Cumulative Frequency Distribution

A cumulative frequency distribution is a table or graph that shows the cumulative frequencies of a set of data. Cumulative frequency is the sum of the frequencies for a given class and all previous classes. It helps in understanding the accumulation of data values over a range. It’s particularly useful for determining the number of observations that fall below a certain value or within a certain range.

Essentially, it allows us to visualize how data builds up over time or as you progress through intervals. This is often depicted in a cumulative frequency table or a cumulative frequency graph (ogive).

Formulae

The core concept involves calculating cumulative frequencies. Let’s define:

• $f_i$: Frequency of the $i^{th}$ class interval.
• $CF_i$: Cumulative Frequency of the $i^{th}$ class interval.

The basic formula is:

$CF_i = f_1 + f_2 + f_3 + … + f_i$

Alternatively:

$CF_i = CF_{i-1} + f_i$

Where $CF_0$ is usually considered to be 0 (since there are no observations before the first interval).

Examples

Let’s look at a couple of examples to illustrate how cumulative frequency works.

Example-1: Heights of Students

Suppose we have the following frequency distribution of heights of students (in cm) in a class:

• 150-155 cm: 5 students
• 155-160 cm: 8 students
• 160-165 cm: 12 students
• 165-170 cm: 7 students
• 170-175 cm: 3 students

To calculate the cumulative frequencies, we build a table:

Height (cm)	Frequency	Cumulative Frequency
150-155	5	5
155-160	8	5 + 8 = 13
160-165	12	13 + 12 = 25
165-170	7	25 + 7 = 32
170-175	3	32 + 3 = 35

The cumulative frequency for the 160-165 cm interval is 25, which means 25 students have a height of 165cm or less.

Example-2: Test Scores

Consider the following data for test scores:

• Scores 0-10: 2 students
• Scores 10-20: 5 students
• Scores 20-30: 8 students
• Scores 30-40: 10 students
• Scores 40-50: 5 students

The corresponding cumulative frequency table is:

Score Range	Frequency	Cumulative Frequency
0-10	2	2
10-20	5	2 + 5 = 7
20-30	8	7 + 8 = 15
30-40	10	15 + 10 = 25
40-50	5	25 + 5 = 30

The cumulative frequency of 25 for the interval 30-40 indicates that 25 students scored less than or equal to 40.

Common mistakes by students

• Adding the wrong frequencies: Students sometimes incorrectly add frequencies. Always ensure you are adding frequencies from the beginning up to, and including, the current interval.
• Confusing cumulative frequency with frequency: Students often mix up the interpretation of frequency and cumulative frequency. Remember that cumulative frequency represents a running total.
• Incorrectly determining the starting point: For the first interval, it’s important that the cumulative frequency is the same as the simple frequency. Students sometimes start with zero, which can be right but it’s more intuitive to start with the frequency of the very first interval
• Not understanding what the cumulative frequency represents: The cumulative frequency is not just a number; it provides information about the number of observations at or below a certain point. For example, a cumulative frequency of 15 means 15 observations have values less than or equal to the upper limit of the corresponding interval.

Real Life Application

Cumulative frequency distributions have numerous real-life applications:

• Analyzing Income Data: Governments use cumulative frequency distributions to understand income distribution and identify income inequality within a population. They can determine what percentage of the population earns below a certain threshold.
• Quality Control: In manufacturing, cumulative frequency is used to analyze the distribution of product measurements (e.g., lengths, weights). This helps determine if a certain percentage of products meet a specific quality standard.
• Medical Research: Cumulative frequencies are vital in medical studies. For example, a cumulative frequency plot can be used to analyze the time to a certain event (like recovery or death) in a group of patients.
• Market Research: Companies use cumulative frequency to analyze customer purchase data. They can identify the percentage of customers who spend less than a certain amount, helping inform pricing and marketing strategies.
• Insurance: Insurance companies use cumulative frequency to assess the frequency of claims based on various factors, like age or location. This data helps to price policies.

Fun Fact

The graph of a cumulative frequency distribution is known as an “ogive” (pronounced oh-jive). The ogive is an S-shaped curve, which helps in quickly visualizing the cumulative distribution of the data.

Recommended YouTube Videos for Deeper Understanding

Q.1 A cumulative frequency distribution table shows the following data for the scores of a math test: Score | Frequency ——- | ——– 40-50 | 5 50-60 | 8 60-70 | 12 70-80 | 7 80-90 | 3 What is the cumulative frequency for the class interval 60-70?

Check Solution

Ans: C

The cumulative frequency for the class interval 60-70 is the sum of frequencies up to and including that interval: 5 + 8 + 12 = 25

Q.2 The following cumulative frequency distribution is given: Marks | Cumulative Frequency ——- | ——– Below 10 | 5 Below 20 | 15 Below 30 | 30 Below 40 | 45 Below 50 | 50 How many students scored marks between 20 and 40?

Check Solution

Ans: C

Students scoring between 20 and 40 is the cumulative frequency below 40 minus cumulative frequency below 20: 45 – 15 = 30

Q.3 The following data represents the weight (in kg) of 20 students: 45, 48, 50, 52, 55, 58, 60, 62, 65, 65, 68, 70, 72, 75, 78, 80, 82, 85, 88, 90. What is the cumulative frequency for the weight 70 kg?

Check Solution

Ans: B

Count the number of students with weight less than or equal to 70: 45, 48, 50, 52, 55, 58, 60, 62, 65, 65, 68, 70. There are 12 students.

Q.4 A cumulative frequency curve (ogive) is plotted for the following data: Class Interval | Frequency ——- | ——– 0-10 | 4 10-20 | 6 20-30 | 8 30-40 | 10 What is the coordinate of a point on the ogive representing the class interval 20-30?

Check Solution

Ans: D

For the class interval 20-30, the upper class limit is 30. The cumulative frequency up to 30 is 4 + 6 + 8 = 18. So the point is (30, 18).

Q.5 A student drew a cumulative frequency curve. The curve shows that the upper quartile ($Q_3$) is 75. This means that:

Check Solution

Ans: D

The upper quartile ($Q_3$) represents the value below which 75% of the data falls.

Next Topic: Ogives: Less Than & More Than Types