In this explainer, we will learn how to find and interpret the mode of a data set.

The mode is an example of a measure of center (or measure of central tendency). If we have a set of numerical data, we would like to find a single number that can represent the whole data set or at least give us some information about typical values in the data set. There are a number of ways to describe typical values. For example, one way to describe a typical value is to see what value is in the middle: this is the median. The mean, which is the average of all the values in a data set, is another example. We could also describe a typical value by looking at the value that occurs most frequently: this is the mode.

The mode, mean, and median are all different examples of measures of center. We will only discuss the mode here.

### Definition: The Mode

The mode of a set of data is the value that appears most often.

Let us look at an example of how to calculate the mode.

### Example 1: Finding the Mode of a Data Set by Ordering

The following data points represent the number of goals scored by a player in 10 consecutive games:

What is the mode score?

### Answer

We have to find the most common value from this data set:

We can do this by writing the data values in order to see which one occurs the most.

The data value 2 appears the most times. Therefore, the mode is 2.

In the above example, it was easy to see which value was most common just by ordering the data values from least to greatest. When we have larger sets of data, however, this may not be a sensible method. Instead, we can use a tally chart or a frequency table to help us. We will do this in the next example.

### Example 2: Finding the Mode of a Data Set by Making a Frequency Table

The table shows the average temperature of a city on each day in April 2016. Find the mode of the temperatures.

24 | 23 | 29 | 29 | 26 | 26 | 23 | 28 | 24 | 26 |

29 | 26 | 26 | 24 | 24 | 23 | 23 | 29 | 27 | 26 |

29 | 24 | 24 | 25 | 29 | 29 | 27 | 28 | 25 | 25 |

### Answer

In this set of data, the data values are the temperatures. The mode is the most common temperature.

To find the mode, we need to count how many times each data value appears. We can use a tally chart or a frequency table to record the results.

Data Value (Temprature) | Tally | Frequency |
---|---|---|

23 | 4 | |

24 | 6 | |

25 | 3 | |

26 | 6 | |

27 | 2 | |

28 | 2 | |

29 | 7 |

We can see that the data value 29 appeared seven times, which is more times than any other data value. Therefore, the mode of the temperatures is 29.

Now that we have seen some examples of how to find the mode of a set of data, let us summarize what we have learned.

### How To: Finding the Mode of a Data Set

To find the data value which appears most often, we can use the following different methods:

- We can write the data values (if they are numbers) in order from least to greatest and
count how often each value appears.

For example, to find the mode of the data set , we could write the numbers in order.

The number 8 appears the most times.

The mode is 8. - We can draw a tally chart and a frequency table to record the number of times each
value occurs.

For example, to find the mode of the data set , we could draw the following tally chart and frequency table. - We can make a bar chart to represent the number of times each value occurs. The mode will be the data value with the tallest bar.

It is important to realize that the mode does not have to be a number. The mode is the only measure of central tendency for which this can be the case. We can have data sets where the data values are not numbers, as in the following example.

### Example 3: Finding the Mode of a Nonnumerical Data Set

A bookstore sold 11 books by Haruki Murakami, 2 books by Henry Thoreau, and 6 books by Carl Jung. Determine the mode for this data.

### Answer

The data values are the authors of the books.

If we were to list the data set, we would have a list of the authors of the books that the bookstore sold. If it helps, you can make a tally chart for the data.

The mode is the most common data value; this is the author who sold the most books.

Therefore, the mode is Haruki Murakami.

Note that some data sets will have more than one mode. For example, the data set

has two modes: 3 and 9. This is because the data values 3 and 9 each appear four times, whereas 1 appears only twice and 7 appears just once.

In addition, some data sets will have no mode. For example, the data set has no mode because each of the different data values appears just once. Hence, there is no data value that appears more often than any other.

The fact that the mode is the most common value can help us to solve problems where there is a missing data value. Here is an example of this type.

### Example 4: Finding an Unknown Value in a Data Set given Its Mode

Rania has the following data: .

If the mode is 6, find the value of .

### Answer

The mode of the data set is the value that appears most often. Without the value , the data set is and all of the data values appear once.

Since we know that when we add the value into the data set, the mode is 6, this means that the most common value is 6. Hence, 6 must appear more than once in this data set:

Therefore, .

We note that in the previous example, if we had chosen to be any value not already in the data set (for example, ), then there would have been five different data values, with each appearing just once. However, as we have stated, if every value in the data set appears the same amount of times, then the data set has no mode. Therefore, is the only option.

Next, we look at an example that shows how the mode can be affected by changes in a data set.

### Example 5: Recognizing How Changing a Data Set Affects the Mode

The line plot shows the number of people who subscribed to an online course last June. The deadline for subscription was the 20th of June. How would the mode of the subscription dates be affected if the administration office forgot to record one person who subscribed on the 14th?

### Answer

Recall that the mode of a data set is the value that appears most often. Here, we need to find the mode of the data shown in the line plot. Then, we must work out if the mode changes when one of the data values is removed.

The data values are the subscription dates. The mode is the most common subscription date, which is represented by the tallest bar in the line plot. We can see that this is the 13th of June, as on that day five people subscribed, which is more than on any other day.

Now, consider what happens if the administration office forgot to record one person who subscribed on the 14th. Since the original line plot shows only one person on the 14th, this data value would be reduced from one to zero. However, this does not affect the mode because the tallest bar remains the same.

We conclude that the mode of the subscription dates would remain unchanged at the 13th of June.

In our final example, we look at how we can work out the mode by analyzing data presented in a frequency table.

### Example 6: Calculating the Mode given a Data Set Presented in a Frequency Table

The table shows the number of goals scored per game by a team in a football season. What is the mode number of goals scored?

Number of Goals Scored per Game | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

Frquency | 4 | 6 | 11 | 5 | 2 | 1 | 1 |

### Answer

Recall that the mode of a data set is the value that appears most often.

In this case, the data values are the different numbers of goals scored per game in a football season. The frequencies represent how often each number was scored. So, for example, there were 0 goals scored on four occasions and 3 goals scored on five occasions.

The mode will be the number of goals that has the highest frequency. Reading along the frequency row, we see that the highest frequency is 11. This appears in the column for 2 goals scored. So, in other words, 2 goals were scored in a game on 11 occasions.

We conclude that the mode number of goals scored is 2.

Let us finish by recapping some key concepts from this explainer.

### Key Points

- The mode of a set of data is the value that appears most often.
- The mode does not have to be a number. The mode is the only measure of central tendency for which this can be the case.
- A data set can have a unique mode, more than one mode, or, in the event when there is no value that appears more often than any other, no mode.
- For small data sets, we can find the mode by ordering the data values from least to greatest. For larger data sets, we can use a tally chart, a frequency table, or a bar chart to help us.
- For a data set presented in a frequency table, the mode is the data value with the highest frequency.
- When given the mode of a data set, we can work backward to find an unknown data value.