3. MEASURE OF CENTRAL TENDENCY
The tendency of the observations to cluster
round some central value is known as
measure of central tendency.
A common word used for measure of
central tendency is ‘average’.
4. ARITHEMETIC MEAN
Arithmetic mean is the simplest but most useful
measure of central tendency.
It is nothing but the ‘average’, which we compute
in our high school arithmetic and therefore can
be easily defined as the sum of all values of the
item in a series divided by the number of items. It
is represented by the symbol ‘M’.
Arithmetic mean is sometimes referred to as’the
mean’ or ‘the average’.
5. CALCULATION OF MEAN IN THE CASE OF UNGROUPED DATA
Let X1,X2,……XN be the N observations of the
Sample. Their mean is defined as
M= (X1+X2+……+XN)/N=ΣX/N
Where Σx stands for the sum of values of the item
and N for the total number of items in a series
or group.
Example
i. Calculate the mean from the following data
3,5,10,7,8,12.
6. UNGROUPED FREQUENCY TABLE
Let the values of the variate be X1,X2,….XK and let
f1,f2,…fk be the number of times they occur or the
corresponding frequencies. Then the mean is
calculated by the formula
M=ΣfX/N
Where N is the total of all frequency
7. example
Find the mean from the following data
score 15 25 35 45 55 65
frequency 7 5 8 4 3 2
Score(x) Frequency(f) fx
15 7 105
25 5 125
35 8 280
45 4 180
55 3 165
65 2 130
N=29 Σfx=985
8. Grouped frequency table
In a grouped frequency table the individual
values of the observations falling in a class are
not known. so the mean can not be found out
without making some assumption regarding
the values of observations falling in each class.
The assumption that is usually made is that, all
the observations falling in a class have their
values equal to the midvalue of the class.so the
mean can be found out as in the case of the
ungrouped frequency table.
10. Shortcut method for mean
Mean for the grouped data can be computed
easily with the help of the following formula
M=A + c*Σfu/N
Where A-assumed mean,
c-class interval,
f-respective frequency of the midvalues of the
class interval,
N-total frequency,and
u=(x-A)/c
12. WHEN TO USE THE MEAN
1. Mean is the most reliable and accurate measure of central
tendency of a distribution in comparison to median and mode. It
has the greatest stability as there are less fluctuations in the mean
of sample drawn from the same population. Therefore , when
reliable and accurate measure of central tendency is needed, we
computed the mean for the given data.
2. Mean can be given an algebraic treatment and is better suited to
further arithmetical computation. Hence it can be easily employed
for he computation of various statistics like standard deviation,
coefficient of correlation,etc.Therefore,when we need to
compute more statistics like these, mean is compued for the
given data.
3. In computation of the mean we give equal weightage to every
item in the series.Therefore it is affected by the value of each item
in that series.
13. Merits
It is rigidly defined
It is easy to understand and simple to
calculate
It depends on the magnitude of all the
observations
It is capable for further algebraic treaatment
14. Demerits
The mean can be an impossible value. For
example, the AM of the number of students
per class in a school may turnout to be a
fraction.
If any observation is missing or its exact
magnitude is not known, the AM can not be
calculated even if its relative position is
known.