 # Calculating the mode

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In a set of data, the mode is the most frequently observed data value. There may be no mode if no value appears more than any other. There may also be two modes (bimodal), three modes (trimodal), or four or more modes (multimodal). In the case of grouped frequency distributions, the modal class is the class with the largest frequency.

#### Mode = the most frequently observed data value

As a set of data can have more than one mode, the mode does not necessarily indicate the centre of a data set. The mode will be close to the mean and median if the data have a normal or near-normal distribution. In fact, if the distribution is symmetrical and unimodal, then the mean, the median and the mode may have the same value.

## Categorical or discrete variables

For categorical or discrete variables, the mode is simply the most observed value. To work out the mode, observations do not have to be placed in order, although for ease of calculation it is advisable to do so.

### Example 1 – Categorical or discrete variables

During a hockey tournament, Anne scored 7, 5, 0, 7, 8, 5, 5, 4, 1, and 5 points in 10 games. The mode of her data set is 5 because this value occurred the most often (four times). This can be interpreted to mean that if one game were selected at random, a good guess would be that Anne would score 5 points.

### Example 2 – Categorical or discrete variables

During Marco's 12-game basketball season, he scored 14, 14, 15, 16, 14, 16, 16, 18, 14, 16, 16 and 14 points. This data set is bimodal; there are two modes, 14 and 16, because both of them occur the most often (five times).

### Example 3 – Categorical or discrete variables

The following data set represents the number of touchdowns scored by Jerome in his high-school football season:

0, 0, 1, 0, 0, 2, 3, 1, 0, 1, 2, 3, 1, 0

First, put the data set in order:

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3

Find and compare the mean, median and mode.

#### Mode = most frequently observed data value = 0

The mode is 0 because this value occurs most often. If one game were selected at random, the mode tells us that a good guess would be that Jerome would not score a touchdown.

Mean = x ÷ n
= 14 ÷ 14
= 1

However, on average (mean), Jerome will score one touchdown per game even though the mode indicates he did not score a touchdown in a lot of games. In this case, the mode does not provide a useful measure of the player's performance.

Median = (n + 1) ÷ 2th value
= (14 + 1) ÷ 2
= 15 ÷ 2
= 7.5

Average = (value below median + value above median) ÷ 2
= (seventh value + eighth value) ÷ 2
= (1 + 1) ÷ 2
= 1

Because the number of values in the data set is even, the median does not fit perfectly in the centre of the data set. Instead, the median had to be found using the above equations. According to the results, the median states that Jerome will score one touchdown per game.

## Grouped variables (continuous or discrete)

When continuous or discrete variables are grouped in tables, the mode is defined as the class interval where most observations lie. This is called the modal-class interval.

In the example of the height of 50 Grade 10 girls, the modal-class interval would be 160 –< 165 cm, as this interval has the most observations in it.

The mode is rarely used as a measure of central tendency for numeric variables. However, for categorical variables, the mode is more useful because the mean and median do not make sense.

Next, you could determine the midrange of the modal class. The midrange is simply the midpoint between the highest and lowest values in a class. The mode is not used very often in conjunction with the midrange because it gives only a very poor estimate of the average.

The mode can be used with categorical data, but the mean and median cannot. The mode may or may not exist, and there may be more than one value for the mode.

## Summary

Circumstances generally dictate which measure of central tendency—mean, median or mode—is the most appropriate. If you are interested in a total, the mean tends to be the most meaningful measure of central tendency because it is the total divided by the number of data. For example, the mean income of the individuals in a family tells you how much each family member can spend on life's necessities. The median measure is good for finding the central value and the mode is used to describe the most typical case.