STATISTICS IN ASSESSMENT OF
LEARNING
Meaning of Statistics:
1. Statistics is a branch of mathematics,
which deals with the collection, analysis and interpretation of data obtained
by conducting a survey or an experimental study.
2. Statistics can be defined as the
collection, presentation and interpretation of numerical data. ----- Croxton
& crowed.
Need and Importance of Statistics
in Education
Statistics, in general,
renders valuable services in the following dimensions:-
(a) In the collection, evidences or facts
(numerical or otherwise).
(b) In the classification organization and
summarization of numerical facts.
(c) In drawing general conclusions and
inferences or making predictions on the basis of particular facts and
evidences.
On account of the above
mentioned services statistics is now regarded as an indispensable instrument in
the field of Education, especially where any sort of measurement or evaluation
is involved. Its need in Education can be summarized as below:-
1. In the construction and standardization of various tests and measures.
Statistical methods helps in the construction and standardization of
various tests and measures like Achievement tests in various subjects.
Intelligence tests, Aptitude tests, Interest, Inventories, Attitude Scales and
various other measures of personality assessment.
2. In making proper use of the results of various tests and measures.
Scores obtained from various tests and
measures are always relative and not absolute. Hence they are meaningless in
themselves. Statistical methods help in their proper presentation, comparison
and interpretation. This services rendered by statistics help us:-
i)
To know
individual differences of our students,
ii)
To render
guidance to the students,
iii)
To compare
the suitability of one method or technique over the other,
iv)
To compare
the results of one system of evaluation with the other,
v)
To compare
the function and working of one institution with the other.
vi)
To make
prediction regarding the future progress of the students.
vii)
To make
selection, classification and promotion of the students.
viii)
To keep
various types of records and furnish Educational statistics etc.
By the above services of
statistics in Education, it can be easily concluded that it is very helpful to
the teachers in the proper functioning of their various duties. It gives a
definite direction to the process of teaching and learning and helps the
teacher to realise the broader aims and objectives of education. By its
increasing popularity in psychological as well as educational researches, it is
now going to be most indispensable to every teacher. Its knowledge not only
helps a teacher in acquainting with the new innovations and researches in the
field of education but also prepares him to become an active participant in
introducing as well as bringing changes in the field of Education.
Uses of Statistics in Education
1. It helps in collecting data either
numerically or otherwise.
2. It also helps in classification,
organization and summarization.
3. It also helps us in drawing general
conclusion and measurement.
4. Statistics also helps in task of
evaluation and measurement.
5. It helps in the construction and
standardization of test as well as using them properly.
6. It helps the teacher to know the
individual differences of the students, comparing the suitability of one method
or technique with another, making predictions for the future etc.
THE FREQUENCY DISTRIBUTION
Frequency distribution may
be considered as a method of presenting a collection of groups of scores in
such a way as to show the frequency in each group of scores or class.
Steps in preparation of frequency
distribution:
1. Finding the Range of the series. i.e.,
R=H-L
2. Determining the class interval.
i.e., i = Range/class desired.
3. Writing the contexts of the frequency
distribution.
i)
Classes of
the scores ii) Tallying the scores in to proper classes. iii) Checking the
table.
4. Preparing the table of frequency
distribution.
|
Class interval (C.I) |
Tallies |
Frequency (f) |
|
|
|
|
Problems:
1) Construct the frequency distribution
table for the following data/scores:
72,
75, 77, 67, 81, 68, 65, 86, 73, 67, 69, 82,
76,
76, 70, 83, 71, 63, 72, 72, 61, 84, 64, 67,
Ans:
R=H-L
R =86-61=25
R=25
i = Range/class desired.
= 25/5 =5
|
Class interval (C.I) |
Tallies |
Frequency (f) |
|
60- 64 |
III |
3 |
|
65 – 69 |
|
6 |
|
70 – 74 |
|
6 |
|
75 – 79 |
IIII |
4 |
|
80 – 84 |
IIII |
4 |
|
85 - 89 |
I |
1 |
N= ∕24
2) Construct the frequency distribution
table for the following data/scores:
Note: In the above example, the
total no. of scores=50,
Highest score=68
Lowest score=21
\Range=68-21=47
Hence, here 10 classes are sufficient.
\i=47¸10=4.7 (approx.5)
3) Construct the frequency distribution
table for the following data/scores:
31, 20, 14, 26, 36, 49, 28, 27, 25, 32,
43, 15,
40, 32, 28, 20, 24, 27, 27, 23, 31, 27,
32, 36,
24, 27, 32, 26, 45, 22, 12, 30, 29, 16,
50, 10,
25, 16, 24, 34, 42, 26, 20, 44, 18, 28,
26, 41,
35, 28, 26, 22, 33, 32, 26, 21, 25, 38,
44, 18,
4) 18, 13, 09, 24, 13, 16, 12, 10, 11, 22,
12, 21, 04, 09, 03, 14, 13, 17, 16, 18,
13, 09, 07, 04, 15
Advantages of Frequency Distribution
1. It makes data easier to interpret.
2. It gives one a fairly clear picture of
how students have performed. For example, the largest group has got the scores
between 25 and 29, while the majority of the group have scored between 20 and
34.
3. If Lakshmi’s score is 46, we can
estimate her relative position in the group and can express our qualitative
judgement about her position.
4. It increases the clarity of the form of
the distribution of scores.
Limitations of Frequency
Distribution
1. Some information is lost. For example,
the scores of 15, 16, 16, 18 and 18 have all been put in the interval 15 – 19
(14.5 – 19.5). Again one cannot say which exact measures have been included,
say, in the interval 30-34.
2. From the table, one does not know the
score of the 25th student in the series.
3. For further calculations, one has to
assume that either (i) the scores are uniformly distributed over the
theoretical limits of the interval, or (ii) all scores fall on the midpoint of
the interval. The degree to which such assumptions affect the accuracy of the graphs
and statistics computed from a class interval depends upon the size and number
of class intervals and the total frequency of the scores.
Limits of Class Intervals
There are various ways in
which the limits of class intervals are expressed are given below,
Method – A Method
– B Method
– C
40 – 50
40 – 49 39.5 –
49.5
50 – 60
50 - 59 49.5 –
59.5
60 – 70
60 – 69 59.5 –
69.5
............ ………… …………….
GRAPHIC REPRESENTATION OF DATA
GRAPH: The diagrammatical representation of numerical data
is called Graph.
Data may also be graphed in various ways. Viz. the histogram, the
frequency polygon, etc.
Advantages of Graphic Representation
1. It helps greatly in enabling us to
understand the data of frequency distributions.
2. It helps us in analyzing numerical data.
3. It helps us in comparing different
frequency distributions to each other.
4. It catches the eyes and holds the
attention which other statistical evidences fail to attract.
5. It helps us to dilute the abstractness
of ideas by translating numerical facts into a more concrete and understandable
form. It permits easy visualization.
The Histogram
It is the graph in which the
frequencies are represented by bars or columns. It appears as a series of bar
graphs placed one next to the other in a vertical array.
Example: Construct the Histogram from the following distribution table
|
Class Interval (C.I) |
Frequency (f) |
|
10- 14 |
3 |
|
15 - 19 |
5 |
|
20 - 24 |
9 |
|
25- 29 |
18 |
|
30 - 34 |
11 |
|
35 -39 |
5 |
|
40 - 44 |
6 |
|
45- 49 |
2 |
|
50 - 54 |
1 |
N =
60
___________________________________________
Steps:
1. Mark the class interval along x=axis.
2. Mark the frequency along y-axis.
3. Heights of the rectangle should be
proportional to the frequency.
4. Construct adjacent rectangle of the same
width.
Properties of the histogram:
1. Frequencies are along the vertical axis
and the scores (C.I) are along the horizontal axis.
2. One assumes that the scores are evenly
distributed within the class interval, thus giving us rectangular bars.
3. The frequencies within each interval of
a histogram are represented by a rectangle, the size of the interval being the
base and the frequency of that interval the height.
4. The area of each rectangle in a
histogram corresponds to the frequency within a given interval. While the total
area of a histogram corresponds to the total frequency (N) of the distribution.
Advantages:
1. It is simple and easily made.
2. All the five advantages of the graphic
representation are applicable here.
Limitations:
1. It is difficult to superimpose more than
one histogram on the same graph.
2. Comparisons of several frequency
distributions cannot readily be made via histograms. Frequency polygons are
much better suited for that purpose.
3. The assumption that the scores are
evenly distributed within the C.I. produces a larger error when N is small than
when N is large.
4. It cannot be smoothed.
The frequency Polygon
A polygon is a many angled
close figure. The frequency polygon is a graphic representation of frequency
distribution in which the mid points of the C.I are plotted against the
frequencies.
Example: Construct the Frequency polygon from the following frequency
distribution table.
|
Class Interval (C.I) |
Frequency (f) |
Mid-point
(x) |
|
10- 14 |
3 |
12 |
|
15 - 19 |
5 |
17 |
|
20 - 24 |
9 |
22 |
|
25- 29 |
18 |
27 |
|
30 - 34 |
11 |
32 |
|
35 -39 |
5 |
37 |
|
40 - 44 |
6 |
42 |
|
45- 49 |
2 |
47 |
|
50 - 54 |
1 |
52 |
N = 60
Properties of the frequency
polygon
1.
Frequencies
are along that vertical axis and the scores (representing the midpoints of C.I)
are along the horizontal axis.
2.
One assumes
that all scores within a class interval fall at the midpoint of that interval.
3.
The midpoint
of the class intervals just above and below the highest and lowest intervals
that contain the actual scores are also marked on the horizontal axis and given
a frequency of zero. This is done to close the figure.
Advantages
1. It is simple and easily made.
2. It is possible to superimpose more than
one frequency polygon on the same graph by using coloured lines, broken lines,
dotted lines, etc.
3. Comparisons of several frequency
distributions can readily be made via frequency polygons.
4. All the advantages of the graphic
representations are applicable here.
5. It can be smoothed.
Limitations
1. The part of the area lying above any
given interval cannot be taken as, proportional to the frequency of that C.I
owing to irregularities in the frequency surface.
2. The assumption that all the scores
within a C.I fall at the midpoint of the interval produces a larger error when
N is large than when N is small.
3. It is less precise than the histogram in
that it does not represent accurately, i.e.
in terms of area, the frequency upon each interval.
