviations from this value must involve two places of decimals .sx Squaring numbers containing two places of decimals is a tedious matter .sx This difficulty can be circumvented by using the deviations from A mentioned above .sx The formula for the variance then becomes- and the standard deviation is The reason we subtract the correction term [FORMULA] is that the sum of squares of deviations from a mean , is smaller than squares about any other point .sx