Cumulative Frequency Table

Earlier in this statistics unit, we learned about frequency tables and frequency distributions. The next lesson will cover two-way frequency tables. Let's look at another type.

Cumulative frequency is just the total frequency at a given interval.

For example, suppose we have the following frequency table: $$\begin{array}{|c|c|} \hline \text{Bin}&\text{Frequency}\\ \hline A&3\\ B&4\\ C&5\\ D&6\\ \hline \end{array}$$

You might remember that the highest frequency represents the mode and the middle value is the median.

To make a cumulative frequency table, we add a column on to the end. $$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Frequency}&\text{Cumulative Frequency}\\ \hline A&3\\ B&4\\ C&5\\ D&6\\ \hline \end{array}$$

The cumulative frequency of bin $A$ is going to be the total frequency up to that point. Since it's the first bin, the cumulative frequency will be the same as the frequency. The cumulative frequency in the first interval will always be the same as its frequency. $$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Frequency}&\text{Cumulative Frequency}\\ \hline A&3&3\\ B&4\\ C&5\\ D&6\\ \hline \end{array}$$

Next, we want to find the cumulative frequency of bin $B$. This is the total frequency of everything up to this point. In other words, the frequency sum of bins $A$ and $B$. $3+4=7$. $$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Frequency}&\text{Cumulative Frequency}\\ \hline A&3&3\\ B&4&3+4=7\\ C&5\\ D&6\\ \hline \end{array}$$

Bin $C$ will be the frequency sum of bins $A$, $B$, and $C$. Likewise, bin $D$ will be the frequency sum of bins $A$, $B$, $C$, and $D$. The cumulative frequency of the last interval will always be the total frequency. $$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Frequency}&\text{Cumulative Frequency}\\ \hline A&3&3\\ B&4&3+4=7\\ C&5&3+4+5=12\\ D&6&3+4+5+6=18\\ \hline \end{array}$$

Using Cumulative Frequency to Find Frequency

We can also work backwards and find a bin's frequency, given its cumulative frequency. Suppose we have the following table: $$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Frequency}&\text{Cumulative Frequency}\\ \hline A&&2\\ B&&11\\ C&&26\\ D&&38\\ \hline \end{array}$$ To find the frequency of a given bin, subtract the previous cumulative frequency from the current cumulative frequency.

Because the first bin's frequency is the same as its cumulative frequency, we know $A$'s frequency is $2$.

To find $B$'s frequency we subtract $A$'s cumulative frequency from $B$'s. $$11-2=9$$ Doing this for the other bins, our table becomes: $$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Frequency}&\text{Cumulative Frequency}\\ \hline A&2&2\\ B&11-2=9&11\\ C&26-11=15&26\\ D&38-26=12&38\\ \hline \end{array}$$

Finding the Cumulative Frequency for a Certain Range

Sometimes, you'll want to find the cumulative frequency not starting at the beginning. For example, you might want the cumulative frequency of the last three bins.

Suppose on the next math test, Marcus's students get the following grades. (Ignore the fact that there are more kids in the class, now.)

$$\begin{array}{|c|c|c|} \hline \text{Bin}&\text{Cumulative Frequency}\\ \hline A&3\\ B&8\\ C&14\\ D&27\\ \hline \end{array}$$

He now wants to know how many kids received at most a $C$. This is the same as asking for the cumulative frequency of just $C$ and $D$. But, we don't know this, since each of their cumulative frequencies are including the frequencies of$A$ and $B$.

To find this, we subtract the last cumulative frequency of the bins we don't need from the last cumulative frequency of the bins we do want. In this problem, we don't care about bins $A$ and $B$. Bin $B$ is the last for these. We do care about bins $C$ and $D$. Bin $D$ is the last for these. Thus, we get:

$$27-8=19$$

The cumulative frequency of $C$ and $D$ is $19$. So $19$ students received at most a $C$ on Marcus's test.

Cumulative Frequency Tables

A cumulative frequency table is a cousin of our regular frequency tables. Their purpose is to show frequencies of individual groups (rows in the table) as well as the sums of those frequencies.

Let's take a look:

The columns in purple and green should look familiar—those are the ones you'll find in a regular old frequency table. It's the column in orange that makes this table cumulative.

So, I might've just snuck something past you. :)

The last question of the $3$ above is a question about cumulative frequency. Anytime you are adding frequencies together, you are calculating a cumulative frequency.

Now, let's do some practice questions that are likely to come up when you're working with cumulative frequency tables.

(Video) How to Make a Cumulative Frequency Chart

by MySecretMathTutor

This video by MySecretMathTutor teaches cumulative frequency.

What is Cumulative Frequency?

Cumulative frequency is the total frequency at that instant. Basically, its just the running total of the frequencies.

He has a table for the age of a child and their weight gain in pounds.

$$\begin{array}{c|c} \text{age}&\text{weight gain}\\ \hline 0&4\\ 1&9\\ 2&2\\ 3&2\\ 4&3\\ 5&4\\ 6&3 \end{array}$$

Cumulative frequency in this case would be finding the total weight of the child at that age. We can start by making a new column on our table.

$$\begin{array}{c|c|c} \text{age}&\text{weight gain}&\text{total weight}\\ \hline 0&4\\ 1&9\\ 2&2\\ 3&2\\ 4&3\\ 5&4\\ 6&3 \end{array}$$

At age $0$, they weigh a total of $4$ pounds. But, at age $1$, they gain $9$ more pounds. So the total weight becomes $4+9=13$.

$$\begin{array}{c|c|c} \text{age}&\text{weight gain}&\text{total weight}\\ \hline 0&4&\color{#8925AE}{4}\\ 1&9&4+9=\color{#8925AE}{13}\\ 2&2\\ 3&2\\ 4&3\\ 5&4\\ 6&3 \end{array}$$

We continue this trend all the way through the table. The total weight will equal the previous row's total weight plus the weight gain of the current row. Filling these out, we get:

$$\begin{array}{c|c|c} \text{age}&\text{weight gain}&\text{total weight}\\ \hline 0&4&4\\ 1&9&13\\ 2&2&15\\ 3&2&17\\ 4&3&20\\ 5&4&24\\ 6&3&27 \end{array}$$