Mean
The mean is the most common measure of
central tendency used by the researchers. It is the measure of central tendency
that is also referred to as the average. A researcher can use the mean to describe
the data distribution of variables measured as intervals or ratios. These
are variables that include numerically corresponding categories or ranges like race, class, gender, etc.
How to Calculate the Mean
1.
If the numbers of observations are less, i.e., less
than 25, then we simply add all the observations and divide the sum by the
total number of observations.
For example, if five
families have 0, 2, 2, 3 and 5 children respectively, then the mean number of
children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five
families have an average of 2.4 children. We can use this formula to calculate
the mean.
Here, ∑xi = Sum of all the
individual values and n= Total number of items.
2.
If the numbers of observations
are more, i.e., more than 25, then we find the frequency (f) of
distinct observations (x) and multiply them to find fx. Now,
add all the values of fx to find ∑fixi. Again, add all the frequencies to get ∑fi and use this formula to
calculate the mean.
Example 1: The height of 10 boys
is measured in centimeters, and the observations are as follows:
150, 139, 128, 152, 132, 146, 149, 141, 143, 135
Find the mean height.
Solution:
Mean height = Sum of
all observations/Number of observations
=
(128 + 132 + 135 + 139 + 141 + 143 + 146 + 149 + 150 + 152)/10
=
1415/10 cm = 141.5 cm
Thus, the mean height of the girls is 141.5 cm.
Example 2: The mean of four numbers 23, x,
17 and
29 is 20.5. Find the value of x.
Solution:
Mean = (23 + x + 17 + 29)/4
20.5 = (69 + x)/4
69 + x = 82
x = 82 – 69
x = 13Median
The median is the value that occurs at
the middle of a distribution of data when the data are organized in the
ascending order. This measure of central tendency can be calculated for
variables that are measured with ordinal, interval or ratio scales.
Calculation of the median is also very
simple. Suppose we have the following list of numbers: 4, 9, 12, 41, 3, 67,
31, 7, 23. First, we must arrange the numbers in ascending order. The result is
this: 3, 4, 7, 9, 12, 23, 31, 41, 67. The median is 12 because it is the exact
middle number. There are four numbers before 12 and four numbers after 12.
If your data distribution has an even
number of data values which means that there is no exact middle number, you simply
adjust the data range slightly in order to calculate the median. For
example, if we add the number 82 to the end of our list of numbers above,
we have 3, 4, 7, 9, 12, 23, 31, 41, 67, 82, so there is no single middle
number. In this case, take the average of the values for the two middle
numbers. In our new list, the two middle numbers are 12 and 23. So, we take the
average of those two numbers: (12 + 23) /2 = 17.5.
How to Calculate Median
In individual series, where data is given in
the raw form, the first step towards median calculation is to arrange the data
in ascending or descending order. Now count the number of observations denoted
by n. The next step is decided by whether the
value of n is even or odd.
1.
If the value of n is odd,
then
2. If the value of n is even, then
Example 1: Find
the median of the following.
24, 22,
18, 23, 26, 17, 28
Solution:
Arranging
the data in ascending order: 17, 18, 22, 23, 24, 26, 28
Here n =
7, which is odd.
Median =
Value of [(7 + 1)/2]th observation
=
Value of 4th observation
Hence,
median = 23
Example 2: Find the median of the following.
38, 42, 41, 40, 35, 37, 45, 36
Solution:
Arranging
the data in ascending order: 35, 36, 37, 38, 40, 41, 42, 45
Here n
= 8, which
is even.
Median = Mean
of (n/2)th and (n/2 +1)th observation
Median =
Mean of 4th and 5th observation
Median = (38
+ 40)/2 = 39
Mode
The mode is the measure of central
tendency that identifies the category or data value that occurs the most
frequently within the distribution of data. In other words, it is the most
common value or the score that appears the highest number of times in a
distribution.
For example, suppose 100 families own
the following pets:
- Dog: 80
- Cat: 25
- Duck:
11
- Parrot: 5
- Fish:
14
Here, the mode is "dog"
since more families own a dog than any other animal. Note that the mode is
always expressed as the category or score, not the frequency of that score. In the
above example, the mode is "dog" not 80, which is the number of times
dog appears.
How to Calculate Mode
In case of raw data, we just have to count
the data items and the item that occurs most frequently in the distribution is
the mode of the series.
Example
1: Find the mode of the following data.
36, 48, 36,
44, 36, 42, 36, 48, 36, 40, 25, 42, 36, 44, 40
Solution:
The observation 36 has maximum
frequency, i.e. 6.
Thus, mode =
36
Example 2: Find the mode of the following data.
14, 15, 13, 15, 16, 15, 13, 15, 16, 15, 13, 15, 16, 15, 13
Solution:
The
observation 15 has maximum frequency, i.e. 7.
Thus, mode =
15