Measures Of Central Tendency

Range

The difference between the highest and the lowest observation gives us an idea of the spread of the observations. This can be found by subtracting the lowest observation from the highest observation. We call the result the range of the observation. Look at the following example:

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

The median is the value of a data set when the numbers are arranged in ascending (small to large) or descending (large to small) order. It represents the central tendency of the data.

First, we arrange the data in order (ascending or descending) and count the total number of values (n).

Find the median position:

If n is , the median is the middle value.

Median position = n+12

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Median for frequency distribution

Find the median from the following frequency table detailing the reading habots of a classroom.

Total number of observations: n =

Find the median position: 202 =

The 10th value lies in the row where CF ≥ 10, which is at x = .

Thus, the median = 3 books.

Mean for ungrouped frequency distribution

The mean (or average) of a dataset is the sum of all values divided by the total number of values. When dealing with an ungrouped frequency distribution table, we use the weighted mean formula:

Mean(x) = ∑fx∑f

Where: x = Data values (observations)

f = Frequency of each value

∑fx = Sum of the products of each value and its frequency

∑f = Total frequency (total number of observations)

A survey records the number of family members in 16 households:

∑f = + + + + =

∑fx = + + + + =

x = ∑fx/∑f = 6016 =

Mean number of family members = 3.75

Mean for ungrouped frequency distribution (by Deviation Method)

Let’s consider a survey that records the number of hours students spend studying per day.

We will use the Deviation Method to find the mean.

Choose an Assumed Mean (A): A good choice for A is a middle value to simplify calculations. Let’s take = 3 (since it is centrally located).

Now, compute the Deviations i.e. d = x−A for each value meaning each value of x is subtracted from A = 3.

∑f = + + + + =

∑fd = + + + + =

x = A + ∑fd∑f = + 525255 = + =