Topic 3: Measures of central tendency, dispersion and shape

Numerical Data Properties & Measures Central Variation Shape Tendency Mean Range Skewness Kurtosis Variance Median Standard Deviation Mode

Measures of Central Tendency

Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents the entire distribution of scores. The goal of central tendency is to identify the single value that is the best representative for the entire set of data.

Central Tendency (cont.) By identifying the "average score," central tendency allows researchers to summarize or condense a large set of data into a single value. Thus, central tendency serves as a descriptive statistic because it allows researchers to describe or present a set of data in a very simplified, concise form. In addition, it is possible to compare two (or more) sets of data by simply comparing the average score (central tendency) for one set versus the average score for another set.

Properties of a good average Easy to understand Simple to Calculate. Should be based on all items in the series Rigidly Defined: Should be properly defined with an algebraic formulae Algebraic Treatment: It can be used for further statistical computation to enhance its usefulness Should have sampling stability Should not be affected by extreme values

Data series Individual data series 3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

Discrete series Continuous series

Arithmetic mean Measure of Central Tendency Most Common Measure Acts as ‘Balance Point’ It should not be used for ordinal or nominal data

Formulae Individual

Formulae Individual Discrete

Formulae Individual Discrete Continuous m= mid point

Example Calculate Arithmetic mean for the data sets given in slide 7 and 8

Properties of Arithmetic mean The sum of the deviations of all values of a distribution from their arithmetic mean is zero. The sum of the squared deviations of the items from the arithmetic mean is minimum If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.

Properties of Arithmetic mean (cont.) Using arithmetic mean and the number of items of two or more related groups , we can compute the combined mean by using this formula

Merits of Arithmetic mean Easy to understand Simple to Calculate. Its based on all items in the series Rigidly Defined It can be used for further statistical computation to enhance its usefulness Should have sampling stability

Demerits Its affected by extreme values Cannot be computed in a distribution with open ended classes

Weighted mean This is arithmetic mean computed by considering relative importance of each items. To give due importance to each item under consideration, we assign number called weight to each item in proportion to its relative importance

The Median The Median is the midpoint of the values after they have been ordered from the smallest to the largest. There are as many values above the median as below it in the data array. For an even set of values, the median will be the arithmetic average of the two middle numbers.

Formulae Individual and Discrete series

EXAMPLES - Median Odd series The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21.

EXAMPLES - Median Odd series Even series The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The heights of four basketball players, in inches, are: 76, 73, 80, 75 Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5

Formulae Continuous series Where L: is the lower class limit of the median class cf: is the cumulative frequency of the class preceeding the median class f: is the frequency of the median class i: is the class interval

exercise Find median for the following data sets (i) 3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

(ii) Discrete series (iii) Continuous series

Properties of the Median It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

Advantages Useful incase of open ended classes Not influenced by extreme values and therefore is preferred to the arithmetic mean in skewed distributions such as income It is most appropriate when dealing with qualitative data i.e where ranks are given or there are other types of items that are not counted or measured but are scored. The median can be determined graphically

Limitations Its computation does not involve all items in the series It is not capable of further algebraic treatment e.g we cannot find the combined median of two data sets. Its value is affected more by sampling fluctuations than the value of the arithmetic mean.

OTHER TYPES OF ‘AVERAGES’ Geometric Mean Harmonic Mean

Geometric Mean This is the nth root of the product of n items or values where x1, x2, x3, etc refer to various items of the series.

Geometric Mean (cont’) To simplify calculations when the number of items is three or more logarithms are used

Properties of geometric mean The product of values of the series remain unchanged when the value of the GM is substituted for each individual value.e.g the GM for the series 2,4,8, is 4

Uses of GM Finding average percent increase in sales, production, population or other economic or business series. Construction of index numbers Is the most suitable average when large weights have to be given to small items and small weights to large items

Merits of Geometric mean Based on all items of the series It is rigidly defined Useful in averaging ratios and percentages and in determining rates of increase and decrease. It is capable of algebraic manipulation. A combined GM can be obtained of two or more series using the formula

Limitations of GM It is difficult to understand It is difficult to compute and interpret It cannot be computed when there are both negative and positive values in a series or one or more of the values is zero. Because of this it has a restricted application

Harmonic Mean It is the reciprocal of the arithmetic mean of the reciprocal of the individual observations

Uses It is useful in situations where the average of rates is required e.g finding average speed.

Merits Its value is based on every item of the series It lends itself to algebraic manipulation. It gives better results than other averages in problems relating to time and rates.

Limitation It is not easily understood It is difficult to compute It gives largest weight to smallest items. This makes it not useful for analysis of economic data. Its value cannot be computed when there are both positive and negative items in a series or when one or more items are zero. Because of these limitations the HM has very few practical applications except in cases where small items need to be given very high weightage.