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Errors in dependent and independent variable

Dependent and independent variable, measurement errors in bothIn the following example we have slats of different lengths and read their length (without bias) in both cm and inch. Every single reading in cm and inch is likely to have reading errors. The data are again plotted in an X-Y diagram, inch on the abscissa; due to the errors points are now displaced both vertically and horizontally.

Correlation and measurement range

If the slats differed little in length, so that the measurement range was quite limited, then the measurement errors would lead to a small cloud which would conceal the linear relationship. In the diagram to the upper right a rather tight relationship (correlation) shows up between inch and cm, but towards the right the correlation is poor. Obviously the coefficient of correlation, a statistical measure of the strength of the relationship between dependent and independent variables measured with errors is larger the greater the data range of data.

In studies in which the relationship between e.g. length and vital capacity is studied from childhood to adulthood, the length ranges between about 100 cm and 200 cm, that is by a factor 2. If we perform the same study in adults length varies between say 160 and 200 cm, a much smaller ranger; for the same measurement error you would expect the correlation to be stronger in children and adolescents than in adults. Random measurement errors, whatever their cause, tend to obscure the relationship between variables; in the case of large errors and a small range the data might even seem entirely unrelated.

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