Median and Mode

Median:

When data are ordered in ascending/descending order of magnitude, the median is defined as the central value or middle item of the collection of data values (or it is occasionally the mean of the values of the two middle items). In the case of an ungrouped frequency distribution, the median is the middle value if a majority of the items are odd, assuming that the 'n' data values are organized in some order. The median is the mean of the two middle values when n is even.

Example 1:

Suppose we have the list of numbers as follows:
13, 17,24,7, 10,32,25,18 and 6
We have to first arrange the numbers in either ascending/descending order. These figures are arranged in an ascending order as follows: 6,7,10,13,17,18,24,25,32
Now as we can see that the series consists of odd number of items, to find out the value of the middle item, we use the formula Where,
Median=\(\frac{n+1}{2}\), Where n is the number of items.

In this case, n =9,

Therefore, median=\(\frac{9+1}{2}\)th item

=5th item.

= 17

Suppose the series consists of one more items 23. We may, therefore, have to include 23 in the above series at its proper place, that is, between 18 and 24 (here). Thus, the series is now 6, 7, 10, 13, 17, 18, and 23,24,25,32. Applying the above formula, the median is the size of the 5.5th item. Here, we have to take the average of the values of the 5th and 6th item. This means an average of 17 and 18, which gives the median as 17.5.
It may be noted that the formula \(\frac{n+1}{2}\) itself is not the formula for the median; it only gives the position of the median, namely, where the middle value lies. In the case of the even number of data items in the series, we have to find the two items whose values have to be averaged to calculate the median.

Median for grouped data:

The following formula is used to get the median in the case of a grouped series (where the numbers are grouped):

\( M_d = L + \frac{\frac{N}{2}- c.f.}{f} \times h \)

Where,

• L = Lower limit of the class where median lies.
• c.f. = Cumulative frequency of class preceding the median class
• f = Frequency of median class.
• h = Width of class-interval.

Example 2:

Calculate the median for the following data:

Soln: In this situation, we must first provide the table the cumulative frequency in order to calculate the median. As a result, the cumulative frequency table is shown as follows:

Now,

Median=\(\frac{n+1}{2}\)

=\(\frac{143+1}{2}\)

=\(\frac{144}{2}\)

=72

It means median lies in the class-interval Rs 1,200 - 1,400.
Again,

\( M_d = L + \frac{\frac{N}{2}- c.f.}{f} \times h \)

=\(1200+\frac{72-43}{30}\times 200\)

=Rs.1393.3

Characteristics Of Median:

• The array's median value corresponds to the array's midpoint.
• For computation, the value must be sorted and may be grouped.
• In the event that a distribution has extreme values, it is favored since it is unaffected by the extreme values.
• It is the most appropriate measure of central tendency for qualitative data, where the items are scored or rated rather than counted or measured.

Mode:

The value that appears in the data set the most frequently is called the mode. It is the value at the location where the things are concentrated most intensively. Consider the following sequence as an illustration:

Example 3:

17,9, 13, 15, 16, 12, 15,3, 7, 15, 37
There are ten observations in the series wherein the figure 15 occurs the maximum number of times three. Thus, the mode for the above case is 15.

Mode for the Grouped data:

\( Mode = L +\frac{f1-f0}{(f1-f0)+(f1-f2)}\times h \)

Where,

• L = The lower value of the class in which the mode lies
• f1 = The frequency of the class where the mode lies
• f0 = The frequency of the class preceding the class containing mode
• f2 = The frequency of the class succeeding the class containing mode
• h= The class-interval of the modal class

Example 4:

Let us take the following example:

Solution:

We can see from Column (2) of the table that the maximum frequency of 12 lies in the class-interval of 60-70. This suggests that the mode lies in this class interval. Applying the formula given earlier, we get:

\( Mode = 60 +\frac{12-8}{(12-8)+(12-9)}\times 10 \)

=65.7 approx.

Characteristics of mode:

• It is the value that occurs the most frequently in the distribution and the point with the highest density.
• There may be more than one mode in a distribution.
• It is the most typical value since it is the most often.

Reference:

• Kunda, Surinder. An Introduction to business statistics. n.d.
• statisticalforecasting.com/characterisitcs-mode-median-mean.php