Statistics – Class 10 Maths Chapter 13 Notes & NCERT PDF

Introduction

In Class IX you found the mean, median and mode of ungrouped data. Real data — marks of a whole school, wages of workers, heights of students — is usually large, so it is grouped into class intervals. This chapter extends the three measures of central tendency to grouped data and introduces cumulative frequency.

Key Concepts

1. Mean of grouped data (Section 13.2)

Each class is represented by its class mark (mid-point): class mark = (upper limit + lower limit) ÷ 2. The book assumes the frequency of each class is centred at its class mark.

Three methods — all give the same mean:

The value of the mean does not depend on the choice of a. The assumed mean and step-deviation methods are just simplified forms of the direct method.

Grouped mean is an approximation: the 30 students' marks give an exact mean of 59.3, but the grouped table gives 62, because of the mid-point assumption.

2. Mode of grouped data (Section 13.3)

The class with the maximum frequency is the modal class. The mode lies inside it:

Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h

where l = lower limit of the modal class, h = class size, f₁ = frequency of the modal class, f₀ = frequency of the class before it, f₂ = frequency of the class after it.

3. Cumulative frequency (Section 13.4)

The cumulative frequency of a class is the sum of the frequencies of all classes up to it. A less than type table uses upper limits (less than 10, less than 20, …); a more than type table uses lower limits (more than or equal to 0, 10, …).

4. Median of grouped data (Section 13.4)

Find n ÷ 2. The median class is the class whose cumulative frequency is greater than (and nearest to) n ÷ 2. Then:

Median = l + [(n/2 − cf) ÷ f] × h

where l = lower limit of the median class, cf = cumulative frequency of the class preceding it, f = frequency of the median class, h = class size.

5. Which measure to use?

Methods Step by Step

Mean (step-deviation): make columns xᵢ, dᵢ = xᵢ − a, uᵢ = dᵢ ÷ h, fᵢuᵢ → add Σfᵢ and Σfᵢuᵢ → substitute.

Mode: find the largest frequency → note l, h, f₁, f₀, f₂ → substitute.

Median: add a cumulative-frequency column → find n ÷ 2 → locate the median class → note l, cf, f, h → substitute.

Make sure classes are continuous before using the mode or median formula (for example, 118–126, 127–135 become 117.5–126.5, 126.5–135.5).

Worked Examples (from the chapter)

Marks of 30 students (classes 10–25, 25–40, …, 85–100 with frequencies 2, 3, 7, 6, 6, 6):

Female teachers (7 classes from 15–25 to 75–85, frequencies 6, 11, 7, 4, 4, 2, 1): Σfᵢ = 35, Σfᵢxᵢ = 1390, so x̄ = 39.71 by all three methods.

Family size (1–3, 3–5, 5–7, 7–9, 9–11 with frequencies 7, 8, 2, 2, 1): Mode = 3 + [1 ÷ (16 − 7 − 2)] × 2 ≈ 3.286.

Marks of 53 students (0–10 to 90–100): n ÷ 2 = 26.5, median class 60–70 (cf 29), l = 60, cf = 22, f = 7, h = 10 → Median = 60 + (4.5 ÷ 7) × 10 ≈ 66.4.

Heights of 51 girls (less-than table): n ÷ 2 = 25.5, median class 145–150, l = 145, cf = 11, f = 18, h = 5 → Median ≈ 149.03 cm.

Missing frequencies (median 525, total 100): cumulative frequency before 500–600 is 36 + x, so 525 = 500 + [(50 − 36 − x) ÷ 20] × 100 → x = 9, and x + y = 24 → y = 15.

Board Exam Focus

Formula Sheet

Common Mistakes