By R. H. Leaver, T. R. Thomas (auth.)
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Example text
The values which divide the histogram into four parts are called quartiles, and those dividing the diagram into ten and a hundred parts are called deciles and percentiles respectively. 3 The mean The mean is the most useful parameter describing central tendency, and is simply the average value of the variate, and is denoted by x. 13) or more briefly, x = 'E xfn where n is the number of values of the variate in the sample. For grouped data, or if the values of the variate are repeated, a more convenient form is x = '1:-fx/'Ef where f is the frequency with which x occurs in the group.
Each of these measurements is itself susceptible to error. How do we combine the errors in our estimate of g? Let us look at the general case of a derived quantity u which is some function of two independent variables x andy, that is u = f(x,y) Now u; = u + &u;, x; = x + &x;,Y; = y + &y; where the delta is used to indicate a residual. So u + &u = f(x + &x,y + &y) 38 39 ERRORS AND THEIR SIGNIFICANCE If this is expanded in a Taylor series we obtain ii. 2 can be extended to include any number of variables.
11) 24 ANALYSIS AND PRESENTATION OF EXPERIMENTAL RESULTS Returning to our analogue, this equation is the equivalent of the parallel-axis theorem for moments of inertia; where a 2 is analogous to the moment of inertia about the centre of gravity and IJ. 9, equivalent to the moment of inertia about the origin. If x is measured from the mean, and 1J. 12) Clearly, the variance is a measure of the spread or dispersion of the variate about the mean of the distribution. Higher moments than the second have a limited use in describing the shape of the distribution, but will not be considered here.