Basic Measures of Central Tendency

1. Isha spends on an average of about 5 hours daily for her studies.

2. The average temperature at this time of the year is about 40 degree celsius.

3. The average age of pupils in my class is 12 years.

4. The average attendance of students in a school during its final examination was 98 per cent.

Many more of such statements could be there. Think about the statements given above.

Do you think that the child in the first statement studies exactly for 5 hours daily? Or, is the temperature of the given place during that particular time always 40 degrees? Or, is the age of each pupil in that class 12 years? Obviously not.

Then what do these statements tell you?

By average we understand that Isha, usually, studies for 5 hours. On some days, she may study for less number of hours and on the other days she may study longer.

Similarly, the average temperature of 40 degree celsius, means that, very often, the temperature at this time of the year is around 40 degree celsius. Sometimes, it may be less than 40 degree celsius and at other times, it may be more than 40°C.

Thus, we realise that average is a number that represents or shows the central tendency of a group of observations or data. Since average lies the highest and the lowest value of the given data so, we say average is a measure of the central tendency of the group of data. Different forms of data need different forms of representative or central value to describe it. **{.m-purple}One of these representative values is the Arithmetic .*

Arithmetic Mean

The most common representative value of a group of data is the arithmetic mean or the mean. To understand this in a better way, let us look at the following example:

Two vessels contain 20 litres and 60 litres of milk respectively.

What is the amount that each vessel would have, if both share the milk equally?

When we ask this question we are seeking the arithmetic mean.In the above case, the average or the arithmetic mean would be

Total quantity of milkNumber of vessels = 20+602 litres = litres.

Thus, each vessels would have 40 litres of milk.

The average or Arithmetic Mean (A.M.) or simply mean is defined as follows:

Mean = Sum of all observationsnumber of observations .

Ashish studies for 3 hours, 7 hours and 2 hours respectively on three consecutive days. How many hours does he study daily on an average?

Solution :

The average study time of Ashish would be

The day 1 is Hours and the day 2 is hours. Day 3 is hours.

Average =Total number of study hoursNumber of days for which he studied = 3+7+2 3 hours = hours per day.

Thus, we can say that Ashish studies for 4 hours daily on an average.

A batsman scored the following number of runs in six innings:36, 35, 50, 46, 60, 48 Calculate the mean runs scored by him in an inning.

Solution :

Total runs = 36 + 35 + 50 + 46 + 60 + 48 = .

To find the mean, we find the sum of all the observations and divide it by the number of observations.

Therefore, in this case, mean = 2756 =

Thus, the mean runs scored in an innings are 45.

Where does the arithmetic mean lie

Consider the data in the above examples and think on the following:

1. Is the mean bigger than each of the observations? NoYes

2. Is it smaller than each observation? NoYes

You will find that the mean lies in between the smallest and the greatest observations.

In particular, the mean of two numbers will always lie between the two numbers. For example the mean of 5 and 11 is (5 + 11)2= which lies between 5 and 11.

Mode

As we have said Mean is not the only measure of central tendency or the only form of representative value. For different requirements from a data, other measures of central tendencies are used.

Look at the following example.

To find out the weekly demand for different sizes of shirt, a shopkeeper kept records of sales of sizes 90 cm, 95 cm, 100 cm, 105 cm, 110 cm.

Following is the record for a week:

If he found the mean number of shirts sold, do you think that he would be able to decide which shirt sizes to keep in stock? Let's see.

Mean of total shirts sold = Total number of shirts soldNumber of different sizes of shirts = 1055 =

Should he obtain 21 shirts of each size? If he does so, will he be able to cater to the needs of the customers? NoYes

He will not be able to cater to customers who need shirts in sizes cm and cm if the same demand comes in next week also. So mean is not a good measure for this use case.

The shopkeeper, on looking at the record, decides to procure shirts of sizes 95 cm,100 cm, 105 cm. He decided to postpone the procurement of the shirts of other sizes because of their small number of buyers.

Look at another example

The owner of a readymade dress shop says, “The most popular size of dress I sell is the size 90 cm".

Observe that here also, the owner is concerned about the number of shirts of different sizes sold. She is however looking at the shirt size that is sold the most. This is another representative value for the data. The highest occuring event is the sale of size 90 cm. This representative value is called the mode of the data.

The mode of a set of observations is the observation that occurs most often.

Continue

Find the mode of the given set of numbers: 1, 1, 2, 4, 3, 2, 1, 2, 2, 4.

Solution : Arranging the numbers with same values together, we get.

= 1, 1, 1, , 2, 2, 2, , 4, 4

Mode of this data is 2 because it occurs more frequently than other observations.

Mode of Large Data

Putting the same observations together and counting them is not easy if the number of observations is large.

Ready for a quick game?

Click on start, enter the number which repeats the most(mode). You have to find the answer in 5 secs.

Start

1, 3, 2, 5, 1, 4, 6, 2, 5, 2, 2, 2, 4, 1, 2, 3, 1, 1, 2, 3, 2, 6, 4, 3, 2, 1, 1, 4, 2, 1, 5, 3, 3, 2, 3, 2, 4, 2, 1, 2

Median

We have seen that in some situations, arithmetic mean is an appropriate measure of central tendency whereas in some other situations, mode is the appropriate measure of central tendency.

Let us now look at another example.

Consider a group of 17 students with the following heights (in cm). Divide the class into two groups so that each group has equal number of students, one group has students with height lesser than or equal to a particular height and the other group has students with heights greater than or equal to the particular height.

How would you do that? Go ahead. Try. It's hard, right?

Let us see the various options you have:

(i) She can find the mean. The mean is.

Instruction

106 + 110 + 123 + 125 + 117 + 120 + 112 + 115 + 110 + 120 + 115 + 102 + 115 + 115 + 109 + 115 + 101 17 = 193017 = 113.5103

So, if the teacher divides the students into two groups on the basis of this mean height, such that one group has students of height less than the mean height and the other group has students with height more than the mean height.

Go ahead, divide the students into two groups on the basis of this mean height, such that one group has students of height less than the mean height and the other group has students with height more than the mean height, then the groups would be of unequal size.

They would have and members respectively.

(ii) The second option you have is to find mode. The observation with highest frequency is 115113 cm, which would be taken as mode. Go ahead and now re distribute the above based on mode.

There are children below the mode and children at the mode and above the mode. Therefore, we cannot divide the group into equal parts.

Let us therefore think of an alternative representative value or measure of central tendency.

For doing this we again look at the given heights (in cm) of students and arrange them in ascending order. We have the following observations:

101, 102, 106, 109, 110, 110, 112, 115, 115, 115, 115, 115, 117, 120, 120, 123, 125

The middle value in this data is because this value divides the students into two equal groups of 8 students each.

This value is called as Median. Median refers to the value which lies in the middle of the data (when arranged in an increasing or decreasing order) with half of the observations above it and the other half below it.

Here, we consider only those cases where number of observations is odd.

Thus, in a given data, arranged in ascending or descending order, the median gives us the middle observation.

Thus we realise that mean, mode and median are the numbers that are the representative values of a group of observations or data. They lie between the minimum and maximum values of the data. They are also called the measures of the central tendency.