Chapter 14: Statistics
Key Formulas:
• Mean (direct): Σf₁x₁/Σf₁
• Mean (assumed mean): a + (Σf₁d₁/Σf₁) where d₁=x₁-a
• Mean (step deviation): a + (Σf₁u₁/Σf₁) × h where u₁=(x₁-a)/h
• Median = l + ((n/2-cf)/f) × h
• Mode = l + ((f₁-f₂)/(2f₁-f₂-f₃)) × h
Exercise 14.1
Q1. Find mean of: 2, 3, 5, 7, 8
SolutionMean = (2+3+5+7+8)/5 = 25/5 = 5
Q2. Find mean of first five multiples of 3.
SolutionMultiples: 3, 6, 9, 12, 15
Mean = (3+6+9+12+15)/5 = 45/5 = 9
Exercise 14.2
Q1. Find mode and mean of: 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 33, 17, 16, 21, 21
SolutionMean = Sum/15
= 14×3+17×2+18×2+21×2+25+28+23+22+33+16
=(42+34+36+42+25+28+23+22+33+16)/15
= 301/15 ≈ 20.07
Mode: 14 occurs most frequently (3 times)
Mode = 14
Exercise 14.3
Q1. Find median of: 2, 3, 4, 4, 5, 5, 5, 6, 6, 7, 8, 9, 10
Solutionn=13 (odd)
Median = ((13+1)/2)th term = 7th term = 5
Q3. Find median from grouped data using formula: l+((n/2-cf)/f)×h
SolutionFor grouped data median:
Median class = class where n/2 lies
Median = l + ((n/2 - cf)/f) × h
where l = lower limit of median class
cf = cumulative frequency of preceding class
f = frequency of median class
h = class size
Exercise 14.4
Q1. Draw ogive for given distribution.
SolutionSteps to draw less than ogive:
1. Mark upper class limits on x-axis
2. Mark cumulative frequencies on y-axis
3. Plot points (upper limit, cf)
4. Join points with smooth curve
Steps for more than ogive:
1. Mark lower class limits on x-axis
2. Mark cumulative frequencies (from top) on y-axis
3. Plot points (lower limit, cf)
4. Join with smooth curve
Median = x-value corresponding to n/2 on ogive