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Home Content 5.1 Introduction 5.2 Table Representation 5.3 Graphical Representation 5.4 Measures of Central Tendency 5.5 Measure of Variability 5.6 Mean, Variance and Standard Deviation for Grouped Data

5.4 Measures of Central Tendency

 
5.4.1
5.4.2
5.4.3

 
We often represent a data set by numerical summary measures, usually called the typical values.  A measure of central tendency gives the centre of a histogram or a frequency distribution curve.

 
5.4.1 Mean
The mean or average is the most frequently used measure of central tendency.  It is obtained by dividing the sum of all values by the number of values in the data set.  Thus, 
Mean for population data: 
Mean for sample data: 
where  is the sum of all values, N is the population size, n is the sample size,  is the population mean, and   is the sample mean.
Example 5.4-1
The following data give the prices of five telephone handsets sold in a shop yesterday. 
 
158
189
265
127
191
Find the mean sale price for these telephone handsets
A major shortcoming of the mean as a measure of central tendency is that it is very sensitive to outliers. (outliers or extreme values are very small or very large relative to the majority of the values in a data set.)
 
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5.4.2 Median
Another important measure of central tendency is the median.  The median is the value of the middle term in a data set that has been ranked in increasing order.  The calculation of the median consists of the following two steps.
1. Rank the given data set in increasing order.
2. Find the middle term.  The value of this term is the median.
The position of the middle term in a data set with n values is obtained as follows.
Position of the middle term =
If the number of observations in a data set is odd, then the median is given by the value of the middle term in the ranked data.  If the number of observation is even, then the median is given by the average of the values of the two middle terms.
Example 5.4-2
Find the median for the following data.
37.1
42.2
53.1
53.2
70.0
71.9
74.9
79.6
93.6
109.6
137.1
168.8
Position of the middle term = 
Therefore, the median is given by the mean of the 6th and 7th values in the ranked data. i.e.
Median = 
The advantage of using the median as a measure of central tendency is that it is not influenced by outliers
 
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5.4.3 Mode
The mode is the value that occurs with the highest frequency in a data set.
Example 5.4-3 
The following data give the speeds (in miles per hour) of eight cars that were stopped on a highway or speeding violations.
 
77
69
74
81
71
68
74
73

Find the mode.
Mode = 74 miles per hour
A major shortcoming of the mode is that a data set may have none or may have more than one mode, whereas it will have only one mean and only one median.
 
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Home Content 5.1 Introduction 5.2 Table Representation 5.3 Graphical Representation 5.4 Measures of Central Tendency 5.5 Measure of Variability 5.6 Mean, Variance and Standard Deviation for Grouped Data