Statistics Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 13 notes: mean of grouped data by direct, assumed-mean and step-deviation methods, and the mode of grouped data.
Statistics — Class 10 Maths Notes
Chapter Snapshot
This chapter finds the average of grouped data in two ways: the mean (by three methods — direct, assumed-mean, step-deviation) and the mode (the most frequent value, via the modal-class formula). These summarise a whole frequency table in a single representative number.
Board relevance: a routine Statistics scorer. Expect a mean calculation and a mode calculation from a frequency table. Set up the table columns neatly — most marks are for correct working.
Syllabus note (rationalised): the median of grouped data and the ogive (cumulative frequency graph) have been removed. Focus on the mean and mode.
Key Concepts & Definitions
Grouped data — data organised into class intervals (e.g. 0–10, 10–20) each with a frequency fᵢ (how many values fall in it).
Class mark (xᵢ) — the midpoint of a class = (upper limit + lower limit)/2. It represents that class in calculations.
Class size (h) — the width of a class interval (upper − lower limit).
Formulas
Mean of grouped data (three methods)
Method Formula Notes
Direct x̄ = Σfᵢxᵢ / Σfᵢ xᵢ = class mark
Assumed mean x̄ = a + (Σfᵢdᵢ / Σfᵢ) a = assumed mean, dᵢ = xᵢ − a
Step deviation x̄ = a + h(Σfᵢuᵢ / Σfᵢ) uᵢ = (xᵢ − a)/h
All three give the same mean. Use step-deviation when the class marks are large — the arithmetic is simplest. Choose a as a class mark near the middle, usually the modal class's mark.
Mode of grouped data
Mode = l + [ (f₁ − f₀) / (2f₁ − f₀ − f₂) ] × h
- l = lower limit of the modal class (the class with the highest frequency).
- f₁ = frequency of the modal class.
- f₀ = frequency of the class before the modal class.
- f₂ = frequency of the class after the modal class.
- h = class size.
Worked Examples
Example 1 — Mean (direct): Find the mean of: classes 0–10, 10–20, 20–30 with frequencies 5, 8, 7.
Class marks: 5, 15, 25. Σfᵢxᵢ = 5(5) + 8(15) + 7(25) = 25 + 120 + 175 = 320. Σfᵢ = 20.
Mean = 320/20 = 16.
Example 2 — Mean (step-deviation): Same data, a = 15, h = 10.
uᵢ = (xᵢ − 15)/10 = −1, 0, 1. Σfᵢuᵢ = 5(−1) + 8(0) + 7(1) = 2. Σfᵢ = 20.
Mean = a + h(Σfᵢuᵢ/Σfᵢ) = 15 + 10(2/20) = 15 + 1 = 16 ✓ (same answer).
Example 3 — Mode: Find the mode of: 0–10 (f=6), 10–20 (f=10), 20–30 (f=8).
Highest frequency 10 → modal class 10–20. So l = 10, f₁ = 10, f₀ = 6, f₂ = 8, h = 10.
Mode = 10 + [(10 − 6)/(2·10 − 6 − 8)]×10 = 10 + (4/6)×10 = 10 + 6.67 = 16.67.
Example 4 — Choosing the method: For classes with marks 500, 1500, 2500, …, the assumed-mean or step-deviation method is far easier than the direct method, because subtracting a and dividing by h turns big numbers into small integers (−2, −1, 0, 1, 2).
Example 5 — Missing frequency: The mean of the following is 18. Find the missing frequency f: classes 11–13, 13–15, 15–17, 17–19, 19–21, 21–23, 23–25 with frequencies 3, 6, 9, 13, f, 5, 4.
Class marks: 12, 14, 16, 18, 20, 22, 24. Σf = 40 + f. Σfx = 36 + 84 + 144 + 234 + 20f + 110 + 96 = 704 + 20f.
Mean = (704 + 20f)/(40 + f) = 18 → 704 + 20f = 720 + 18f → 2f = 16 → f = 8. This "given the mean, find the missing frequency" set-up is a very common 3-mark question.
Example 6 — Reading a modal class: In a distribution, the frequencies rise, peak at one class, then fall. The class where the frequency is highest is the modal class; you then read off f₀ (the class just before) and f₂ (the class just after) to plug into the mode formula. If two classes tie for the highest frequency, the data is bimodal and the simple formula does not apply — but board questions always give a single clear modal class.
Important Question Patterns
1. Mean, direct method (2–3 marks): build the fᵢxᵢ column, sum, divide by Σfᵢ.
2. Mean, step-deviation (3 marks): choose a and h, build the uᵢ and fᵢuᵢ columns, apply the formula — asked especially when class marks are large.
3. Missing frequency (3 marks): given the mean, set up the mean equation with the unknown frequency and solve.
4. Mode (3 marks): identify the modal class (highest frequency), then apply the mode formula with the right f₀, f₁, f₂.
5. Interpret (1–2 marks): which average is most suitable; find the class mark; convert a "less than" table into class intervals.
⚡ Quick Revision
- Class mark = (upper + lower)/2; class size h = width.
- Mean — direct: x̄ = Σfᵢxᵢ/Σfᵢ. Assumed mean: a + Σfᵢdᵢ/Σfᵢ (dᵢ = xᵢ − a). Step deviation: a + h·(Σfᵢuᵢ/Σfᵢ) (uᵢ = (xᵢ − a)/h). All give the same mean.
- Use step-deviation when class marks are large (simplest arithmetic).
- Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]·h; modal class = highest frequency; f₀ before, f₂ after.
- Missing frequency: plug into the mean formula and solve for the unknown.
- Removed from syllabus: median of grouped data and the ogive.
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