Statistics – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

For grouped data we find the mean, mode and median of frequency distributions, and use the relation between the three.

Mean of grouped data

• Direct method: x̄ = Σfᵢxᵢ / Σfᵢ, where xᵢ is the class mark.
• Assumed mean method: x̄ = a + Σfᵢdᵢ / Σfᵢ with dᵢ = xᵢ − a.
• Step deviation: x̄ = a + h × Σfᵢuᵢ / Σfᵢ with uᵢ = (xᵢ − a)/h.

Mode and median

• Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h (modal class has the highest frequency).
• Median = l + [(n/2 − cf)/f] × h (median class contains the n/2-th observation).
• Empirical relation: 3 Median = Mode + 2 Mean.

Key terms

Class mark

(upper limit + lower limit)/2

Modal class

the class with the highest frequency

Cumulative frequency

running total of frequencies

Formulas

• Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
• Median = l + [(n/2 − cf)/f] × h
• 3 Median = Mode + 2 Mean

Important questions with answers

Quick revision

• Mean uses class marks.
• Modal class = highest frequency; median class = where cf first reaches n/2.
• Mode = 3 Median − 2 Mean.

Exam tips

• In the median formula, cf is the cumulative frequency of the class BEFORE the median class.
• Make classes continuous (e.g. 9.5–19.5) before using formulas if they are given as 10–19, 20–29.

Practice MCQs

1. Class mark of 10–20 is

   1. 10
   2. 15
   3. 20
   4. 30
   Show answer

   Answer: (b) 15. (10 + 20)/2 = 15.

2. The class with the highest frequency is called the

   1. median class
   2. mean class
   3. modal class
   4. first class
   Show answer

   Answer: (c) modal class. Mode is the most frequent value.

3. If mean = 20 and median = 18, then mode =

   1. 14
   2. 16
   3. 22
   4. 24
   Show answer

   Answer: (a) 14. 3 × 18 − 2 × 20 = 14.

4. Mean of 2, 4, 6, 8, 10 with frequencies 1, 2, 3, 2, 1 is

   1. 7
   2. 5
   3. 6
   4. 6.5
   Show answer

   Answer: (c) 6. Σfx = 2 + 8 + 18 + 16 + 10 = 54; Σf = 9; mean = 6.

5. In the median formula, cf stands for

   1. cumulative frequency of the class before the median class
   2. frequency of the median class
   3. total frequency
   4. class width
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   Answer: (a) cumulative frequency of the class before the median class. Standard notation.

6. Which measure is found graphically from the point where the two ogives meet?

   1. mean
   2. mode
   3. median
   4. range
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   Answer: (c) median. The x-coordinate of the meeting point is the median.

7. For a distribution, l = 40, f₁ = 12, f₀ = 8, f₂ = 6, h = 10. Mode =

   1. 45
   2. 44
   3. 46
   4. 42
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   Answer: (b) 44. 40 + (4/10) × 10 = 44.

8. n = 50, median class 30–40, cf before it 22, its frequency 10. Median =

   1. 30
   2. 32
   3. 35
   4. 33
   Show answer

   Answer: (d) 33. 30 + [(25 − 22)/10] × 10 = 33.