Measures of Central Tendency: Mean, Median, Mode, examples, definition, formulas
Measures of central tendency is an important measure in descriptive statistics. Mean, median, mode in descriptive statistics play a crucial role for data analysis and research. We will learn what is central tendency in statistics and why is matters, definition of central tendency, measures of central tendency, how to calculate mean median mode, mean median mode formulas and examples of central tendency, advantages and disadvantages of mean median mode, Real Life Examples of Central Tendency.
Measures of Central Tendency: Quantitative data in the mass exhibits the tendency to concentrate at certain values , usually somewhere in the center of the distribution. measures of this tendency are called measures of central tendency or averages.
Central Tendency Definition: According to Bowley, averages are "statistical
constants which enables us to comprehend in a single effort the significance of
the whole"
Why central tendency?
Measures of central tendency give us an idea about the concentration of
the values in the central part of the distribution and hence name measures
of central tendency.
How to Calculate Measures of Central Tendency: following are the five measures of central
tendency that are in common use:
1) Arithmetic Mean or Simple Mean
2) Median
3) Mode
4) Geometric Mean
5) Harmonic Mean
Let’s us now understand Definition and
examples of Mean Median Mode, How to Calculate Mean Median and Mode and Advantages
and Disadvantages of Mean Median Mode.
Mean is an important measure of central
tendency. It is also known as ‘ averages’.
There are three types of mean:
1.
Arithmetic
mean
2.
Geometric
mean
3. Harmonic mean
d Definition of Arithmetic mean: arithmetic mean of a set of observations is their sum divided by the number of observations . Arithmetic mean is simply called mean.
Mean Formula
For the mean (or average) of 𝑛 variables x1,x2,...xn :
In case of frequency distribution:
For xi I fi
where
i=1,2,…,n
xi=
variable
fi= frequency of the variable xi
In case of grouped or continuous frequency distribution:
x Is taken as the mid- value of the
corresponding class.
STEP DEVIATION METHODS OF MEAN
If the values of x and/or f are large,
calculation of mean by above formula is quite time consuming. The arithmetic is
reduced to a great extent by taking deviations of the given values from any
arbitrary point ‘A’ .
Summing both sides over I from 1 to n, we get
then mean :
In case of grouped or continuous frequency distribution, the arithmetic mean is reduced to still greater extent by taking di= (xi-A)/h
Where A = an arbitrary point
h
= the common magnitude of class interval
then we have h*di=xi-A
formula :
|
Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
|
No. of students |
12 |
18 |
27 |
20 |
17 |
6 |
Solution:
|
Marks |
No. of students (f) |
Mid- points (x) |
fx |
|
0-10 |
12 |
5 |
60 |
|
10-20 |
18 |
15 |
270 |
|
20-30 |
27 |
25 |
675 |
|
30-40 |
20 |
35 |
700 |
|
40-50 |
17 |
45 |
765 |
|
50-60 |
6 |
55 |
330 |
|
Total |
100 |
|
2800 |
Advantages and Disadvantages of Mean
Advantages :
1. It is rigidly defined.
2. it is
easy to understand and easy to calculate.
3. It is based upon all the observations.
4. It is amenable to arithmetic treatment.
5. It is stable average as it is affected least
by fluctuations of sampling.
Disadvantages :
1.
It cannot
be determined by inspection, nor it can be located graphically.
2.
It can not
be used if we are dealing with qualitative characteristics, which cannot be
measured quantitatively; such as honesty, beauty.
3.
It cannot
be obtained if a single observation is mission or lost unless we drop it out.
4.
It is affected
very much by extreme values.
5.
Arithmetic mean
cannot be calculated if the extreme class is open, e.g. below 10 or above 90.
6.
In extremely
asymmetric distribution, usually arithmetic mean is not a suitable measure of
location.
Median: Definition, Examples, formulas, Advantages
and Disadvantages
Median definition:
Median is
the value of the variable which divides it into two equal parts. It is the value
which exceeds and is exceeded by the same number of observations. Such that number
of observations above it is equal to the number of observations below it. Median
is thus a positional average.
Median formula:
In case of ungrouped data:
If number of observations is odd then median is the
middle value after the values have been arranged in ascending or descending
order of magnitude.
If number of observations is even then , there are
two middle terms and median is obtained by taking the arithmetic mean of middle
terms.
In case of discrete frequency distribution:
Median is obtained by considering the cumulative
frequencies. The steps for calculating median are given below:
i)
Find N/2 ,
where N=∑fi
ii)
See the
less then cumulative frequency (c.f.) just greater than N/2.
iii)
The corresponding
value of x is median.
In case of continuous frequency distribution:
In case of continuous frequency distribution,
the class corresponding to the c.f. just greater than N/2 is called the median
class and the value of median is obtained by the following formula:
f
= frequency of the median class
h
= magnitude of the median class
c
= c.f. of the class preceding the median
class.
N
= ∑f
Advantages and Disadvantages of Median:
Advantages:
1.
It is rigidly
defined.
2.
It is
easily understood and is easy to calculate. In some cases it can be located
merely by inspection.
3.
It is not at all affected by extreme values.
4.
It can be
calculated for distribution with open-end classes.
Disadvantages:
1.
In case of even
number of observations, median cannot be determined exactly. We merely estimate
it by taking the mean of two middle terms.
2.
Median is
not based on all the observations. This property is sometimes described by saying
that median is insensitive.
3.
It is not
amenable to algebraic treatment.
4.
As compared
with mean, it is affected much by fluctuations of sampling.
Mode: Definition, Examples, formulas, Advantages
and Disadvantages
Mode definition:
Mode is the value which occurs most frequently in a
set of observations and around which the other items of the set of cluster
densely. In other words , mode is the value of the variable which is predominant
in the series.
In case of discrete frequency distribution:
Mode is the value of x corresponding to maximum
frequency.
Example:
|
x |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
f |
4 |
9 |
16 |
25 |
22 |
15 |
7 |
3 |
Value of x corresponding to the maximum frequency ,
viz, 25 is 4 .
Hence, mode is 4.
In case of continuous frequency distribution:
In case of continuous frequency distribution,
mode is given by the formula:
h
= magnitude
f1
= frequency of modal class
f0
= frequency of the class preceding modal class
f2
= frequency of the class succeeding the modal class.
Advantages and Disadvantages of Mode:
Advantages:
1.
Mode is readily
comprehensive and easy to calculate. Like median, mode can also be located in
some cases merely by inspection.
2.
Mode is not
at all affected by extreme values.
3.
Mode can be
located even if frequency distribution has unequal class interval. Open end
classes also do not pose any problem in the location of mode.
Disadvantages:
1.
Mode is ill-defined.
It is not always possible to find a clearly defined mode.
2.
It is not
based upon all the observations.
3.
It is not
capable of further mathematical treatment.
4.
As compared
with mean, mode is affected to a greater extent, by fluctuations of sampling.
Real Life Examples of Central Tendency: if a company wants to produce shoes for particular market. if company want profit in the business then company has to produce the shoes of that size which has more demand in market. so that they can fulfill the demand and make profits . in this case , company has to collect data and check which size of shoes is in demand. this can be measures by measures of central tendency. like the size of shoes is in huge demand which is shown by mode of data. then company will get profit by producing that size of shoes in more quantity to meet the supply and demand conditions.
