What is Cumulative Frequency?

S3-SA3-0021

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Cumulative frequency is the running total of frequencies. It tells us how many data points fall at or below a certain value in a dataset, by adding up frequencies step-by-step.

Simple Example

Quick Example

Imagine you're checking how many students scored certain marks in a science test. If 5 students scored 30 marks, and 8 students scored 40 marks, then the cumulative frequency for 40 marks would be 5 + 8 = 13. This means 13 students scored 40 marks or less.

Worked Example

Step-by-Step

Let's find the cumulative frequency for the number of samosas sold at a shop each day for a week.

Day | Samosas Sold (Frequency)
Monday | 15
Tuesday | 20
Wednesday | 10
Thursday | 25
Friday | 30

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Step 1: For Monday, the cumulative frequency is just the frequency itself. So, 15.

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Step 2: For Tuesday, add Monday's cumulative frequency to Tuesday's frequency. 15 + 20 = 35.

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Step 3: For Wednesday, add Tuesday's cumulative frequency to Wednesday's frequency. 35 + 10 = 45.

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Step 4: For Thursday, add Wednesday's cumulative frequency to Thursday's frequency. 45 + 25 = 70.

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Step 5: For Friday, add Thursday's cumulative frequency to Friday's frequency. 70 + 30 = 100.

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Answer: The cumulative frequencies are: Monday (15), Tuesday (35), Wednesday (45), Thursday (70), Friday (100). This means by Friday, 100 samosas were sold in total.

Why It Matters

Understanding cumulative frequency helps in data science and AI to see trends and distributions quickly. Economists use it to analyze income distribution, and engineers might use it to understand component failure rates. It's a foundational skill for anyone working with data!

Common Mistakes

MISTAKE: Adding the current frequency to the *original* frequency of the previous class. | CORRECTION: Always add the current frequency to the *cumulative* frequency of the previous class.

MISTAKE: Forgetting to start the first cumulative frequency as just the first frequency itself. | CORRECTION: The cumulative frequency for the first data point is always equal to its own frequency.

MISTAKE: Calculating cumulative frequency downwards but reading it as 'at least' instead of 'at or below'. | CORRECTION: Standard cumulative frequency (adding frequencies from top to bottom) always means 'at or below' a certain value.

Practice Questions

Try It Yourself

QUESTION: A bus stop recorded passengers waiting: 5 (8 AM), 12 (9 AM), 7 (10 AM). What is the cumulative frequency for 9 AM? | ANSWER: 17

QUESTION: The number of books read by students in a month: 1 book (10 students), 2 books (15 students), 3 books (8 students), 4 books (5 students). What is the cumulative frequency for students who read 3 books or less? | ANSWER: 33

QUESTION: The marks obtained by 20 students in a test are given: 10 marks (3 students), 20 marks (7 students), 30 marks (6 students), 40 marks (4 students). Create a cumulative frequency column for this data. What is the cumulative frequency for 40 marks? | ANSWER: Cumulative Frequencies: 3, 10, 16, 20. The cumulative frequency for 40 marks is 20.

MCQ

Quick Quiz

What does a cumulative frequency of 50 for '40 kg' mean in a weight survey?

Exactly 50 people weigh 40 kg.

50 people weigh more than 40 kg.

50 people weigh 40 kg or less.

The average weight is 40 kg.

The Correct Answer Is:

Cumulative frequency represents the total count of data points up to and including a specific value. So, 50 people weigh 40 kg or less.

Real World Connection

In the Real World

When you check your mobile data usage, sometimes apps show you how much data you've used in total over a period, or how many messages you've sent. This 'running total' is like cumulative frequency. Even cricket commentators use it when they say 'India has scored 150 runs by the 20th over,' giving a cumulative score.

Key Vocabulary

Key Terms

FREQUENCY: How often something happens or appears. | DATASET: A collection of related information or numbers. | RUNNING TOTAL: A total that is updated continuously as new numbers are added. | DISTRIBUTION: How data values are spread out.

What's Next

What to Learn Next

Great job understanding cumulative frequency! Next, you can explore how to create a 'Cumulative Frequency Curve' or 'Ogive'. This graph visually represents cumulative frequency and helps us find important values like the median easily.