Let’s say I walk into a class of fifth graders, and start asking them their ages one by one and note them down. There are 20 students in the class and this is what my list looks like
Roll. No.
Age
1. 10
2. 10
3. 10
4. 9
5. 10
6. 10
7. 9
8. 11
9. 10
10. 10
11. 12
12. 10
13. 9
14.
10
15. 10
16. 11
17. 10
18. 10
19. 9
20. 10
As I look at their ages, do I see something common or worth noting. More specifically do I find a number around which most of their ages tend to concentrate?
Yes, I find that most of the fifth graders are 10
years old, and, that single number tells the value around which most of the ages
in that class tend to move around. Even though, not all of the 20 students have
the same age, yet I can say that my average fifth grader is a 10-year-old. This
is what a measure for central tendency does, it tries to describe the whole set
of data with a single value in such a way that that single value represents the
middle or central position in that dataset.
Definition of Measure of Central Tendency: Measure of central tendency is a
statistical measure which gives the summary of the data by a single value. It
tries to describe the whole set of data with a single value in such a way that
that single value represents the middle or central position in that dataset.
There are three main measures of central tendency
- Arithmetic Mean
- Median
- Mode
Which measure out of the three methods to choose
depends upon the type of data you have and how you want to finally use and
interpret the value.
1. Mean :
It is one of the most popular and commonly used measure of central tendency. It is also known as Arithmetic Average, and it is calculated by adding up the value each observation in the data set and dividing this sum by the total number of observations.
So, if you have a set of
5 observations say 56, 58, 65, 59, 62, its mean is found out by adding these five
values (56+58+65+59+62) and dividing by the total number of observations i.e 5.
So, the mean = (56+58+65+59+62)/5
= 300/5
=60
In our example
above the total ages in the class of 20 students comes out to be 200. Thus the
mean is the total of values divided by the number of students which is 20 here.
Mean = 200/20
= 10.
Some Important Points Regarding Mean
- The mean may not necessarily be one of the values in our dataset. In our example 56, 58, 65, 59, 62, the mean comes out to be 60, which is a value lying between the dataset but the value is not one from the data.
- The value of the mean gets influenced by the presence of outliners. An outliner is any value which is either extremely high or low relative to our given data. So say in our example, a pupil aged 65 decides to enroll in fifth grade in this case. Now, the sum of ages comes out to be 265 and when we divide it by the number of students 11 now, the mean comes out to be 255/11, which is 24.09. The outliner has significantly changed the value of our average age to 24. By merely looking at the data we know that none of the 20 students has an age over 12 and one outliner of 60 is representing a different picture altogether.
2. Median
Median is that value which lies exactly in the middle
of the distribution, when the values are arranged in either ascending or
descending order.
So, by definition the median divides a series into
half such that 50 percent of the values lie on each side of the median value.
Now, if a series has odd number of values, the median is the middle value. For
an even series, we get two middle values. So, the median in this case would be
the mean of these two middle values.
Odd Series Example
Lets say we have the score of 7 students in an exam
|
Marks |
|
56 |
|
70 |
|
49 |
|
82 |
|
53 |
|
59 |
|
61 |
To find median,
the first step is to arrange the series in either ascending or descending
order.
|
Marks |
|
49 |
|
53 |
|
56 |
|
59 |
|
71 |
|
70 |
|
82 |
In the above series, 59 lies in the middle of the
series. So, the median is 59 dividing the series in two halves, 50 percent of
values being less than 56 and 50 percent values more than 56.
Even Series Example
In case the series has even number of values, we will
get two middle terms. In this case, the average of the two values gives us the
median value.
|
Marks |
|
56 |
|
70 |
|
49 |
|
82 |
|
53 |
|
59 |
|
61 |
|
58 |
Arranging the series in ascending order, we get the
following series
|
Marks |
|
49 |
|
53 |
|
56 |
|
58 |
|
59 |
|
71 |
|
70 |
|
82 |
In the above series, the values 58 and 59 both lie in
the middle. So, their average values gives the median
=58+59/2
=58.5
Some Important Points of Median
- Median is not influenced by the presence of extreme values, known as outliners. So, if we have a couple of extreme values, the median value will not change unlike mean and present a more accurate picture of the data.
- All the values are not considered while calculating median.
3. Mode
Mode is that value which comes the most number of
times in a distribution. Or it is the most commonly occurring value in a
series. For example, in our example of a class of fifth graders, 10 years is the value which is coming the
most number of times i.e 13 times out of a total of 20 values.
Some Important Points of Mode
- Sometimes, a series may have more than 1 mode i.e 2 (bi-modal series) or 3 modes
- There may also be a case where a series has all different values, in this case the mode doesn’t exist, and we have to use a mean or median to find the measure of central tendency.