Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S7-SA3-0071
What is the Mode of Grouped Data?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The Mode of Grouped Data is the value that appears most frequently in a dataset that has been organised into class intervals. It tells us which range of values has the highest concentration of observations, helping us understand the most common category.
Simple Example
Quick Example
Imagine a shop selling different sizes of kurtas. If they record how many kurtas of each size (Small, Medium, Large, XL) they sell in a day, the size that sells the most is the mode. For grouped data, instead of exact sizes, we look at ranges like 'age 10-20', 'age 20-30', etc.
Worked Example
Step-by-Step
Let's find the mode for the following data showing marks of 50 students in a test:
Marks (Class Interval) | Number of Students (Frequency)
----------------------|-------------------------------
0-10 | 5
10-20 | 12
20-30 | 18
30-40 | 10
40-50 | 5
---1. Identify the modal class: The class interval with the highest frequency is 20-30, with a frequency of 18. This is our modal class.
---2. Identify the lower limit (L) of the modal class: L = 20.
---3. Identify the frequency of the modal class (f1): f1 = 18.
---4. Identify the frequency of the class preceding the modal class (f0): f0 = 12.
---5. Identify the frequency of the class succeeding the modal class (f2): f2 = 10.
---6. Calculate the class size (h): h = Upper limit - Lower limit = 30 - 20 = 10.
---7. Apply the mode formula: Mode = L + [ (f1 - f0) / (2*f1 - f0 - f2) ] * h
Mode = 20 + [ (18 - 12) / (2*18 - 12 - 10) ] * 10
Mode = 20 + [ 6 / (36 - 12 - 10) ] * 10
Mode = 20 + [ 6 / 14 ] * 10
Mode = 20 + (0.42857...) * 10
Mode = 20 + 4.2857...
Mode = 24.29 (approximately)
---The mode of the grouped data is approximately 24.29.
Why It Matters
Understanding the mode helps scientists and engineers identify the most common occurrences or preferences. For example, in AI/ML, it helps find the most frequent pattern in data. In FinTech, it can show the most common investment bracket, guiding financial products.
Common Mistakes
MISTAKE: Confusing the modal class frequency with the mode itself. | CORRECTION: The mode is a value within the modal class, calculated using a specific formula, not just the highest frequency number.
MISTAKE: Incorrectly identifying f0 or f2 (frequencies of preceding/succeeding classes). | CORRECTION: f0 is ALWAYS the frequency of the class *before* the modal class, and f2 is ALWAYS the frequency of the class *after* the modal class.
MISTAKE: Making calculation errors in the denominator (2*f1 - f0 - f2). | CORRECTION: Always calculate 2*f1 first, then subtract f0, and then subtract f2. Use brackets carefully.
Practice Questions
Try It Yourself
QUESTION: Find the modal class for the data: Class: 0-5 (Freq: 3), 5-10 (Freq: 7), 10-15 (Freq: 10), 15-20 (Freq: 5). | ANSWER: 10-15
QUESTION: A survey on daily mobile data usage (in GB) showed: 0-1 GB (20 users), 1-2 GB (35 users), 2-3 GB (15 users). Calculate the mode. | ANSWER: L=1, f1=35, f0=20, f2=15, h=1. Mode = 1 + [ (35-20) / (2*35 - 20 - 15) ] * 1 = 1 + [ 15 / (70 - 35) ] * 1 = 1 + [ 15 / 35 ] = 1 + 0.428 = 1.43 GB (approx).
QUESTION: For a dataset with modal class 30-40, L=30, f1=25, f0=20, f2=10, and h=10. If a student calculates the mode as 38, is this correct? Show your working. | ANSWER: Mode = 30 + [ (25-20) / (2*25 - 20 - 10) ] * 10 = 30 + [ 5 / (50 - 30) ] * 10 = 30 + [ 5 / 20 ] * 10 = 30 + 0.25 * 10 = 30 + 2.5 = 32.5. So, the student's calculation of 38 is incorrect.
MCQ
Quick Quiz
Which of the following is the correct formula for calculating the mode of grouped data?
Mode = L + [ (f1 - f0) / (2*f1 - f0 - f2) ] * h
Mode = L + [ (f1 + f0) / (2*f1 + f0 + f2) ] * h
Mode = L + [ (f1 - f0) / (f1 - f0 - f2) ] * h
Mode = L + [ (f1 - f2) / (2*f1 - f0 - f2) ] * h
The Correct Answer Is:
A
Option A is the correct formula for the mode of grouped data. The numerator uses the difference between the modal class frequency and the preceding frequency, and the denominator accounts for the frequencies of the modal, preceding, and succeeding classes.
Real World Connection
In the Real World
Imagine a company like Zomato or Swiggy analyzing delivery times. They might group delivery times (e.g., 0-10 mins, 10-20 mins, 20-30 mins). Finding the mode helps them know the most common delivery time range, allowing them to optimize routes or offer better estimated times to customers. This is crucial for their business in India.
Key Vocabulary
Key Terms
Modal Class: The class interval with the highest frequency | Lower Limit (L): The lower boundary of the modal class | Frequency (f): The number of observations in a class | Class Size (h): The width of a class interval | Preceding Frequency (f0): Frequency of the class before the modal class | Succeeding Frequency (f2): Frequency of the class after the modal class
What's Next
What to Learn Next
Now that you understand the mode, you should explore how to calculate the Mean and Median for grouped data. These are other important measures of central tendency that give a complete picture of your dataset, essential for advanced topics in statistics.
