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Lecture 3

Measures of Central Tendency

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To be able to determine which of the three measures(mean, median and mode) to apply to a given set of data with

the given purpose of information.

Objective

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Introduction Definition of Measures of Central Tendency Mean Arithmetic Mean

Ungrouped data Grouped data

Median Ungrouped data Grouped data Graphical Method

Content

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Mode Ungrouped data Grouped data

Content

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Measures of central tendency are single values that are typical and representative for a group of numbers.

They are also called measures of locations.

A representative values of location for a group of numbers is neither the biggest nor the smallest but is a number whose values is somewhere in the middle of the group.

Such measures are often used to summarize a data set and to compare one data set with another.

Measures ofCentral Tendency

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The average value of a set of data.

Appropriate for describing measurement data, eg. heights of people, marks of student papers.

Often influenced by extreme values.

Mean

Nx

nx

nxxx n

x

........... x 21

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of summation thesize population N

size, sample variable, of value

where,

nxx

• For Ungrouped data : Arithmetic Mean

• Usually we seldom use population mean, µ, because the population is very large and it would be troublesome to gather all the values. We usually calculate the sample mean, and use it to make an estimation of µ.

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Example: Find the average age of five students whose ages are 18, 19, 19, 19, 20 and 22 respectively.

Solution

old. years 19.6 598

nx

x

Ungrouped Data

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nfx

x

Method Basic 1)

Grouped Data

The distances traveled by 100 workers of XYZ

Company from their homes to the workplace are

summarized below (next slide). Find the mean

distance traveled by a worker.

For Example:

Distances (km) No. of workers

0 and under 2 12

2 and under 4 35

4 and under 6 24

6 and under 8 18

8 and under 10 11

TOTAL 100

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Grouped DataKilometers

travelledNo of

WorkersMid

Point(X)

Total distance travelled

ƒ(x)

0 and under 2 12 1 12

2 and under 4 35 3 105

4 and under 6 24 5 120

6 and under 8 18 7 126

8 and under 10

11 9 99

total 100 Total 462Mean = 462/100 = 4.62

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It is easy to calculate and understand.

It makes use of all the data points and can be determined with mathematical exactness.

The mean is useful for performing statistical procedures like comparing the means between data sets.

Advantages of the mean

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Can be significantly influenced by extreme abnormal values.

It may not be a value which correspond with a single item in the data set.

Every item in the data set is taken into consideration when computing the mean. As a result it can be very tedious to compute when we have a very large data set.

It is not possible for use to compute the mean with open ended classes.

Weaknesses of the mean

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The weighted mean enables us to calculate an average which takes into account the relative importance of each value to the overall total.

ExampleA lecturer in XYZ Polytechnic has decided to use weighted average in awarding final marks for his students. Class participation will account for 10% of the student’s grade, mid term test 15%, project 20%, quiz 5% and final exam 50%.

Weighted mean

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From the information given, compute the final average for Zaraa who is one of the students.

Zaraa’s marks are as follows: class participation 90quiz 80 project 75mid term test 70final examination 85

Weighted mean

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Solution

Subjects Marks (x) Weight (w) wx

Participation 90 10 900

Quiz 80 5 400

Project 75 20 1500

Mid term test 70 15 1050

Final exam 85 50 4250

total 100 8100

81 1008100

x

wwx

Weighted mean

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The median of a set of values is defined as the value of the middle item when the values are arranged in ascending or descending order of magnitude.

Median

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If the data set has an odd number of observations, the middle item will be the required median after the data has been arranged in either ascending or descending order.

If the data array has an even number of observations, we will take the average of the two middle items for the required median as shown at the next slide.

Ungrouped Data

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a) odd number of values data set: 2,1,5,2,10,6,8 array: 1,2,2,5,6,8, 10 median = value of fourth item.

b) Even number of values data set:9,6,2,5,18,4,12,10

array: 2,4,5,6,9,10,12,18 median = (6+9) / 2

= 7.5

Ungrouped DataExample :

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ExampleKilometers travelled

No of workers

Cumulative frequency

0 and under 2

12 12

2 and under 4

35 47

4 and under 6

24 71

6 and under 8

18 89

8 and under 10

11 100

Total 100

Grouped Data

Middle term = n/2 = 100/2 = 50th term

We identify the median class by examining the cumulative frequencies

10

cf

50

42 6 8

ogives

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We can also find the median using a graph. Here, we will first plot the “more than” and “less than” ogives. The median is the value of the intersection point of the two ogives.

Graphical Method

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Example :

Distance (km)

No of workers

Less than cf

More than cf

0 and under 2 12 12 882 and under 4 35 47 534 and under 6 24 71 296 and under 8 18 89 118 and under

1011 11 0

total 100

Graphical Method

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It is easy to compute and simple to understand.

It is not affected by extreme values.

Can take on open ended classes.

It can deal with qualitative data.

Advantages of the median

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It can be time consuming to compute as we have to first array the data.

If we have only a few values, the median is not likely to be representative.

It is usually less reliable than the mean for statistical inference purpose. It is not suitable for arithmetical calculations, and has limited use for practical work.

Disadvantages of the median

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The most frequent or repeated value.

In the case of a continuous variable, it is possible that no two values will repeat themselves. In such a situation, the mode is defined as the point of highest frequency density, i.e., where occurrences cluster most closely together.

Mode

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Like the median, the mode has very limited practical use and cannot be subjected to arithmetical manipulation.

However, being the value that occurs most often, it provides a good representation of the data set.

Mode

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The mode can be obtained simply by inspection.

Example 1,4,10,8,10,12,13 Mode=10 1,3,3,7,8,8,9 Mode= 3 and 8 1,2,3,4,9,10,11 No mode

Mode for Ungrouped Data

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In the case of grouped data, finding the mode may not be as easy. Since a grouped frequency distribution does not show individual values, it is obviously impossible to determine the value which occurs most frequently.

Here we can assign a mode to grouped data that have the highest frequency even though we may not know whether or not any data value occurs more than once.

Mode for Grouped Data

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Advantages It can take on open ended classes. It cannot be affected by extreme values. It is also applicable to qualitative data.

Disadvantages Not clearly defined. Some data may have no mode It is difficult to interpret and compare if data set has

more than one mode

Advantage and Disadvantage of the Mode

References :

Lecture & Tutorial Notes from Department of Business & Management, Institute Technology Brunei, Brunei Darussalam.