# Frequency table

A frequency table is a type of chart that is used to summarize the number of times an object (event, observation, etc.) occurs in a set of data. It typically takes the form of a two-column table, where the left column indicates the object being counted while the right column shows the number of occurences. This is often done using tally marks when actually counting from a given set of data. When displaying the frequency table however, it is more common to have already counted the data so as to present it in numerical form for readability. Below is an example of a frequency table that counts the number of instances of the numbers below; it includes the intermediary step of using tally marks to count each instance of each number.

1, 12, 6, 4, 7, 1, 12, 8, 3, 1, 7, 9

Number Count Frequency
1 ||| 3
3 | 1
4 | 1
6 | 1
7 || 2
8 | 1
9 | 1
12 || 2

The objects being counted can be just about any qualitative or quantitative variable (above example). An example of a frequency table involving qualitative values is one that depicts a count of how students get to school (walking, bus, car, bike, train, etc.).

Frequency tables are useful for providing a quick representation of data that can be used to identify patterns. In the example below, we can see that most of the students in this data set either ride their bike or walk to school.

How students get to school
Walking Riding the bus Riding in a car Riding a bicycle From this information, we can see that most students do not get to school in a motorized vehicle, and may start forming ideas about why this is the case. We could guess (but would need to verify) that the students who participated in the survey all live very close to the school. This is not necessarily the reason, but demonstrates one way to interpret the pattern depicted by the frequency table.

## Frequency table vs dot plot

Frequency tables and dot plots are two ways of representing the same data. Both have their own merits. Frequency tables are generally more concise, but it can be easier to identify patterns using dot plots.

While frequency tables typically show frequency using numerical values, a dot plot shows frequency using dots over a number line.

Example

Use a frequency table and dot plot to show the frequency distribution of the scores received by the 20 students in the class on their past quiz: 10, 8, 7, 7, 6, 8, 10, 7, 9, 8, 7, 10, 5, 9, 3, 10, 8, 7, 10, 6.

Frequency table:

Score # of students
3 1
5 1
6 2
7 5
8 4
9 2
10 5

Dot plot:

While both display the same data, certain observations we can make about the data are more easily noticeable using a dot plot. For example, if we wanted to know what the most common score was on the quiz using the frequency table, we would need to find the largest number(s) in the table. In this case, scores of 7 and 10 were the most common, with 5 students each. However, we needed to look through the entire table to determine the most common score (highest number of occurences) while also noticing that two of the scores tied. This is something we can immediately see from the dot plot. We don't need to look at 3, 5, 6, 8, or 9, since we can see that the highest frequency values are 7 and 10, and that they occurred 5 times. We only need to count the number of dots.

On the other hand, if our data included 200 students, the frequency table would be much more efficient than the dot plot. If for example, there were 150 students who scored an 8, even if we could see that 8 had the largest frequency on the dot plot, we would have to count the number of dots. Even if each dot represented a larger quantity, such as each dot representing 10 students, we would still need to count a significant number of dots. In contrast, we could just scan the frequency table for the largest score; since there are only 10 possible scores this will likely be more efficient than using the dot plot.

The above is only one example. There is plenty of other information that we can glean from the data, so whether or not to use a frequency table, dot plot, or something else requires a good understanding of the various tools available.