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Educational Statistics

EPS 211

Prof. Y. K. A. Etsey

Dept. of Educational Foundations

August 2010

UNIT 1

NATURE OF STATISTICS

DEFINITIONS OF STATISTICS

There are three basic definitions of the term, Statistics.

1. Statistics (plural) is the body of numbers or data collected in any field. For example industrial statistics – number of employees in an industry, number of products, value of products; vital statistics – measurements of bust, waist, hips; population statistics – number of people in a region, number of people with secondary education
2. Statistics (singular) is the study of methods and procedures used in collecting, organizing, analyzing, and interpreting a body of numbers for information and decision making.
3. Statistics ( plural; statistic - singular) are the values computed from a body of numerical data. For example the “average” age of Level 200 students in UCC, the proportion of EPS 211 students who are males.

From the definitions above, Educational Statistics can be defined as;

1. the body of numbers or data in the field of education. For example, school statistics – number pupils in a school, number of teachers in a school, number of textbooks in a school; number of teachers in a region, number of pupils in a district.
2. the study of the methods and procedures used in collecting, organizing, analyzing and interpreting a body of numbers related to education for information and decision making.

The main concern of this course is the second definition.

given the values 0, 1, 2, 3, 4 etc. Height can take on values such as 1.2 metres, 1. metres, 2.0 metres and 2.2 metres. Religious affiliation can be broken down to categories and given values as: Christian – 1, Moslem – 2, Traditionalist – 3, Buddhist - 4.

Variables can be classified as ordered or unordered****.

For ordered variables , the attributes differ in magnitude along a quantitative dimension. For example, the number of pupils in a class can be 20, 25, 32, 40, 45 etc where 25 is greater than 20 and 40 is less than 45.

For unordered variables, the attributes are classified into two or more mutually exclusive categories that are qualitatively different. For example, gender is classified into male and female. Undergraduate halls of residence on UCC campus are Oguaa, Adehye, Atlantic, Casely Hayford, VALCO, Kwame Nkrumah.

Variables can also be classified as discrete or continuous.

Discrete variables have values which in theory assume only certain distinct values or whole numbers on a number line. These variables usually represent counts of indivisible entities, for example, 8, 12, 20, 45, 100. For example the number of goals scored in a soccer game or the number of students in a class.

Continuous variables have values which in theory assume any value on a number line between two points. The values can differ by infinitesimal amounts, for example, 10.5, 14.16, 42.001, 56.2222278. For example the height of a student or the weight of a car is a continuous variable.

Descriptive and Inferential Statistics

Descriptive statistics provides summary data about a group. A single number

is often used to represent a group. For example, a mean age of 30 years can be used to represent the ages of 60 Masters in Education students. A median age

of 21 years can be used to represent the ages of all Level 300 students in the

University of Cape Coast.

Inferential statistics uses data from a small group called a sample to make statements or generalizations about a much larger group called a population. For example, to know the mean age of first year university students (i.e. population) in Ghana, a small group (i. e. sample) say 200, of first year students can be used. Their mean age could be used as an estimate of the mean age of all first year university students in Ghana.

Scales of Measurement

Depending upon the traits/attributes/characteristics and the way they are measured, different kinds of data result representing different scales of measurement. There are 4 types of measurement scales. These are Nominal, Ordinal, Interval and Ratio.

Nominal Scales : A nominal scale classifies persons or objects into two or more categories. Whatever the classification, a person can be in one and only one category, and members of a given category have a common set of characteristics. For identification purposes, categories are numbered. e.g. Gender: Male - 1, Female - 2. All males have a common characteristic and all females have a common characteristic which is different from males.

Ordinal Scales: An ordinal scale not only classifies subjects but also ranks them in terms of the degree to which they possess a characteristic/attribute of interest. An ordinal scale puts subjects in order from highest to lowest, or from most to least. With respect to height, 5 students can be ranked from 1 to 5, the subject with rank 1 being the shortest. Though ordinal scales do indicate that some subjects are higher or better than others, they do not indicate how much higher or better. The intervals between the ranks are not equal.

Practice Exercises

1. Statistics is important for classroom teachers because it A. enables them to write appropriate objectives. B. helps them to construct good test items. C. helps them to evaluate students’ grades. D. is useful for promotion and certification. 2. Which of the following variables can be classified as unordered? A. Enrolment in a course B. Halls of residence C. Height of a student D. Scores in a quiz 3. A district director of education measures many variables on a sample of schools. An example of a variable measured in an ordinal scale is the A. enrolment of the classes in each school. B. income in cedis of the teachers. C. professional qualification of the teachers. D. years of service for each teacher.

4. Which of the following variables is best measured using an interval scale?

A. Favourite food

B. Musical ability

C. Religious affiliation

D. Year of birth

Use the information below to answer questions 5 – 6.

A study was conducted to see how well reading success in primary three could be predicted from various kinds of information obtained in kindergarten (reading readiness, age, gender, and socio-economic status).

5. Which of the variables represents a nominal scale?

A. Age

B. Gender

C. Reading readiness

D. Reading success

6. Which of the variables represents a ratio scale? A. Age

B. Gender

C. Reading success

D. Socio-economic status

7. Which of the following variables can be classified as continuous? A. Age of students in a psychology class.

B. Colour of dresses students wear to class.

C. Number of photocopy machines on a college campus.

D. Percentage of female students in the University of Cape Coast.

8. Which of the following variables can be classified as ordered? A. Parents occupation

B. Region of country

C. Religious affiliation

D. Statistics achievement

A. Ages of students in the EPS 211 class in UCC. B. Names of houses of residence in a Senior High School. C. Number of students in each class in a Senior High School. D. Points obtained by houses in inter-house athletics competition.

UNIT 2

DATA REPRESENTATION

Raw scores are often represented by graphics/pictures or tables. The representation of data in these forms enables more information to be derived from the scores. In educational statistics, more emphasis is placed on pictorial representations.

Nsowah-Nuamah (2005) gave 4 purposes of pictorial representations of data.

1. To make it easier to hold in mind the mass of figures they represent.
2. To show more clearly than tables any relations which may exist between figures.
3. To show degrees of difference between items.
4. To make a quick, lasting and accurate impression of relations, trends and comparisons. Pictorial representations of data should have the following properties.
5. They should be easier to understand than the figures they represent.
6. They should be simple and clear in their meaning.
7. They should carry all the information necessary to appreciate then fully.
8. They should be adequately labelled.
9. They should be titled.
10. They should have a key which may be included within or below the body of the chart.
11. They should have footnotes or source notes, when appropriate, and these should be given below the diagram. Bar graph/Charts

Data that are from nominal scales are represented in graphic form with the use of bar graphs. Bar graphs give a pictorial description of the data and emphasize how groups compare with one another. They are used to compare the sizes of the

various parts. The height of the bars is the basis for the comparisons and not the area of the bars.

Bar graphs are either column or horizontal. Column graphs are more popular in education. Column bar graphs are simple, compound (multiple) or component. Examples are shown below.

Table 1 Performance in Inter-House Athletics at BASS House Total Points

One 120

Two 100

Three 150

Four 60

Five 170

The figure below is a simple column bar graph showing performance in an Inter- House Athletics competition at BASS.

The figure below is a compound column bar graph showing school enrolment at Texas JSS by gender.

School Enrolment at Texas JSS by Gender

0

5

10

15

20

25

30

35

40

45

1A 1B 1C 2A 2B 2C Forms

Enrolment

Male

Female

Component Bar Chart

It is known as composite or stacked bar chart. It is used when a set of data combines to form a total. The total is the length/height of the bar. It allows for visual comparisons between different components ie how components contribute to the total of the category.

School enrolment at Texas JSS

0

10

20

30

40

50

60

70

1A 1B 1C 2A 2B 2C

Female Male

Constructing bar graphs/charts

1. Draw two axes, a vertical and horizontal. Label the vertical axis by the source of the values/scores e.g. enrolment, points etc. Label the horizontal axis by the names of the categories.
2. Divide vertical scale by points considering the lowest value and the highest value. Choose appropriate scales such that the bars are not too tall or too short and must start with zero.
3. Construct equally wide and equally spaced bars for each category with the height of the bar being the value/score for the category on the horizontal axis, which has the names of the categories as the label.
4. Where computer softwares such as Microsoft Excel and SPSS are not available, it is recommended that graph sheets be used.
5. Shade/colour the bars to differentiate bars and components.

Strengths and Limitations

1. Bar charts are easy to draw
2. Values can be read easily from the vertical axis.
3. Comparisons can be made easily and the significance of the information is easily grasped.
4. It cannot be effective with interval and ratio scales of measurement

Inter-House Athletics at BASS

One Two Three Four Five

Constructing pie charts

1. Calculate the degree equivalents for the value of each category/group by dividing the total point for each group by the overall total points and multiple the result by 360^0. For example, for House One above we have:

3600 720 600

(^120)   and for House Two, we have 3600 600 600

2. Use a pair of compass and protractor to draw the circle and the sectors based on the degrees calculated.
3. Shade/Colour the sectors to differentiate one from the other.

Strengths and limitations

1. Individual parts of the whole are seen and can be compared.
2. It provides a visual impression of the proportion that each part contributes to the overall total.
3. Angles are harder to compare.
4. They are not easy to draw especially where statistical softwares and computers are not available.

5. Data that are in continuous form and of ratio and interval scales are not appropriate.
6. The values of each component cannot be read from the chart but must be provided.
7. It is not useful where there are many parts as these parts become too small.
8. It only gives a visual impression but not the details of the data.

Uses

Pie charts can be used by teachers and educational practitioners for examination results by number of passes in various subjects, school enrolment by class, form or subjects.

Line Graphs

Data that are related to time are best used for line graphs. Time could be days, weeks, months and years. Line graphs show changes in the data over a period of time. Data from interval and ratio scales are most appropriate. Line graphs could be simple or compound. Simple line graphs give a pictorial description of the data. Compound line graphs compare group data over a period of time. Examples are shown below.

Table 4

Attendance at monthly teachers’ workshops

Month Total

January 120

February 85

March 100

April 150

May 90

Month

Attendance

Female Male

Jan 60 60

Feb 45 40

March 60 40

April 80 70

May 50 40

June 40 45

July 50 50

August 25 35

Sep 40 50

Oct 30 35

Nov 50 50

Dec 90 60

Compound line graph showing attendance at a monthly teachers’ workshop by gender.

Constructing line graphs

1. Draw two axes, a vertical and horizontal. Label the vertical axis by the source of the values/scores e.g. attendance, enrolment, points etc. Label the horizontal axis by the time period e.g. months, days, weeks etc.
2. Divide vertical scale by points considering the lowest value and the highest value. Choose appropriate scales such that the graph is not too tall or too flat and must start with zero.
3. Plot the value/quantity for each time period on the graph and join all the points by a straight line.
4. Where computer softwares such as Microsoft Excel and SPSS are not available, it is recommended that graph sheets be used.

Strengths and Limitations

1. Values can be read easily from the vertical axis.
2. Comparisons can be made easily and the significance of the information easily grasped.
3. They are less appropriate for nominal scale data.

Attendance at monthly teachers’ workshops by gender

0

10

20

30

40

50

60

70

80

90

100

Jan Feb Mar Apr May ay

Jun Jul Augt Sep Oct Nov Dec Month

Female Male

Attendance