DESCRIPTIVE MEASURES
It is useful to summarize data by methods that lead to numerical results, called descriptive measures. We discuss two types of descriptive measures: measures of central tendency and measures of dispersion. They may be computed from the data contained in a sample or from the data of a finite population. A descriptive Thomas Sabo Bracelets measure computed from or used to describe a sample of data is called a statistic while a descriptive measure computed from or used to describe a population of data is called a parameter.
Measures of Central Tendency
Even when you draw a collection of data from a common source, individual observations are not likely to have the same value. It is impractical to keep in mind all the values that may be present in a set of data. What we need is some single value that we may consider typical of the set of data as a whole. The need for such a single value is usually met by one of the three measures of central tendency: the arithmetic mean, the median, and the mode.
The Arithmetic Mean. The most familiar measure of central tendency is the arithmetic mean. Popularly known as the average, it is sometimes called the arithmetic average, or simply the mean. We find it by adding all the values in a set of data and dividing the total by the number of values that were summed.
The properties of the arithmetic mean include the following: (1) For a given set of data, there is one, and only one, arithmetic mean. (2) Its meaning is easily understood. (3) Since every value goes into its computation, it is affected by the magnitude of each value. Because of this property, the arithmetic mean may not be the best measure of central tendency when one or
two extreme values are present in a set of data. And (4) the mean, unlike some descriptive measures whose values may be determined by inspection, is a computed measure, and therefore it can be manipulated algebraically. This property makes it an especially useful measure for statistical inference purposes.
The Weighted Mean. When the frequency of occurrence of the individual measurements to be averaged varies, we may refer to the frequencies as weights and to the resulting mean as a weighted mean. Sometimes the measurements to be averaged vary in importance rather than frequency of occurrence. In such cases, a weighted mean will provide an average that reflects the relative importance of the individual measurements.
The Median. The median is that value above which half the values lie and below which the other half lie. If the number of items is odd, the median is the value of the middle item of an ordered array, when the items are arranged in ascending (or descending) order of magnitude. If the number of items is even, none of the items has an equal number of values above and below it. In this event, the median is equal the mean, or average, of the two middle values.
The Mode. The mode for ungrouped discrete data is the value that occurs most frequently. If all the values in a set of data are different, there is no mode.
The Geometric Mean
There are some problems requiring the calculation of an average for which none of the averages discussed so far is appropriate. For example, when we wish to obtain the average value of a series of ratios, percentages, or rates of change, the arithmetic mean proves to be an inadequate choice for the job. The measure needed in these situations is the geometric mean.
The geometric mean of a series of n measurements is the nth root of the product of the n measurements.
Calculating the geometric mean according to its basic definition can be a laborious task. By the use of logarithms, however, the measure may be computed with relative ease.
The following are some characteristics of the geometric mean: (1) It is not unduly influenced by extreme values. (2) It is always smaller Thomas Sabo Charms than the arithmetic mean. (3) It is a meaningful measure only when all of the measurements are positive. And (4) the product of a series of measurements remains unchanged if the geometric mean of the measurements is substituted for each measurement in the series.
The Harmonic Mean mother average that is preferred over other such measures in certain situations is the harmonic mean. The harmonic mean of a series of measurements is the reciprocal of the arithmetic mean) f the reciprocals of the individual measurements.
The harmonic mean is the average of choice when the average of time rates is required. It lass decided advantages when the data to be averaged are certain types of price data, ^consequently, the harmonic mean finds frequent use in the field of economics.
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