Mode

The Concept of Mode

The mode is the value that appears most frequently in a set of data. It is another type of measure of central tendency, just like the mean and the median.

A set of data can have:

1. One mode – called unimodal
2. Two modes – called bimodal
3. More than two modes – called multimodal
4. No mode – if all values occur with the same frequency

Why mode is useful:

• In a shoe shop: You may want to know the most common shoe size.
• In a restaurant: You may want to know the most ordered dish.

Example 1

Find the mode in the following sets of numbers:

a) 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5

b) 58, 57, 60, 59, 50, 56, 62

c) 5, 10, 10, 10, 15, 15, 20, 20, 20, 25

Solution:

a) 1 appears 3 times → Mode is 1

b) All values appear only once → No mode

c) Both 10 and 20 appear 3 times each → Modes are 10 and 20 (Bimodal)

Mode from Grouped Data (Using Frequency Table or Histogram)

When data is grouped into classes, we no longer have individual values. Instead of a single number as the mode, we use a modal class — the class with the highest frequency.

Example 2

The examination results for a group of students are:

Mark (%)	30–39	40–49	50–59	60–69	70–79
Frequency	5	3	20	2	10

(a) Draw a histogram

Table for histogram:

Class Interval	Frequency	Class Boundaries
30–39	5	29.5–39.5
40–49	3	39.5–49.5
50–59	20	49.5–59.5
60–69	2	59.5–69.5
70–79	10	69.5–79.5

Graph is to be placed here.

(b) State the modal class

The highest frequency is 20 → Modal class is 50–59

Estimating the Mode from a Histogram

Method 1: Using drawing

1. Draw the histogram.
2. Identify the modal class (the tallest bar).
3. Draw diagonal lines from:
   • Top left of modal class to top left of the next right class.
   • Top right of modal class to top right of the next left class.
4. The point where the lines intersect gives the mode on the horizontal axis.

Graph is to be placed here.

Method 2: By Calculation

Use the following formula:

Estimated Mode = L + [(fM − fL) / (2fM − fL − fR)] × W

Where:

• L = Lower class boundary of the modal class
• fM = Frequency of the modal class
• fL = Frequency of the class before the modal class
• fR = Frequency of the class after the modal class
• W = Class width

For Example 2:

Modal class = 50–59 → L = 49.5

fM = 20, fL = 3, fR = 2

W = 10

Estimated Mode = 49.5 + [(20 − 3) / (2×20 − 3 − 2)] × 10 = 49.5 + (17 / 35) × 10 = 49.5 + 4.86 = 54.36 (approx)