1. Topic 3: Measures of central tendency, dispersion and shape

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

3. Measures of Central Tendency

4. 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.

5. 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.

6. 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

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

8. Discrete series
Continuous series

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

10. Formulae
Individual

11. Formulae
Individual
Discrete

12. Formulae
Individual
Discrete
Continuous
m= mid point

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

14. 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.

15. 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

16. 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

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

18. 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

19. 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.

20. Formulae
Individual and Discrete series

21. 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.

22. 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

23. 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

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

25. (ii) Discrete series
(iii) Continuous series

26. 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.

27. 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

28. 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.

29. OTHER TYPES OF ‘AVERAGES’
Geometric Mean
Harmonic Mean

30. 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.

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

32. 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

33. 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

34. 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

35. 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

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

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

38. 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.

39. 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.