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Measures of Central Tendency

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

  1. Arithmetic Mean
  2. Median
  3. 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.

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