Statistics — Exercise 13.2

Use the direct method: represent each class by its midpoint xᵢ, multiply by fᵢ, and divide the product total by the frequency total. These are grouped-data estimates.

\[\bar{x}=\frac{\sum f_i x_i}{\sum f_i}=\frac{2830}{80}\approx 35.375\]

Mean ≈ 35.375 years.

The modal class is 35–45, with largest frequency 23. Use l=35, h=10, f₁=23, f₀=21, f₂=14.

\[\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}h=35+\frac{2}{11}\times 10\approx 36.818182\]

Modal estimate ≈ 36.818182 years.

The modal age is around 36.82 years in the most frequent class 35–45; the mean around 35.38 averages all ages.

The modal class is 60–80, with largest frequency 61. Use l=60, h=20, f₁=61, f₀=52, f₂=38.

\[\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}h=60+\frac{9}{32}\times 20\approx 65.625\]

Modal estimate ≈ 65.625 hours.

Use the direct method: represent each class by its midpoint xᵢ, multiply by fᵢ, and divide the product total by the frequency total. These are grouped-data estimates.

\[\bar{x}=\frac{\sum f_i x_i}{\sum f_i}=\frac{532500}{200}\approx 2662.5\]

Mean ≈ 2662.5 rupees per month.

The modal class is 1500–2000, with largest frequency 40. Use l=1500, h=500, f₁=40, f₀=24, f₂=33.

\[\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}h=1500+\frac{16}{23}\times 500\approx 1847.826087\]

Modal estimate ≈ 1847.826087 rupees per month.

The most frequent spending class is ₹1500–2000; the mean also reflects families spending larger amounts.

Use the direct method: represent each class by its midpoint xᵢ, multiply by fᵢ, and divide the product total by the frequency total. These are grouped-data estimates.

\[\bar{x}=\frac{\sum f_i x_i}{\sum f_i}=\frac{1022.5}{35}\approx 29.214286\]

Mean ≈ 29.214286 students per teacher.

The modal class is 30–35, with largest frequency 10. Use l=30, h=5, f₁=10, f₀=9, f₂=3.

\[\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}h=30+\frac{1}{8}\times 5\approx 30.625\]

Modal estimate ≈ 30.625 students per teacher.

The modal estimate describes the most frequent class of states/UTs, while the mean averages the grouped state ratios; this is supplied exercise data, not a current national statistic.

The modal class is 4000–5000, with largest frequency 18. Use l=4000, h=1000, f₁=18, f₀=4, f₂=9.

\[\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}h=4000+\frac{14}{23}\times 1000\approx 4608.695652\]

Modal estimate ≈ 4608.695652 runs.

The modal class is 40–50, with largest frequency 20. Use l=40, h=10, f₁=20, f₀=12, f₂=11.

\[\text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}h=40+\frac{8}{17}\times 10\approx 44.705882\]

Modal estimate ≈ 44.705882 cars per three minutes.