ICSE Class 10 Maths: Mean, Median, Mode from Grouped Data

Why grouped data matters in everyday life

Ever wondered how a school can tell the average height of its students without measuring each kid individually? They group heights into ranges and work with those groups. That’s exactly what we do in statistics when we talk about "grouped data".

💡 In Simple Words: Grouped data is just a way of bundling many numbers into intervals, like sorting marbles by colour. From those bundles we can still figure out the average (mean), the middle point (median) and the most common size (mode) without looking at every single marble.

What is grouped data?

Grouped data is a collection of numbers arranged into class intervals (the ranges) and a frequency (how many numbers fall in each range). Think of it as a histogram you might have seen – each bar tells you how many items belong to that bar’s range.

Mean (average) for grouped data

The mean is the total sum of all values divided by the number of values. With groups we can’t add every single value, so we use the mid‑point of each class (the number exactly in the middle of the interval) as a stand‑in for every item in that class.

Steps to find the mean:

• Write down the class intervals and their frequencies.
• Find the midpoint (also called class‑mark) of each interval: midpoint = (lower limit + upper limit) / 2.
• Multiply each midpoint by its frequency (this gives f·x where f is frequency and x is midpoint).
• Add up all the f·x values to get Σf·x.
• Add up all the frequencies to get Σf.
• Mean = Σf·x ÷ Σf.

Example:

Midpoints: 14.5, 24.5, 34.5, 44.5.

Now calculate f·x:

• 4 × 14.5 = 58
• 7 × 24.5 = 171.5
• 5 × 34.5 = 172.5
• 3 × 44.5 = 133.5

Σf·x = 58 + 171.5 + 172.5 + 133.5 = 535.5

Σf = 4 + 7 + 5 + 3 = 19

Mean = 535.5 ÷ 19 ≈ 28.2

Median (middle value) for grouped data

The median is the value that splits the data into two equal halves. For groups we first locate the median class – the class where the cumulative frequency crosses half of Σf.

Steps to find the median:

• Find Σf (total frequency) and compute N/2 where N = Σf.
• Make a cumulative frequency column (add frequencies as you go down).
• Identify the class whose cumulative frequency first exceeds N/2 – that’s the median class.
• Use the formula: Median = L + [(N/2 – CF_before) / f] × h, where:
• L = lower limit of median class
• CF_before = cumulative frequency just before the median class
• f = frequency of median class
• h = class width (upper – lower, usually same for all classes)

Example (same table as above):

• Total N = 19, so N/2 = 9.5.
• Cumulative frequencies: 4, 11, 16, 19.
• 9.5 is first exceeded by 11, so the median class is 20‑29.
• L = 20, CF_before = 4, f = 7, h = 10.

Median = 20 + [(9.5 – 4) / 7] × 10 = 20 + (5.5/7) × 10 ≈ 20 + 7.86 ≈ 27.9

Mode (most frequent value) for grouped data

The mode is the value that appears most often. In grouped data we look for the modal class – the class with the highest frequency.

Steps to find the mode:

• Identify the class with the largest frequency.
• Use the formula: Mode = L + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h, where:
• L = lower limit of modal class
• f₁ = frequency of modal class
• f₀ = frequency of the class before modal class (if none, use 0)
• f₂ = frequency of the class after modal class (if none, use 0)
• h = class width

Example (same table):

• Modal class = 20‑29 (frequency 7).
• f₁ = 7, f₀ = 4 (class 10‑19), f₂ = 5 (class 30‑39), L = 20, h = 10.

Mode = 20 + [(7 – 4) / (2×7 – 4 – 5)] × 10 = 20 + (3 / (14 – 9)) × 10 = 20 + (3/5) × 10 = 20 + 6 = 26

Quick comparison of formulas

📝 Likely Exam Questions

• Given the grouped data below, calculate the mean, median and mode.
  Answer: Show step‑by‑step as in the examples, arriving at Mean ≈ 28.2, Median ≈ 27.9, Mode = 26.
• Explain why we use the class midpoint when finding the mean of grouped data.
  Answer: The midpoint represents every value in that class, allowing us to treat the whole class as a single number for averaging.
• In a frequency table, the cumulative frequency just before the median class is 22 and the total frequency is 50. The median class is 30‑39 with width 10 and frequency 15. Find the median.
  Answer: N/2 = 25, Median = 30 + [(25‑22)/15]×10 = 30 + (3/15)×10 = 32.
• For the same table, the modal class is 30‑39 with frequencies of the preceding and succeeding classes 12 and 9 respectively. Find the mode.
  Answer: L = 30, f₁ = 15, f₀ = 12, f₂ = 9, h = 10 → Mode = 30 + [(15‑12)/(2·15‑12‑9)]×10 = 30 + (3/9)×10 = 33.3.
• State one advantage of using grouped data over raw data when dealing with large samples.
  Answer: It simplifies calculations and presents the distribution clearly, saving time and space.