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I am trying to take three independent indicators and combine them into a single composite measure.
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For discussion, let's say that this is a composite measure for 'economic performance', and that the measure contains three independent indicators: worklessness, qualifications and unemployment. Let's also assume that there is a complete data set, with scores for each of the 50 local areas.
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To compare and contrast those 50 local areas, we believe that the most robust way is to work out the standard deviation for each indicator (i.e. working out the standard deviation [s.d.] for each of the three indicators, for each of the 50 local areas).
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The problem that we face, however, is when we come to combine the indicators together into a single composite measure. I understand from reading elsewhere (including on stats.stackexchange.com) that standard deviations cannot simply be added together; and that the correct method is to square each of the standard deviations individually to obtain the variance; add the variances together, and then divide this by the number of variances (i.e. divide by three in this case) and then taking the square root of this number.
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If it was okay to add standard deviations together, there would be no problems, as we would simply be able to add standard deviations together, rank the resulting value, and quintile them.
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However, if we follow (4.) above to combine standard deviations, the majority of the results now fall within one standard deviation of the mean. The smallest level of granularity is now 1 s.d., making it impossible to rank / quintile them, except assume that anything within 1 s.d. is in the mid quintile.
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Is there any way I can justify simply adding together standard deviations, so that they can actually be ranked and quintiled, instead of following the 'proper' method?
Solved – Combining and ranking standard deviations
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Best Answer
The solution, it turns out, is to normalise the variables first (i.e. by calculating standard scores -- in this case, z-scores / z-values) which can be added together -- and that solves the problem in (4.) above.
Sets of data, each containing 50 observations of 3 variables - to classify the spread of the 50 observations into equal groups, so: