Elements of Statistics for Economists Session 2 Presentation of - - PowerPoint PPT Presentation

College of Education School of Continuing and Distance Education

2014/2015 – 2016/2017

ECON 214 Elements of Statistics for Economists

Session 2 – Presentation of Data: Graphical Methods Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh

Session Overview

• The aim of descriptive statistical methods is to present

information in clear, concise and accurate manner. This session will illustrate the methods for summarizing data in a very informative way, using tables and graphs.

• At the end of the session, the student will

– Be able to organize data into frequency distribution – Be able to portray a frequency distribution in histogram, frequency polygon, and cumulative frequency polygon – Be able to present data using bar charts, line graphs and pie charts

Slide 2

Session Outline

The key topics to be covered in the session are as follows:

• Frequency distributions
• Graphic presentation of frequency distributions
• Other graphic forms of presenting data

Slide 3

Reading List

• Michael Barrow, “Statistics for Economics, Accounting and

Business Studies”, 4th Edition, Pearson

• R.D. Mason , D.A. Lind, and W.G. Marchal, “Statistical

Techniques in Business and Economics”, 10th Edition, McGraw- Hill

Slide 4

FREQUENCY DISTRIBUTIONS

Topic One

Slide 5

FREQUENCY DISTRIBUTIONS

• Raw data in itself is meaningless unless it can be

presented in an informative way.

• Descriptive statistics summarizes raw

information in a comprehensible way.

• We may use tables, graphs and/or numeric

values.

• A frequency distribution is grouping of data into

categories showing the number of observations in each mutually exclusive category.

Slide 6

FREQUENCY DISTRIBUTIONS

• The purpose of constructing a frequency distribution

is to condense raw data into a table that is more readily comprehended.

• Although we lose the identification of the specific

value of each measurement, the advantage of a frequency distribution is that such a table makes it easier to interpret the reported values.

Slide 7

FREQUENCY DISTRIBUTIONS

• Some definitions are worth knowing:

– Class limits: They are the smallest and largest

• bservations (values) in each class of a frequency
• distribution. Each class has two limits; we have the lower

class limit and the upper class limit. – Class frequency: The number of observed values in each class. – Class boundaries: They denote specific points along a measurement scale separating adjoining classes. The lower class boundary is obtained by subtracting 0.5 from the lower class limit, while the upper class boundary is

• btained by adding 0.5 to the upper class limit.

Slide 8

FREQUENCY DISTRIBUTIONS

– Class interval or width (or size): The number of measurement units or range of values included in each

• class. It is obtained as upper boundary minus lower
• boundary. It is also obtained by subtracting the lower

limit of a class from the lower limit of the next class. – Class mark or midpoint: The value that divides a class into two equal parts. This is the simple average of the upper and lower class limits.

• Note that in some frequency distributions,

class boundaries class limits overlap.

Slide 9

FREQUENCY DISTRIBUTIONS

• Illustration

– The Dean of the School of Continuing and Distance Education wishes to determine the hours of study Distance Education students do. He selects a random sample of 30 students and determines the number of hours each student studies per week as follows: – 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 9.1, 26.1, 15.7, 14.0, 17.8, 36.3, 23.2, 12.9, 27.1, 16.6.

• Organize the data into a frequency distribution.

Slide 10

FREQUENCY DISTRIBUTIONS CONT’D

• Assume the number of classes is pre-determined to be 6.
• The class intervals used in the frequency distribution

should be equal.

• Determine the class interval by using the formula:

i = (highest value-lowest value)/number of classes i = 36.3-9.1/6 = 4.53 ≈ 5.

• Starting the lower limit of the first class at 8, we can have

the following classes: 8-12; 13-17; 18-22; 23-27; 28-32; 33-37.

• Count the number of values in each class and indicate

the number against each class.

Slide 11

FREQUENCY DISTRIBUTIONS CONT’D

Slide 12

The class interval is 5 (i.e. 13 minus 8).

Hours of studying Frequency, f 8-12 1 13-17 12 18-22 10 23-27 5 28-32 1 33-37 1 Total 30

FREQUENCY DISTRIBUTIONS CONT’D

Slide 13

• The relative frequency of a class is obtained by

dividing the class frequency by the total frequency.

Hours of studying Frequency, f Relative frequency, rf 8-12 1

1/30

13-17 12

12/30

18-22 10

10/30

23-27 5

5/10

28-32 1

1/30

33-37 1

1/30

Total 30 1

GRAPHIC PRESENTATION OF A FREQUENCY DISTRIBUTION

Topic Two

Slide 14

Graphic Presentation of a Frequency Distribution

• The three commonly used graphic forms are

histograms, frequency polygons, and a cumulative frequency curve (ogive).

Slide 15

Histogram

• Histogram: A graph in which the classes are marked on the horizontal

axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other and without spaces/gaps among the bars.

Slide 16

2 4 6 8 10 12 14 10 15 20 25 30 35 Hours spent studying Frequency

Frequency Polygon

• A frequency polygon consists of line segments

connecting the points formed by the class midpoint and the class frequency.

Slide 17

2 4 6 8 10 12 14 10 15 20 25 30 35 Hours spent studying Frequency

Cumulative Frequency Distribution (Ogive)

• A cumulative frequency curve (ogive) is used to determine how many or

what proportion of the data values are below a certain value (or the upper boundary of each class).

• The cumulative frequency for a given class is obtained by adding the

frequency of that class to the cumulative frequency of the preceding class.

• The cumulative frequency for the last class always equals the total frequency.

Slide 18

Hours of studying less than Cumulative frequency 7.5 12.5 1 17.5 13 22.5 23 27.5 28 32.5 29 37.5 30

Slide 19

5 10 15 20 25 30 35 10 15 20 25 30 35 Hours Spent Studying Frequency

• The Ogive is obtained as line segments connecting the

points formed by the class midpoint and the cumulative frequency.

– Note: alternatively the upper class boundary is used.

Slide 20

OTHER GRAPHIC FORMS OF PRESENTING DATA

• Topic Three

The Bar Chart

• A bar chart depicts frequencies for different categories

(of data) by a series of bars (separated by spaces/gaps).

– Consider the data on the education level and employment status data of the labour force of a country (measured in ’000s). – Note that the data is already summarised (cross-tabulated). – We can graphically represent the data by bar charts.

Slide 21

Higher A levels Other No Total education qualification qualification In work 8,224 5,654 11,167 2,583 27,628 Unemployed 217 231 693 303 1,444 Inactive 956 1,354 3,107 2,549 7,966 Total 9,397 7,239 14,967 5,435 37,038

Bar chart by education qualification of people who work

• The bar chart is for education levels of only people who work.
• The height of each bar is determined by the associated frequency.
• The first bar is 8224 units high, the second is 5654, and so on. The
• rdering of the bars could be reversed (‘no qualifications’ becoming

the first category) without altering the message.

Slide 22 2000 4000 6000 8000 10000 12000 Higher education Advanced level Other qualifications No qualifications Number of people (000s)

Multiple bar chart: Educational qualifications by employment category

• Note that the bars for unemployed and inactive are constructed in

the same way as for those in work: the height of the bar is determined by the frequency or number of persons in each category.

Slide 23

2000 4000 6000 8000 10000 12000

Higher education Advanced level Other qualifications No qualifications

Number of people (000s)

In work Unemployed Inactive

Stacked bar chart: Educational qualifications and employment status

• Note: The overall height of each bar is determined by the sum of

the frequencies of the category, given in the final raw of the Table containing the data.

Slide 24 2000 4000 6000 8000 10000 12000 14000 16000 Higher education Advanced level Other qualifications No qualifications

Number of people (000s)

Inactive Unemployed In work

Stacked bar chart: Educational qualifications and employment status (percentages)

• Percentages in each employment category, by educational qualification.

– That is, number of persons in each employment status in each educational qualification category has been converted into a percentage of the total persons in each educational qualification category.

• Instead of bars, we could have used lines in each of the 4 charts to obtain

line charts.

Slide 25

0% 20% 40% 60% 80% 100% Higher education Advanced level Other qualifications No qualifications Inactive Unemployed In work

The Pie Chart

• The pie chart is another useful way of presenting data

graphically.

• It is particularly useful for showing how a variable is

distributed between different categories.

• Just as the stacked (or component) bar chart, a pie chart

portrays the contributions from different sources to some aggregate (relative frequency distribution).

• Depicted in circular form, the circle is divided

proportionally to the relative frequency allocated to the different groups.

Slide 26

The pie chart: Educational qualifications of those in work

• Since a circle is 360 degrees, the angle represented by each category is given

by: angle= (frequency in category/total frequency)x100

• So for Higher Education, angle = (8224/27628)x100 = 107.2 degrees.
• And so forth.

Slide 27

30% 20% 41% 9% Higher education Advanced level Other qualifications No qualifications

References

• Michael Barrow, “Statistics for Economics, Accounting and

Business Studies”, 4th Edition, Pearson

• R.D. Mason , D.A. Lind, and W.G. Marchal, “Statistical

Techniques in Business and Economics”, 10th Edition, McGraw-Hill

Slide 28