Statistics: Making Sense of Grouped Data

Understand mean, median, and mode for grouped data with clear formulas and worked examples tailored to the Class 10 Statistics syllabus.

By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026

Part of Class 10 (CBSE) prep

🧠 The cricket scorecard

When a commentator says a batsman 'averages 52', they have squeezed an entire career of scores into one number that captures typical performance. Statistics is exactly this skill: taking a big, messy pile of numbers - marks, heights, incomes - and summarising them with a single representative value so we can compare and decide. Mean, median, and mode are three different ways to find that 'typical' value.

In Class 10, data is usually 'grouped' - arranged in class intervals like 10-20, 20-30, and so on, each with a frequency (how many items fall in it). We learn to find the three measures of central tendency for such grouped data.

The three measures of central tendency

mean ABCDE mean = sum of values ÷ how many
The mean is the balancing height — the level the bars would settle to if levelled out.
MeasureWhat it tells you
MeanThe arithmetic average - total shared equally
MedianThe middle value when data is arranged in order
ModeThe value (or class) that occurs most often
Mean (direct method): Mean = (sum of fi xi) / (sum of fi), where xi is the class mark (midpoint) and fi is the frequency of each class.
Median = L + [ ((n/2) - cf) / f ] x h, where L = lower limit of median class, n = total frequency, cf = cumulative frequency before the median class, f = frequency of median class, h = class size.
Mode = L + [ (f1 - f0) / (2 f1 - f0 - f2) ] x h, where L = lower limit of modal class, f1 = frequency of modal class, f0 = frequency of the class before it, f2 = frequency of the class after it, h = class size.
ℹ️ There is a neat empirical relationship linking all three: Mode = 3 Median - 2 Mean. If a question gives you any two of them, you can estimate the third with this formula.

📝 Finding the mean of grouped data

Marks 0-10 have 4 students, 10-20 have 6, 20-30 have 10.

Class marks (midpoints) xi are 5, 15, 25 respectively.

fi xi products: 4 x 5 = 20, 6 x 15 = 90, 10 x 25 = 250.

Sum of fi xi = 20 + 90 + 250 = 360.

Sum of fi = 4 + 6 + 10 = 20.

Mean = 360 / 20 = 18 marks.

How to find the median class

  1. 1Make a cumulative frequency (cf) column, adding frequencies down the list.
  2. 2Compute n/2, where n is the total frequency.
  3. 3The median class is the first class whose cumulative frequency is greater than or equal to n/2.
  4. 4Read off L, cf (of the class before), f and h, then apply the median formula.
⚠️ For the median you use n/2 (not (n+1)/2) because grouped data is treated as continuous. Also make sure your classes are continuous - if they are like 0-9, 10-19, first convert them to boundaries like -0.5 to 9.5, 9.5 to 19.5 (subtract 0.5 from each lower limit and add 0.5 to each upper limit) before using the formulas.

Quick revision

  • ✓Class mark = (lower limit + upper limit) / 2.
  • ✓Mean uses class marks weighted by frequency (direct, assumed-mean, or step-deviation methods).
  • ✓Median needs a cumulative frequency table and the n/2 position.
  • ✓Mode uses the modal class - the one with the highest frequency.
  • ✓Empirical relation: Mode = 3 Median - 2 Mean.

⚡ Quick check

If the mean of a distribution is 24 and the median is 26, what is the mode using the empirical relationship?

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