Question

Find the mode of the following data:
(i) 10, 8, 4, 7, 8, 11, 15, 8, 6, 8.
(ii) 27, 23, 39, 18, 27, 21, 27, 27, 40, 36, 27.

Answer 1

Hint: We first define what mode is. Then we discuss how to order the given observations in a particular way to find the mode. We use a formula of mode for discrete data to find the solution. First, we calculate the number of appearances of a particular frequency and we take the one which has the most appearances. If there are more than one such frequencies then we take the one with the highest value.

Complete step-by-step solution:
The mode of a variable is defined as the value which has the highest frequency or frequency density of magnitude according to the variable which is discrete or continuous. It also has to be the greater or equal number of appearances than any other frequencies.
We first need to arrange the given observations in order, preferably in ascending order.
If the latter case happens then a unimodal distribution can be fitted with a smooth frequency curve. The mode will be the highest point of the curve.
There can be cases where it has more than one mode value. The mode will be undefined if all the values are equal.
Now for discrete data, we just take the highest value.
For our given first problem the given data is 10, 8, 4, 7, 8, 11, 15, 8, 6, 8.
There are ten given observations. These are all discrete data.
We arrange them in ascending order and get 4, 6, 7, 8, 8, 8, 8, 10, 11, 15.
The highest value is 15. But 8 appears four times which is greater than any other frequencies.
So, the mode is 8.
For our given second problem the given data is 27, 23, 39, 18, 27, 21, 27, 27, 40, 36, 27.
There are eleven given observations. These are all discrete data.
We arrange them in ascending order and get 18, 21, 23, 27, 27, 27, 27, 27, 36, 39, 40.
The highest value is 40. But 27 appears five times which is greater than any other frequencies. So, the mode is 27.

Note: We need to be careful about the data being grouped data or discrete data. As for different types of data, the formula will be different. We also find the curve for the grouped data situations. We also need to remember that the number of appearances of a frequency is the first and foremost thing to consider and then comes the value part.