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STATISTICAL MEASURES OF DATA

Parameter.Any numerical value describing a characteristic of a population.The parameter is a constant value describing the population.

Statistic. Any numerical value describing a characteristic of a sample.

A. MEASURES OF CENTRAL LOCATION

Any measure indicating the center of a set of data.

Population Mean(). If the set of data x1, x2,, xN, not necessarily alldistinct, represents a finite population of size N, then the population mean is

Sample Mean(x). If the set of data x1, x2,, xN, not necessarily all distinctrepresents a finite sample of size n, then the sample mean is

Median(x). The median of a set of observations arranged in an increasing ordecreasing order of magnitude is the middle value when the number of

observations is odd or the arithmetic mean of the two middle values whenthe number of observations is even.

Mode(x). The mode of a set of observations is that value which occurs mostoften or with the greatest frequency. The mode does not always exist. Thisis certainly true when all observations occur with the same frequency. Forsome sets of data there may be several values occurring with the greatestfrequency in which case we have more than one mode.

Midrange. The midrange is defined as the mean of the largest and smallest valuesin a set of data.

Weighted mean. Often, we wish to average the k quantities x1, x2, xk byattaching more significance to some of the numbers than to others. We accomplishthis by assigning weights w1, w2, ,wk to the kquantities, where weights representmeasures of their relative importance. The corresponding weighted mean, w or xw,is given by

Combined mean. Suppose that k finite populations having N1, N2, , Nkmeasurements, respectively, have means 1, 2, , k. The combined populationmean,c , of all the population is

If random samples of size n1, n2, , nk, selected from these k populations,have the meansx1, x2, , xk, respectively, the combined sample mean,xc, ofall the sample data is

Geometric mean. The geometric mean, G, of k positive numbers x1, x2, , xk isthe kth root of their product; that is,

Note that the logarithm of the geometric mean of the k positive numbersequals the arithmetic mean of their logarithms. The geometric mean is used

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primarily to average data for which the ratio of consecutive terms remainsapproximately constant. This occurs, for example, with such data as rates ofchange, ratios, economic index numbers, population sizs over consecutivetime periods, and like.