In
mathematics, we have to come across a large set of data. To get valuable
information about this large set of data, we get help from measures of central
tendency. The measures of central tendency provide us with statistical
information about this set of information. Mean, median, mode and range are
known as four primary measures of central tendency. These four measures of
information provide us with individual information about the set of data. After
combining this information, we can easily know how these data points are
connected. The brief explanation about the mean, median, mode and range are
given below;
What Is Mean?
Mean is also
known as the average value of the given set of data. If you have a discrete set
of numbers, its central value is known as mean. Mean is denoted by ‘X̄’. If you
have X1, X2, ..., Xn numbers, mathematical formula to find mean of the data is
given below;
X̄ = X1+X2+…+Xn/n
X̄ = ΣX/n
Steps to Calculate the Mean
To calculate
the mean of a discrete set of numbers, you will have to follow the following
steps;
Ø First of all, you should determine
the numbers of the set. These numbers should consist of the real numbers only.
It means that these numbers should not be variables. You should also try to
arrange these numbers in the ascending order.
Ø Secondly, you should add these
numbers to find the sum. For this reason, you can use a calculator or you can
also add these numbers by hand.
Ø Thirdly, you should count all the
values. If a specific value is repeated in your set, you should count each
value to get the total.
Ø At last, you will have to divide the
sum of the values on the total number of values. In this way, you will get mean
of the set of discrete numbers.
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Types of Mean
Mean is
known as the most commonly used central tendency measure. Mean is further
divided into three types. These three types of the mean are explained below;
Ø
Arithmetic Mean
The
arithmetic mean is simply known as the average of the data because we can
easily obtain the arithmetic mean by dividing the sum of the values over the
total number of the values. The arithmetic mean is denoted by ‘X̄’. The formula
to calculate the arithmetic mean is given below;
X̄ = X1+X2+…+Xn/n
X̄ = ΣX/n
Ø
Example of Arithmetic Mean
Find the
arithmetic mean of the data 4, 10, 22 and 32.
A.M =
4+10+22+32/4
A.M = 68/4
A.M = 17
Ø
Weighted Arithmetic Mean
If we have
such data in which values don’t have equal importance, we have to provide
certain numerical values to provide them with relative importance. These
numerical values are known as weights. If X1, X2, ……, Xn have values and their
weights are W1, W2,….., Wn. We can find the weighted mean by using the
following formula;
X̄w = W1X1+ W2X2+…+ WnXn/ΣW
X̄w = ΣWX/ ΣW
Ø
Example of Weighted Mean
If a student
obtains marks 78, 98 and 96 out of 100 from English, Physics and Chemistry and
weights for these subjects are 2, 3 and 4 respectively, find its weighted mean.
X̄w = 78x2+98x3+96x4/9
X̄w = 156+294+384/9
X̄w = 834/9
X̄w = 92.67
Ø
Geometric Mean
Geometric
Mean is denoted by ‘G’. If we want to find the geometric mean of a set of n
positive values X1, X2, ……, Xn, first of all, we have to find their product.
After that, we have to take the nth root of their product. The formula to find
Geometric Mean is given below;
Ø
Example of Geometric Mean
Find
geometric mean of the set of the numbers 1,3 and 9.
Ø
Harmonic Mean
Harmonic
Mean is denoted by ‘H’. To find the harmonic mean of a set of n positive values
X1, X2, ……, Xn, we have to find the reciprocal mean of the reciprocal of the
values. The formula to find harmonic mean is given below;
Ø
Example of Harmonic Mean
Find the harmonic
mean of the values 7, 9, 11 and 12.
What is the Median?
The median
is a value in the data set which separates the higher half from the lower half.
Therefore, the middle value in a data set is known as the median. The most important
benefit of the median is that it provides us with a better idea about the
typical value. For example, if you want to get an idea about household income, you
can get its value in two ways. First, you can find it by mean but the problem
with the mean is that it can either provide you with the lowest value or the
highest value. The second way is to find it by the median. As median provides
us middle value, therefore, it is the best way to find the typical value of a
data set.
Steps to Find Median
Based on the
total numbers of values in a data set, we use two formulas to find the median
of the data set.
Ø
Finding the Median of the Odd Values
To find the
median of the odd values, you should follow the following steps;
Ø You should arrange the values in the
ascending order.
Ø You should separate the half values
from the left side and half values from the right side. For example, if you
have 7 values, you should separate three values from the left side and three
values from the right side.
Ø The middle value will be the median
of the data set.
Ø
Example to Find Median of the Odd Values
Find the median
of the data set 2, 3, 9, 7, 0, 6 and 5.
First of
all, we should arrange the values in the ascending order.
0, 2, 3, 5,
6, 7, 9.
As the total
number of values is 7. Therefore, we should separate three values from the
right side and three values from the left side.
0,
2, 3, 5, 6, 7, 9.
As ‘5’ is
the middle value. Therefore, ‘5’ is the median of the data set.
Ø
Finding the Median of the Even Values
To find the median
of the even numbers, you should follow the following steps;
Ø Like the odd values, you should
arrange the values in the ascending order.
Ø You should separate the half values
from the left side and half values from the right side by leaving two values in
the middle.
Ø By taking the mean of the two middle
values, we can get the median of the data set.
Ø
Example to Find Median of the Even Values
Find the
median of the data set 2, 7, 8, 9, 5, 0, 7 and 3.
By arranging
the values in the ascending order.
0, 2, 3, 5,
7, 7, 8, 9.
As there are
eight values in this data set, therefore, we should separate three values from
the right side and three values from the left side.
0,
2, 3, 5, 7, 7, 8, 9.
As 5 and 7
are in the middle. So, by taking mean of these two values, we get ‘6’.
Therefore, ‘6’ is the median of this data set.
What is Mode?
In a data
set, the value that appears more often is known as a mode. Like mean and
median, the mode is also an essential way to express some specific information
about a data set.
Rules to
Find Mode of a Data Set
Like mean
and median, there are also some rules to find the mode of the data. These rules
are explained below;
Ø First of all, you should arrange the
values in a data set in the ascending order.
Ø If one value is appearing more often
in the data set, this value is known as the mode of the data set.
Ø If more than one values are appearing
more often in the data set, these values are known as the mode of the data set.
It means that a data set can also have more than one mode.
Ø If no value is repeating in the data
set, this data set doesn’t have a mode.
Examples to Find Mode of the Data Set
Ø
Example 1
Find Mode of
the data set 2, 3, 4, 2, 6, 7, 8, 2 and 0.
0, 2, 2, 2, 3, 4, 6, 7, 8.
As ‘2’ is repeated
more often three times, therefore, ‘2’ is the mode of this data set.
Ø
Example 2
Find Mode of
the Data Set 2, 3, 4, 2, 3, 8, 9, 0 and 6.
0, 2, 2, 3, 3, 4, 6,
8, 9.
As ‘2’ and ‘3’
are repeated for two times, therefore’ ‘2’ and ‘3’ are the modes of the data
set.
Ø
Example 3
Find Mode of
the Data Set 1, 42, 26, 39, 10, 5, 6, 7 and 8.
1, 2, 5, 7,
8, 10, 26, 39, 42.
As no value
is repeated for more than one time, therefore, there is no mode of this data
set.
What is Range?
The
difference between the largest and the smallest values in a data set is known
as its range. The range of a data set provides us with specific value about the
data set because we have to subtract the smallest value from the largest value.
Steps to Find Range of a Data Set
To find out
the range of a data set, you should follow the following steps;
Ø First of all, you should arrange the
values of the data set in the ascending order.
Ø Secondly, you should mark the smallest
value in the data set.
Ø Thirdly, you should make the largest
value in the data set.
Ø At last, you should subtract the smallest
value from the largest value.
Ø The result of this subtraction is
known as the range of the data set.
Example to Find Range of the Data Set
Find the range
of the data set 32, 22, 38, 40, 50 and 12.
12, 22, 32,
38, 40, 50.
Smallest
value = 12
Largest
value = 50
Range = Largest
value – The smallest value
Range = 50 –
12
Range = 38
 |
Example to Find Mean, Median, Mode and Range of the Data Set
Find Mean,
Median, Mode and Range of the data set 3, 8, 9, 5, 12, 15, 6, 0, 3.
The
ascending form of the data set is given below;
0, 3, 3, 5,
6, 8, 9, 12, 15.
Mean
X̄ = ΣX/n
X̄ = 0+3+3+5+6+8+9+12+15/9
X̄ = 61/9
X̄ = 6.78
Median
0,
3, 3, 5, 6, 8, 9, 12, 15.
Median = 6
Mode
0, 3, 3, 5,
6, 8, 9, 12, 15.
As ‘3’ is
repeated more often, therefore, ‘3’ is known as the median of the data set.
Mode = 3
Range
Smallest
value = 0
Largest
value = 15
Range = 15 –
0
Range = 15.
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