CBSE Class 10 · Maths · board exam 2027
Chapter 13: Statistics — notes
The chapter’s 8 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. First, a quick recall: mean, median, mode
Mean, median and mode are called the measures of central tendency.
- • Measures of central tendency: mean, median, mode
- • Mean = (sum of observations)/(number of observations)
- • 4, 6, 9, 3, 8
- • Sum = 4 + 6 + 9 + 3 + 8 = 30
- • Mean = 30/5 = 6
- • Median: arrange in order first
- • 7, 2, 9, 4, 5 → 2, 4, 5, 7, 9
- • n = 5 (odd) ⇒ ((n + 1)/2)th = 3rd value
- • Median = 5
- • n even ⇒ median = average of the (n/2)th and (n/2 + 1)th values
- • 3, 5, 6, 8, 10, 12: 3rd = 6, 4th = 8
- • Median = (6 + 8)/2 = 14/2 = 7
- • Mode = the value that occurs most often
- • 3, 5, 3, 6, 5, 3, 7: 3 → 3 times, 5 → 2 times
- • Mode = 3
- • Range = largest − smallest = 7 − 3 = 4 (spread, not centre)
2. Grouped data and the class mark
In grouped data we do not see each value.
- • Grouped data: classes and their frequencies fᵢ
- • Class 10-25: lower limit = 10, upper limit = 25
- • Class size h = 25 − 10 = 15
- • xᵢ = (lower limit + upper limit)/2
- • Class mark of 10-25 = (10 + 25)/2 = 35/2 = 17.5
- • Classes with gaps: 118-126, 127-135, …
- • Gap = 127 − 126 = 1, half = 0.5
- • 118-126 → 117.5-126.5, 127-135 → 126.5-135.5
- • Mid-values 115, 125, 135, … ⇒ h = 125 − 115 = 10
- • 115 − 5 = 110, 115 + 5 = 120
- • Class: 110-120
3. Mean: the direct method
The mean of grouped data.
- • Marks: 10-25, 25-40, 40-55, 55-70, 70-85, 85-100
- • Students (fᵢ): 2, 3, 7, 6, 6, 6
- • x̄ = Σfᵢxᵢ/Σfᵢ
- • | Class | fᵢ | xᵢ | fᵢxᵢ |
- • | 10-25 | 2 | 35 ÷ 2 = 17.5 | 2 × 17.5 = 35 |
- • | 25-40 | 3 | 65 ÷ 2 = 32.5 | 3 × 32.5 = 97.5 |
- • | 40-55 | 7 | 95 ÷ 2 = 47.5 | 7 × 47.5 = 332.5 |
- • | 55-70 | 6 | 125 ÷ 2 = 62.5 | 6 × 62.5 = 375 |
- • | 70-85 | 6 | 155 ÷ 2 = 77.5 | 6 × 77.5 = 465 |
- • | 85-100 | 6 | 185 ÷ 2 = 92.5 | 6 × 92.5 = 555 |
- • Σfᵢ = 2 + 3 + 7 + 6 + 6 + 6 = 30
- • Σfᵢxᵢ = 35 + 97.5 + 332.5 + 375 + 465 + 555 = 1860
- • x̄ = 1860/30 = 186/3 = 62
- • Mean = 62 marks
4. Mean: the assumed mean method
Multiplying 7 by 47.5 and adding big numbers is tiring.
- • Assumed mean method: x̄ = a + Σfᵢdᵢ/Σfᵢ
- • dᵢ = xᵢ − a
- • a = 47.5
- • | xᵢ | fᵢ | dᵢ = xᵢ − 47.5 | fᵢdᵢ |
- • | 17.5 | 2 | 17.5 − 47.5 = −30 | 2 × (−30) = −60 |
- • | 32.5 | 3 | 32.5 − 47.5 = −15 | 3 × (−15) = −45 |
- • | 47.5 | 7 | 47.5 − 47.5 = 0 | 7 × 0 = 0 |
- • | 62.5 | 6 | 62.5 − 47.5 = 15 | 6 × 15 = 90 |
- • | 77.5 | 6 | 77.5 − 47.5 = 30 | 6 × 30 = 180 |
- • | 92.5 | 6 | 92.5 − 47.5 = 45 | 6 × 45 = 270 |
- • Σfᵢdᵢ = (−60 − 45) + (0 + 90 + 180 + 270)
- • = −105 + 540 = 435
- • 435/30 = 14.5
- • x̄ = 47.5 + 14.5 = 62
- • Direct method = assumed mean method = 62
- • Σfᵢ(xᵢ − x̄) = 0 (always)
5. Mean: the step deviation method
Look at the deviations: minus 30, minus 15, 0, 15, 30, 45.
- • uᵢ = (xᵢ − a)/h
- • x̄ = a + h × Σfᵢuᵢ/Σfᵢ
- • | xᵢ | fᵢ | uᵢ = dᵢ ÷ 15 | fᵢuᵢ |
- • | 17.5 | 2 | −30 ÷ 15 = −2 | 2 × (−2) = −4 |
- • | 32.5 | 3 | −15 ÷ 15 = −1 | 3 × (−1) = −3 |
- • | 47.5 | 7 | 0 ÷ 15 = 0 | 7 × 0 = 0 |
- • | 62.5 | 6 | 15 ÷ 15 = 1 | 6 × 1 = 6 |
- • | 77.5 | 6 | 30 ÷ 15 = 2 | 6 × 2 = 12 |
- • | 92.5 | 6 | 45 ÷ 15 = 3 | 6 × 3 = 18 |
- • Σfᵢuᵢ = (−4 − 3) + (0 + 6 + 12 + 18) = −7 + 36 = 29
- • x̄ = 47.5 + 15 × 29/30
- • 15 × 29/30 = 29/2 = 14.5
- • x̄ = 47.5 + 14.5 = 62
- • Direct = assumed mean = step deviation = 62
- • Σfᵢuᵢ/Σfᵢ = ū ⇒ x̄ = a + h × ū
6. Mode of grouped data
The mode is the value that occurs most often.
- • Mode = l + ((f₁ − f₀)/(2f₁ − f₀ − f₂)) × h
- • l = lower limit of the modal class, h = class size
- • f₁ = frequency of the modal class
- • f₀ = class before it, f₂ = class after it
- • Frequencies: 2, 3, 7, 6, 6, 6 ⇒ modal class = 40-55
- • l = 40, h = 15, f₁ = 7, f₀ = 3, f₂ = 6
- • f₁ − f₀ = 7 − 3 = 4
- • 2f₁ − f₀ − f₂ = 14 − 3 − 6 = 5
- • Mode = 40 + 4/5 × 15 = 40 + 12 = 52
- • Mode = 52, mean = 62
7. Cumulative frequency and the median
For the median we need one new idea, the cumulative frequency, cf.
- • cf = all frequencies added up to this class
- • | Class | f | cf |
- • | 0-10 | 5 | 5 |
- • | 10-20 | 3 | 5 + 3 = 8 |
- • | 20-30 | 4 | 8 + 4 = 12 |
- • | 30-40 | 3 | 12 + 3 = 15 |
- • | 40-50 | 3 | 15 + 3 = 18 |
- • | 50-60 | 4 | 18 + 4 = 22 |
- • | 60-70 | 7 | 22 + 7 = 29 |
- • | 70-80 | 9 | 29 + 9 = 38 |
- • | 80-90 | 7 | 38 + 7 = 45 |
- • | 90-100 | 8 | 45 + 8 = 53 |
- • n = 53, n/2 = 53/2 = 26.5
- • First cf greater than 26.5 = 29 ⇒ median class = 60-70
- • Median = l + ((n/2 − cf)/f) × h
- • l = 60, cf = 22, f = 7, h = 10
- • Median = 60 + ((26.5 − 22)/7) × 10 = 60 + 45/7
- • ≈ 60 + 6.43 ≈ 66.43
- • Less than 10: 5, less than 20: 8, less than 30: 12, …
- • Frequency of 10-20: 8 − 5 = 3, of 20-30: 12 − 8 = 4
8. How the three measures are related, and which to use
There is a useful relation between the three: 3 median equals mode plus 2 mean.
- • 3 Median = Mode + 2 Mean
- • Mean = 30, mode = 24
- • 3 Median = 24 + 2 × 30 = 24 + 60 = 84
- • Median = 84/3 = 28
- • Symmetric distribution: mean = median = mode
- • Mean: uses all observations, pulled by extreme values
- • Median: a typical value when extremes are there (e.g. wages)
- • Mode: the most frequent value (e.g. the most popular TV programme)
Summary
Very good! We found the mean of grouped data in three ways: the direct method, the assumed mean method and the step deviation method. All three use the class mark, and all three give the same answer. Then the mode, from the modal class with l, h, f 1, f 0 and f 2. Then the cumulative frequency, and with it the median class and the median formula. Then the relation 3 median equals mode plus 2 mean. Write the three formulas, for the mean, the mode and the median, in your notebook once, without looking, with the meaning of every letter. Then try the board questions that follow. If anything was not clear, press the Ask button and ask me. Done, you worked very well today!
AskMedha · Medha Ma’am — CBSE Class 10 Maths, Chapter 13. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Statistics" notes as a PDF?
- Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
- Where are these notes written from?
- From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.
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