The Mode of Ungrouped Data

The Mode

This is the final type of average, and the easiest one to work out… so long as you remember how! The mode is the number or value that appears most frequently in the data set.

How to work out the Mode:
Find the most common piece of data (number or letter) and this is your model
NOTE: You can have no modes or more than one mode, and you must write them all down!

For discrete numerical data, the mode is the most frequently occurring value in the data set.
For continuous numerical data, we cannot talk about a mode in this way because no two data Values will be exactly equal. Instead we talk about a modal class, which is the class or group that occurs most frequently.

Definition: Mode
The mode of a data set is the value that occurs most often in the set. The mode can also be described as the most frequent or most common value in the data set.

To calculate the mode, we simply count the number of times that each value appears in the data set and then find the value that appears most often.

A data set can have more than one mode if there is more than one value with the highest count. For example, both 2 and 3 are modes in the data set {1; 2; 2; 3; 3}. If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

Worked example: Finding the mode.
QUESTION
Find the mode of the data set {2; 2; 3; 4; 4; 4; 6; 6; 7; 8; 8; 10; 10}.
SOLUTION
Step 1: Count the number of times that each value appears in the data set


Step 2: Find the value that appears most often
From the table above we can see that 4 is the only value that appears 3 times. All the other values appear less than 3 times. Therefore the mode of the data set is 4.

One problem with using the mode as a measure of central tendency is that we can usually not compute the mode of a continuous data set. Since continuous values can lie anywhere on the real line, any particular value will almost never repeat. This means that the frequency of each value in the data set will be 1 and that there will be no mode. We will look at one way of addressing this problem in the section on grouping data.

Q1. In a Mathematics class, 23 learners completed a test out of 25 marks. Here is a list of their results:


Find the mode of this data.
solution:
The mode of the results of the test is 13 (13 appears 4 times).

Q2. Find the mode of the following data:
(i) 7, 9, 8, 7, 7, 6, 8, 10, 7 and 6
(ii) 9, 11, 8, 11, 16, 9, 11, 5, 3, 11, 17 and 8
solution:
(i) Mode =7 since 7 occurs 4 times.
(ii) Mode =11 since it occurs 4 times

Q3. The following table shows the frequency distribution of heights of 50 boys:


Find the mode of heights.
solution:
Mode is 122 cm because it occur maximum number of times. i.e. frequency is 18.

Q4. Find the median and mode for the set of numbers:

2, 2, 3, 5, 5, 5, 6, 8 and 9

solution:
Median =(9+1)/2=5th term which is 5.
Mode =5 because it occurs maximum number of times.

Let’s read the post ‘Examples of The Median for Ungrouped Data‘.

Q5. Calculate the mode of the following data set:

{10; 10; 10; 18; 7; 10; 3; 10; 7; 10; 7}

solution:
We first sort the data set: {3; 7; 7; 7; 10; 10; 10; 10; 10; 10; 18}. The mode is the value that occurs most often in the data set.
Therefore the mode is: 10.

Q6. The learners in Ndeme’s class have the following ages:

{5; 6; 7; 5; 4; 6; 6; 6; 7; 4}

Find the mode of their ages.
solution:
We first sort the data set: {4; 4; 5; 5; 6; 6; 6; 6; 7; 7}. The mode is the value that occurs most often in the data set. Therefore the mode is: 6.

Q7. Calculate the mode of the following data set:

{6; 10; 6; 6; 13; 12; 12; 7; 13; 6}

solution:
We first sort the data set: {6; 6; 6; 6; 7; 10; 12; 12; 13; 13}. The mode is the value that occurs most often in the data set.
Therefore the mode is: 6

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