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What is a 'Less Than' Cumulative Frequency?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
A 'Less Than' Cumulative Frequency tells us how many data points are *less than* or *equal to* a particular value in a dataset. It's like counting how many items fall below a certain limit as you go up the list.
Simple Example
Quick Example
Imagine your school principal wants to know how many students scored less than 70 marks in the last science test. A 'less than' cumulative frequency for 70 marks would give exactly that count. It sums up all students who scored 0-10, 10-20, ..., all the way up to 60-70 marks.
Worked Example
Step-by-Step
Let's find the 'Less Than' Cumulative Frequency for mobile data usage (in GB) for 100 users.
Data groups: 0-5 GB (20 users), 5-10 GB (35 users), 10-15 GB (30 users), 15-20 GB (15 users).
Step 1: Identify the upper limits of each group. These are 5 GB, 10 GB, 15 GB, 20 GB.
---Step 2: For 'Less than 5 GB', the frequency is 20 users.
---Step 3: For 'Less than 10 GB', we add users from 0-5 GB and 5-10 GB. So, 20 + 35 = 55 users.
---Step 4: For 'Less than 15 GB', we add users from 0-5 GB, 5-10 GB, and 10-15 GB. So, 20 + 35 + 30 = 85 users.
---Step 5: For 'Less than 20 GB', we add all users. So, 20 + 35 + 30 + 15 = 100 users.
Answer: The 'Less Than' Cumulative Frequencies are: Less than 5 GB (20), Less than 10 GB (55), Less than 15 GB (85), Less than 20 GB (100).
Why It Matters
Understanding 'Less Than' Cumulative Frequency is key in fields like AI/ML to analyze data distributions, in FinTech to assess risk by knowing how many customers fall below a certain income, and in Medicine to track how many patients show symptoms below a certain age. It helps engineers design better products by understanding user demographics and preferences.
Common Mistakes
MISTAKE: Using the lower limit of a class interval instead of the upper limit for 'Less Than' cumulative frequency. | CORRECTION: Always use the *upper limit* of the class interval to define the 'less than' point. For example, for 10-20, you calculate 'less than 20'.
MISTAKE: Forgetting to add the previous cumulative frequency to the current frequency. | CORRECTION: Cumulative frequency means 'total up to this point'. So, you must add the frequency of the current class interval to the cumulative frequency of the *previous* class interval.
MISTAKE: Confusing 'Less Than' with 'More Than' cumulative frequency. | CORRECTION: 'Less Than' counts items below a value, starting from the lowest and adding up. 'More Than' counts items above a value, usually starting from the highest and subtracting, or summing from the bottom up and then subtracting from total.
Practice Questions
Try It Yourself
QUESTION: In a class, marks in Maths are: 0-20 (5 students), 20-40 (8 students), 40-60 (12 students). What is the 'less than' cumulative frequency for 'less than 40 marks'? | ANSWER: 13 students (5 + 8)
QUESTION: A survey of auto-rickshaw distances covered in a day: 0-10 km (10 drivers), 10-20 km (15 drivers), 20-30 km (20 drivers), 30-40 km (5 drivers). What is the 'less than' cumulative frequency for 'less than 30 km'? | ANSWER: 45 drivers (10 + 15 + 20)
QUESTION: For the data: 0-10 (4 items), 10-20 (6 items), 20-30 (7 items), 30-40 (3 items). Create a complete 'Less Than' Cumulative Frequency table. | ANSWER: Less than 10 (4), Less than 20 (10), Less than 30 (17), Less than 40 (20).
MCQ
Quick Quiz
If a data set has class intervals 0-10 (freq=3), 10-20 (freq=7), 20-30 (freq=5), what is the 'less than' cumulative frequency for 'less than 20'?
3
7
10
15
The Correct Answer Is:
C
The 'less than' cumulative frequency for 'less than 20' includes all items in the 0-10 interval (3) and the 10-20 interval (7). So, 3 + 7 = 10.
Real World Connection
In the Real World
When you see reports about smartphone sales, they often use 'less than' cumulative frequencies. For instance, a report might say 'X% of users spend less than Rs. 15,000 on a phone'. This helps companies like Jio and Samsung understand market segments and plan new products or data plans. Even in cricket, analysts use it to see how many players score 'less than' 50 runs in a season.
Key Vocabulary
Key Terms
FREQUENCY: The number of times a particular value or item appears in a dataset. | CLASS INTERVAL: A range of values used to group data, e.g., 0-10, 10-20. | CUMULATIVE: Building up or increasing by successive additions. | UPPER LIMIT: The highest value in a class interval. | DATASET: A collection of related data points.
What's Next
What to Learn Next
Great job learning 'Less Than' Cumulative Frequency! Next, you should explore 'More Than' Cumulative Frequency. It's the opposite concept and understanding both will give you a complete picture of how data is distributed and help you draw powerful conclusions.
