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# Cumulative Frequency

Cumulative Frequency
• Calculus • Decision Maths • Geometry • Mechanics Maths • Probability and Statistics • Pure Maths • Statistics Frequency refers to the number of times an event or outcome occurs. The cumulative frequency at a point x is the sum of the individual frequencies up to and at point x. Typically, cumulative frequency gives us the number of times an outcome occurs that is above or below a given value.

Consider the following table:

 Number of pizzas eaten in January (x) Number of persons (frequency) 0 3 1 1 2 4 3 2 4 0 5 2

The above frequency table can tell us how many people had a certain number of pizzas in January. For example, the number of people who had exactly 2 pizzas in January was 4. Now, suppose we want to know how many people had a maximum of 2 pizzas in January. This would be represented by the cumulative frequency at x = 2, which would be equal to the sum of the number of people who had 0, 1 and 2 pizzas in January ie. 3 +1 +4 = 8.

## Cumulative frequency table

A cumulative frequency table is a very useful statistical tool to help us deal with frequencies and cumulative frequencies. To construct a cumulative frequency table from the above example just add another column for the cumulative frequency. The cumulative frequency for each value of x is equal to the sum of the frequency for that value of x and the cumulative frequency for the previous value of x (the cumulative frequency for the first value of x will be the same as the frequency).

Thus, we get the following cumulative frequency table:

 Number of pizzas eaten in January (x) Number of persons (frequency) Cumulative frequency 0 3 3 1 1 3 + 1 = 4 2 4 + 4 = 8 3 2 8 + 2 = 10 0 10 + 0 = 10 5 2 10 + 2 = 12

## Cumulative frequency graph

Another very commonly used tool to deal with cumulative frequencies is a cumulative frequency graph.

Let's draw the cumulative frequency graph for the above example. The value of x is represented on the x-axis and the cumulative frequency on the y-axis. Example cumulative frequency graph, Nilabhro Datta - StudySmarter Originals

## Cumulative frequency for grouped frequency distribution

In statistics, data are very often grouped into classes which represent a continuous range of values. This is a very common practice in the case of frequency distribution.

For example, consider the following frequency distribution table:

 Restaurant ratings (x) Number of restaurants (frequency) 0.0 - 1.0 1.0 - 2.0 28 2.0 - 3.0 45 3.0 - 4.0 40 4.0 - 5.0 35

To obtain the cumulative frequency table from the above data, we can follow the same steps that we did for the earlier example with discrete values.

 Restaurant ratings (x) Class Mark Number of restaurants (frequency) Cumulative frequency (y) 0.0 - 1.0 0.5  1.0 - 2.0 1.5 28 40 2.0 - 3.0 2.5 45 85 3.0 - 4.0 3.5 40 125 4.0 - 5.0 4.5 35 160

Now, to create the cumulative frequency graph, we need to use the class mark for each class. The class mark is the middle value of each class. Therefore, the class mark for the class, 1.0 - 2.0 will be (1.0 + 2.0) / 2 = 1.5. Similarly, the class mark for the class, 4.0 - 5.0 will be 4.5.

Thus the cumulative frequency graph obtained will be the following: The cumulative frequency curve for the given frequency distribution, Nilabhro Datta - Study Smart Originals

As you can see, the graph has been plotted using the respective class mark of each class (0.5, 1.5, 2.5 ...). Note that the lowest possible value is 0, therefore the graph starts from (0, 0)

For the following frequency table showing the mass of mangoes in grams, construct the cumulative frequency table and cumulative frequency graph.

 Mass in grams (x) Frequency 50 ≤ x < 70 22 70 ≤ x < 90 23 90 ≤ x < 110 47 110 ≤ x < 130 18 130 ≤ x < 150 7

#### Solution

Create the resultant cumulative frequency table.

 Mass in grams (x) Class Mark Frequency Cumulative frequency 50 ≤ x < 70 60 22 22 70 ≤ x < 90 80 23 45 90 ≤ x < 110 100 47 92 110 ≤ x < 130 120 18 110 130 ≤ x < 150 140 7 117

Now you can draw the corresponding cumulative frequency graph. The cumulative frequency curve for the given grouped frequency distribution, Nilabhro Datta - Study Smarter Originals

## Estimating medians, quartiles, and percentiles using cumulative frequency

In the case of grouped frequency distribution, it is usually not possible to calculate the exact values of medians, quartiles and percentiles. Using cumulative frequency graphs, it is possible to estimate these values.

Tip: The values obtained are approximations and are not going to be exact values.

Here is a rough outline of the process that you can follow to obtain the value of medians, quartiles and percentiles from a grouped frequency distribution using cumulative frequency graphs.

#### Steps:

1) Given a grouped frequency distribution table, obtain the cumulative frequency table.

2) On the graph, plot the points obtained from the cumulative frequency table using the upper class boundary (not the class mark), and the corresponding cumulative frequency.

3) Draw an approximate best fit curve through the plotted points.

4) Estimate the required median/quartile/percentile values from the graph. For example, in a graph plotted from a frequency distribution with 200 noted outcomes:

• 200/2 = 100th value is the media

• 200/4 = 50th value is the lower quartile and 200 × 3/4 = 150th value is the upper quartile

• 200 × 90/100 = 180th value is the 90th percentile and 200 × 30/100 = 60th value is the 30th percentile.

Consider the following frequency table showing the mass of mangoes in grams, construct the cumulative frequency table and cumulative frequency graph.

Estimate the value(s) of the a) median b) upper and lower quartiles c) 43rd percentile d) 85th percentile

 Mass in grams (x) Frequency 50 ≤ x < 70 17 70 ≤ x < 90 23 90 ≤ x < 110 30 110 ≤ x < 130 18 130 ≤ x < 150 12

#### Solution

Create the resultant cumulative frequency table.

 Mass in grams (x) Frequency Cumulative frequency 50 ≤ x < 70 17 17 70 ≤ x < 90 23 40 90 ≤ x < 110 30 70 110 ≤ x < 130 18 88 130 ≤ x < 150 12 100

Now plot the points on a graph taking mass along the X-axis and cumulative frequency along the Y-axis, and draw the best fit curve through those points. Example best fit cumulative frequency curve to estimate median, quartiles and percentiles, Nilabhro Datta - StudySmarter Originals

From the above graph, we can obtain our estimates for the necessary median, quartiles and percentiles.

1. Median = value of (100/2 = 50)th data point = 95.78

2. Upper quartile = value of (100 × 3/4 = 75)th data point = 115.53 Lower quartile = value of (100 × 1/4 = 25)th data point = 77.88

3. Value of 43rd percentile = value of (100 × 43/100 = 43)rd data point = 90.87

4. Value of 85th percentile = value of (100 × 85/100 = 85)th data point = 125.95

## Cumulative Frequency - Key takeaways

• Frequency refers to the number of times an event or outcome occurs. The cumulative frequency at a point x is the sum of the individual frequencies up to and at point x.

• Two commonly used methods for representing cumulative frequency information are cumulative frequency graphs and cumulative frequency tables.

• For grouped frequency distribution, it is usually not possible to calculate the exact values of medians, quartiles and percentiles. Using cumulative frequency graphs, it is possible to estimate these values.

• Medians, quartile and percentile values obtained from cumulative frequency graphs are usually best approximations and not exact values.

For a frequency distribution on a domain of values, the cumulative frequency for each value x is equal to the sum of the frequency for that value of x and the cumulative frequency for the previous value of x.

Frequency refers to the number of times an event or outcome occurs. The cumulative frequency at a point x is the sum of the individual frequencies up to and at point x.

How to draw a cumulative frequency graph: first construct the cumulative frequency table from the given frequency distribution. Then plot the corresponding points on the graph.

How to find the median from the cumulative frequency graph: find the value of the n/2th data point from the graph, where n is the total number of data points.

An example of how cumulative frequency could be useful would be finding out the number of students in a class who have got less than 80/100 on an exam from a frequency distribution of the total marks obtained.

## Final Cumulative Frequency Quiz

Question

What is cumulative frequency?

The cumulative frequency at a point x is the sum of the individual frequencies up to and at the point x.

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Question

Which of the following can you obtain from a cumulative frequency distribution? a) median b) quartiles c) percentiles d) all of the above

d

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Question

If a cumulative frequency for the (n-1)th value is 85 in discrete frequency distribution with 110 data points, what is the raw frequency for the nth value?

25

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Question

For a grouped frequency distribution, what is the class mark for the class 0.5 - 1.0?

0.75

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Question

For a grouped frequency distribution, what is the class mark for the class 2.5 - 3.5?

3.0

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Question

For a grouped frequency distribution, what is the class mark for the class 8 - 12?

10

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Question

State whether the following statement is true or false : the curve for a cumulative frequency graph is never decreasing.

True

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Question

The cumulative frequency curve for an experiment with 200 trials is given by x = y/5, where the cumulative frequency is represented on the y-axis. Find the median.

x = (200/2)/5 = 20

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Question

The cumulative frequency curve for an experiment with 200 trials is given by x = y/5, where the cumulative frequency is represented on the y-axis. Find the upper quartile.

x = (200 × 3/4)/5 = 30

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Question

The cumulative frequency curve for an experiment with 200 trials is given by x = y/5, where the cumulative frequency is represented on the y-axis. Find the 43rd percentile.

x = (200 × 43/100)/5 = 17.2

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Question

The cumulative frequency curve for an experiment with 200 trials is given by x = y/5, where the cumulative frequency is represented on the y-axis. Find the 70th percentile.

x = (200 × 70/100)/5 = 28

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Question

The cumulative frequency curve for an experiment with 100 trials is given by x = 2y + 3, where the cumulative frequency is represented on the y-axis. Find the median.

x = 2 × (100/2) + 3 = 103

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Question

The cumulative frequency curve for an experiment with 100 trials is given by y = 2x + 3, where the cumulative frequency is represented on the y-axis. Find the lower quartile.

x = 2 × (100/4) + 3 = 53

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Question

A grouped frequency distribution has been made for the length of 500 snakes. The cumulative frequency of a class (8.0 - 8.5) inches is 320. How many snakes are more than 8.5 inches long?

180

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Question

A grouped frequency distribution has been made for the length of 500 snakes. The cumulative frequency of a class (8.0 - 8.5) inches is 320. Which of the following is the correct conclusion?

There are 320 snakes shorter than than or equal to 8.5 inches

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