Such a curve is perfectly bell shaped curve in which case the value of X or M or Z isjust the same and skewness is altogether absent.From what has been stated above, we can say that there are several types of statistical averages.Researcher has to make a choice for some average.
There are no hard and fast rules for theselection of a particular average in statistical analysis for the selection of an average mostly dependson the nature, type of objectives of the research study. ![]() Kothari 2004 Sample Size Determination Series Only AsThe chief characteristics and the limitations of thevarious averages must be kept in view; discriminate use of average is very essential for soundstatistical analysis.MEASURES OF DISPERSIONAn averages can represent a series only as best as a single figure can, but it certainly cannot revealthe entire story of any phenomenon under study. Specially it fails to give any idea about the scatter ofthe values of items of a variable in the series around the true value of average. In order to measurethis scatter, statistical devices called measures of dispersion are calculated. Important measures ofdispersion are (a) range, (b) mean deviation, and (c) standard deviation.(a) Range is the simplest possible measure of dispersion and is defined as the difference betweenthe values of the extreme items of a series. Thus, HF KI HF IKRange Highest value of an Lowest value of an item in a series item in a seriesThe utility of range is that it gives an idea of the variability very quickly, but the drawback is thatrange is affected very greatly by fluctuations of sampling. Its value is never stable, being based ononly two values of the variable. As such, range is mostly used as a rough measure of variability andis not considered as an appropriate measure in serious research studies.(b) Mean deviation is the average of difference of the values of items from some average of theseries. In calculating mean deviation weignore the minus sign of deviations while taking their total for obtaining the mean deviation. Meandeviation is, thus, obtained as under: Processing and Analysis of Data 135c hMean deviation from mean X Xi X, if deviations, X i X, are obtained from n or arithmetic average.b gMean deviation from median m Xi M, if deviations, Xi M, are obtained n or from medianb gMean deviation from mode z Xi Z, if deviations, X i Z, are obtained from n mode.where Symbol for mean deviation (pronounced as delta); Xi ith values of the variable X; n number of items; X Arithmetic average; M Median; Z Mode.When mean deviation is divided by the average used in finding out the mean deviation itself, theresulting quantity is described as the coefficient of mean deviation. Coefficient of mean deviationis a relative measure of dispersion and is comparable to similar measure of other series. Meandeviation and its coefficient are used in statistical studies for judging the variability, and therebyrender the study of central tendency of a series more precise by throwing light on the typicalness ofan average. It is a better measure of variability than range as it takes into consideration the values ofall items of a series. Even then it is not a frequently used measure as it is not amenable to algebraicprocess.(c) Standard deviation is most widely used measure of dispersion of a series and is commonlydenoted by the symbol (pronounced as sigma). Standard deviation is defined as the square-rootof the average of squares of deviations, when such deviations for the values of individual items in aseries are obtained from the arithmetic average. It is worked out as under: b g d iStandard deviation Xi X 2 n If we use assumed average, A, in place of X while finding deviations, then standard deviation would be worked out asunder: b g F b gI Xi A 2 Xi A 2 HG KJn n Orb g FHG b gIKJ 2 fi Xi A 2 fi Xi A fi fi, in case of frequency distribution.This is also known as the short-cut method of finding. Research Methodology Orb g d iStandard deviation fi Xi X 2 fi, in case of frequency distribution where fi means the frequency of the ith item.When we divide the standard deviation by the arithmetic average of the series, the resulting quantityis known as coefficient of standard deviation which happens to be a relative measure and is oftenused for comparing with similar measure of other series. When this coefficient of standard deviationis multiplied by 100, the resulting figure is known as coefficient of variation. Sometimes, we workout the square of standard deviation, known as variance, which is frequently used in the context ofanalysis of variation. The standard deviation (along with several related measures like variance, coefficient of variation,etc.) is used mostly in research studies and is regarded as a very satisfactory measure of dispersionin a series. It is amenable to mathematical manipulation because the algebraic signs are not ignoredin its calculation (as we ignore in case of mean deviation). These advantages make standard deviation and its coefficient a very popular measure ofthe scatteredness of a series. It is popularly used in the context of estimation and testing of hypotheses.MEASURES OF ASYMMETRY (SKEWNESS)When the distribution of item in a series happens to be perfectly symmetrical, we then have thefollowing type of curve for the distribution: (X M Z ) Curve showing no skewness in which case we have X M Z Fig. Such a curve is technically described as a normal curve and the relating distribution as normaldistribution.
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