Statistics

Board: GSEB | Class: Std 10

Comprehensive study notes for Statistics by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Mean of Grouped Data

Three methods: Direct method: x̄ = Σfx/Σf. Assumed mean method: x̄ = a + Σfd/Σf. Step-deviation method: x̄ = a + h(Σfu/Σf).

Mode of Grouped Data

Mode = l + ((f&#8321–f&#8320)/(2f&#8321–f&#8320–f&#8322)) × h, where l = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency of class before, f₂ = after.

Median of Grouped Data

Median = l + [(n/2 – cf)/f] × h, where l = lower limit of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class size, n = total frequency.

Empirical Relationship

Mode = 3Median – 2Mean. This relationship helps verify calculations and estimate one measure if the other two are known.

Ogive (Cumulative Frequency Curve)

Less than ogive: plot upper limits vs. cumulative frequency. More than ogive: plot lower limits vs. cumulative frequency. The intersection of both ogives gives the median.

Important Formulas

Direct Mean	x̄ = Σfˡxˡ/Σfˡ
Mode	Mode = l + ((f&#8321-f&#8320)/(2f&#8321-f&#8320-f&#8322))h
Median	Median = l + [(n/2-cf)/f]h
Empirical	Mode = 3Median - 2Mean

Solved Examples

Example 1: Find mean: CI 0-10(2), 10-20(5), 20-30(8), 30-40(3).

Solution: x̄ = Σfx/Σf = (5×2+15×5+25×8+35×3)/(2+5+8+3) = 330/18 = 18.33.

Example 2: Find median: CI 0-10(2), 10-20(5), 20-30(8), 30-40(3), 40-50(2).

Solution: n=20, n/2=10. Median class = 20-30. Median = 20 + [(10-7)/8]10 = 20 + 3.75 = 23.75.

Example 3: If mean=25 and median=26, find mode.

Solution: Mode = 3(26) – 2(25) = 78 – 50 = 28.

Practice Questions

1. Find the mean of: CI 0-20(5), 20-40(12), 40-60(15), 60-80(8), 80-100(5).
2. Find the median: Marks 0-10(6), 10-20(10), 20-30(18), 30-40(22), 40-50(14), 50-60(8).
3. Find the mode: CI 0-10(8), 10-20(12), 20-30(20), 30-40(15), 40-50(5).
4. Draw a less than ogive and find median from it for: CI 0-10(3), 10-20(7), 20-30(12), 30-40(8), 40-50(5).
5. The mean and median of a distribution are 14.2 and 15.1. Find mode.

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