Hickey
(1990) defined measure of central tendency as an index of central location in
the distribution. That is, they are potential scores which tend to balance all
other scores on other side. They are, therefore the measures of typical or
average performance. They tell us about the general level of performance.
Measures of central tendency explain how clustered data are around a central
point of distribution. The central tendency it look about three things that
are;-
i.
Mean
ii.
Median
iii.
Mode
I.
THE
MEAN
Is the arithmetic averaged of all
scores, and is the most important and most frequently used measure of central tendency.
(McCall, 1984)
OR
Mean is defined as the sum (total)
of all the scores divided by their number (Lewis, 1967).
It is mathematically mean
is represented by
which
read as “x bar”. In calculating mean there are two formulas
that can be used to find mean these are as follows;
·
.Population mean
·
Sample mean
And all of these
two formulas can be calculated both in
grouped and ungrouped data
The formula for calculating Population mean
for ungrouped data
µ=
Where by:-
µ=
Represents population mean
x=
represents individual scores
N= represents
number of scores
Represents
the last indefinite score in the distribution
Also
this formula can be shorted to read as;-
µ=
Where by ∑
(read as sigma or summation of ×)
represents sum or total of all scores.
Example
no. 1
The scores in EDU 210 in a senior
one class at Mwenge University are 7, 6, 4, 3, 5, 7, 4, and4.Find mean from the
following scores.
From the formula
µ=
Where by
µ=Population
sample
∑×=7+6+4+3+5+7+4+4=40
N=8
µ=
=5
Therefore the mean
will be 5 score
The
formula for calculating Sample mean for ungrouped data
From the formula
=
Whereby; -
=
Sample mean
∑×=summation of x
n==
represents number of scores for individual
Example
no.2
Find mean of the following scores, 3, 6, 4,2and5
0
= 20/5=4
= 4
Therefore
the mean is 4
The mean of grouped frequency distribution
Also for grouped frequency
distribution the mean can be calculated by considering;
·
The formula for population mean
·
The formula for sample mean
The approach to this is essentially
the same as that employed with ungrouped frequency distribution.
The
formula for population mean
Whereby;-
µ=Population
Mean
= summation
= midpoint of each group
= Number of scores
f=frequency
Example
no.1
The following scores were recorded in a
religious Education test at Ipinda secondary school. Compute the mean.
|
Score interval
|
f
|
|
|
|
20-24
|
4
|
22
|
88
|
|
15-19
|
8
|
17
|
136
|
|
10-14
|
10
|
12
|
120
|
|
5-9
|
6
|
7
|
42
|
|
0-4
|
2
|
2
|
4
|
|
Total
|
∑f=30
|
|
∑fx=390
|
|
|
|
|
|
Therefore
µ=
=
µ=
13
Therefore population mean will
be 13
THE
MEDIAN
The median refers to
the set, listing the number in ascending order and the selecting the values
that lies half-way along the list (Croft et al, 1997) or
Median refers to the
point that divides the distribution in two parts such that an equal number of
score or data values fall above and below that point (TIE, 1993).
These scores must be
routed in their order of magnitude. It is therefore a point which is most central
and divides the distribution into two equal halves such that they are as many
scores above or below. Median is easy to determine if the number of the scores
is odd. But if the number of scores
is even it is complex to find
median. Therefore in calculating median we have both grouped data and ungrouped
data.
Median for ungrouped
data.
Example
no.1
Find median of the
following number of scores, 12, 10, 8, 8, 5, 3, and 2.
Before calculating you
have to arrange the number in ascending
order or descending order
In this example it is
easily to find median because odd number are involved then median can be
determined by observing or inspection. Therefore the median from the above
equation is 8.
If odd numbers are
involved the median can be determined by observing or inspection.
However
the mathematical formulae for odd numbers is;-
Where by
N=
Number of scores
=
shows positions of the score
Example No, 2
Use
the following data to find median 12, 10, 8, 8, 5, 3 and 2.
From
Whereby
N=7
=
This
indicates that the median is the 4th
position. Therefore the score which is in the 4th position (either way) is 8.
Example no 3.
Find
median for the following number of scores.
12, 10, 8, 8, 5, 3.
This
score is for even scores, and then you have to take the numbers that obtained
at the center then divide by two.
8+8 =8
2
Therefore
the median will be 8
The
median for grouped data frequency.
Formulae
Median
= L +
Where
L=
exact lower limit of scores interval upon which the median falls
Size
of the score interval (number of score in an interval)
Cumulative frequency below the interval
containing the median
Frequency of the class interval containing the
median.
N=
number of scores in the distribution
Example
No, 3
|
Score
interval
|
|
|
|
20-24
|
4
|
30
|
|
15-19
|
8
|
26
|
|
10-14
|
10
|
18
|
|
5-9
|
6
|
8
|
|
0-4
|
2
|
2
|
|
Total
|
∑f=30
|
|
|
|
|
|
The
median class is the class interval containing the 30th scores out
with the help of cumulative frequency column. So out of 30 scores in an array
the 15th score is the median. By case the 15th score
falls within class (10-14)
Therefore
Median
class= (10-14)
L
= Lower limit (10-0.5) =9.5
=5
Cfb=8
Median = L +
Median=9.5 +
=9.5+
= 9.5 +3.5
= 13
THE
MODE
The
mode is defined as the scores with the highest frequency. OR
Mode
refers to the value that occurs most often (Croft et...al 1997).
Mode
refers to the most frequent score in a set of scores (Lenin and Gronlund,
2000).
So
mode is the value that occurs most frequently in a set of scores. Mode can be
one or more than two. If a distribution has two modes, it is known as Bimodal; also if a distribution has
three modes, it is known as trimodal
and those modes have more than two distributions are known as multimodal. Also if all scores occur
with equal frequency then there is no mode.
Mode can be in form of grouped or ungrouped data
Mode for ungrouped data.
Example
1:
Find mode of the following data.8, 6, 7, 5, 6, 5, 7, 8, and 5
The
mode is 5 because it occurs three times and such distribution is known as Unmodal (one mode).
Example
2; -
Find mode of the following data.2, 3, 2, 1, 3 and 5
The
mode here is 2 and 3 because two and three occur two times. And such
distribution is known as Bimodal.
Mode for grouped
distribution,
If the distribution is grouped, the mode
obtained by the formulae;-
Mode
=L +
Where
by
L=
exact lower limit of class interval containing the mode (modal class)
=absolute value of the difference between
frequency of modal class and pre- modal class.
=absolute value of difference between
frequency of modal class and post modal class.
=number of scores in class interval.
|
Score
interval
|
F
|
|
20-24
|
4
|
|
15-19
|
8
|
|
10-14
|
10
|
|
5-9
|
6
|
|
0-4
|
2
|
∑f=30
In
this case class interval 10-14 is the modal class because it occurs most times
(10 times)
Class
5-9 is pre- modal class
Class
15- 19 is post modal class.
Therefore
L=
10 – 0.5 = 9.5
= 10 – 6 = 4
= 10 – 8
= 5
Mode
= 9.5 +
Mode=
9.5 + 4/6 × 5
Mode
= 9.5 20/6
Mode = 12.8
INTERPRETATION
For
example 3, 4, 4, 4, 5, 6,7and 7
Mean from
the equation is 5
Median
is 4.5
Mode
is 4
The
interpretation of above scores
·
If the calculated mean is greater than
median it shows that the performance of the students was poor. Take the example
from the equation above the mean of above equation is 5 and the median is 4.5. Therefore this
indicates poor performance. That is to say those 4 students performed below the
mean
or average and indicate poor performance. In other side if mean
is less than median it indicates the performance of the students to be good
performance.
·
Mode used to explain the contents, and then if
calculated modes is less than mean value it indicates poor performance to the
students. In other side if the calculated mode is greater than the mean it
indicates good performance. Take the example above, the mean
from the question above is 5 and mode is 4 this indicates poor performance.
GRAPHICAL
INTERPRETATION
If
the mean
value is greater than median it indicates poor performance this is because the
value of mean
is greater than the median. In the graph the direction of the tail will shift
to right and the graph known as positive
skewed in the symmetrical curve. Take the example above the mean
is 5 and median is 4.5, therefore the calculated mean is greater than calculated median this
indicates poor performance
