What is the Median of Grouped Data?

S7-SA3-0069

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The median of grouped data is the middle value of a dataset that has been organized into classes or groups. It's the point where half the observations are below it and half are above it, even when you don't know the exact value of each observation.

Simple Example

Quick Example

Imagine a school survey showing how many hours students spend studying daily, grouped like '0-2 hours', '2-4 hours', '4-6 hours'. We want to find the median study time. We can't know each student's exact time, but we can estimate the middle study time for the whole group.

Worked Example

Step-by-Step

Let's find the median for the daily pocket money (in Rupees) of students in a class:

Class Interval (Pocket Money) | Frequency (Number of Students)
-------------------------------|----------------------------------
0-50 | 5
50-100 | 10
100-150 | 15
150-200 | 8
200-250 | 7

---1. Calculate the cumulative frequency (cf):
Class Interval | Frequency (f) | Cumulative Frequency (cf)
----------------|---------------|---------------------------
0-50 | 5 | 5
50-100 | 10 | 5 + 10 = 15
100-150 | 15 | 15 + 15 = 30
150-200 | 8 | 30 + 8 = 38
200-250 | 7 | 38 + 7 = 45

---2. Find N (total frequency). Here, N = 45. Then calculate N/2 = 45/2 = 22.5.

---3. Identify the median class. This is the class whose cumulative frequency is just greater than or equal to N/2. Here, N/2 = 22.5. The cf just greater than 22.5 is 30, which belongs to the class 100-150. So, the median class is 100-150.

---4. Identify the values for the median formula:
l = lower limit of the median class = 100
h = class size (width) = 150 - 100 = 50
f = frequency of the median class = 15
cf = cumulative frequency of the class *preceding* the median class = 15 (from the 50-100 class)

---5. Apply the median formula for grouped data: Median = l + [(N/2 - cf) / f] * h
Median = 100 + [(22.5 - 15) / 15] * 50

---6. Calculate:
Median = 100 + [7.5 / 15] * 50
Median = 100 + [0.5] * 50
Median = 100 + 25
Median = 125

---The median daily pocket money is Rs. 125.

Why It Matters

Understanding the median of grouped data helps scientists in Climate Science analyze temperature ranges, engineers in EVs understand battery performance distribution, and doctors in Medicine study patient recovery times. It's crucial for making fair decisions and understanding the 'typical' value in large datasets, impacting careers from data analysis to financial planning.

Common Mistakes

MISTAKE: Using the frequency of the median class instead of the cumulative frequency of the *preceding* class in the formula. | CORRECTION: The 'cf' in the median formula refers to the cumulative frequency of the class *before* the median class. Always double-check this value.

MISTAKE: Forgetting to calculate N/2 before finding the median class. | CORRECTION: The first step after finding cumulative frequency is to calculate N/2. This value is critical for locating the correct median class.

MISTAKE: Confusing the lower limit of the median class (l) with the upper limit or simply picking a value from within the class. | CORRECTION: 'l' in the formula specifically means the *lower boundary* of the median class interval.

Practice Questions

Try It Yourself

QUESTION: For a dataset, N=80 and N/2=40. The median class is 60-70, with frequency 20. The cumulative frequency of the preceding class is 30. What is the median? | ANSWER: 65

QUESTION: A survey of mobile data usage (in GB) showed: 0-5 GB (12 users), 5-10 GB (18 users), 10-15 GB (25 users), 15-20 GB (15 users). Find the median data usage. | ANSWER: 10.8 GB

QUESTION: In a factory, the daily wages (in Rs.) of workers are: 100-200 (10 workers), 200-300 (25 workers), 300-400 (30 workers), 400-500 (15 workers). If the median wage is Rs. 316.67, what is the frequency of the class 200-300? (Hint: Work backwards from the formula.) | ANSWER: 25 workers

MCQ

Quick Quiz

Which term in the median formula for grouped data represents the cumulative frequency of the class *preceding* the median class?

The Correct Answer Is:

The term 'cf' in the median formula for grouped data specifically stands for the cumulative frequency of the class just before the median class. 'f' is the frequency of the median class, 'h' is class size, and 'l' is the lower limit.

Real World Connection

In the Real World

In cricket analytics, the median can be used to understand the 'middle' range of scores batsmen achieve over a season, even if their scores are grouped (e.g., 0-50 runs, 50-100 runs). This helps coaches understand typical performance better than just an average, especially when there are extreme scores.

Key Vocabulary

Key Terms

MEDIAN: The middle value in a dataset when arranged in order | GROUPED DATA: Data organized into classes or intervals | CUMULATIVE FREQUENCY: The running total of frequencies | CLASS INTERVAL: A range of values in which data is grouped | FREQUENCY: The number of times an observation occurs in a class

What's Next

What to Learn Next

Now that you understand the median, explore the 'Mode of Grouped Data'. It's another way to find a central value and will help you get an even clearer picture of data distributions, which is super useful in data science and AI!