Class 10 - Chapter 13: Statistics

Exercise 13.2

Q1. The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and the mean of the data. Compare and interpret the two measures of central tendency.

Mean

Mean = Σfx / Σf

= 2830 / 80

= 35.375

≈ 35.38 years

Mode

Modal Class = 35-45

l = 35, h = 10

f₁ = 23, f₀ = 21, f₂ = 14

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

= 35 + [(23 − 21)/(46 − 21 − 14)] × 10

= 35 + (2/11) × 10

= 35 + 1.82

= 36.82 years

Mean = 35.38 years

Mode = 36.82 years

Interpretation: Most patients admitted were around 37 years of age. The mean age is about 35 years. Both values are close, indicating that the data is fairly consistent.

Q2. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components. Determine the modal lifetime of the components.

Modal Class = 60-80

l = 60, h = 20

f₁ = 61, f₀ = 52, f₂ = 38

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

= 60 + [(61 − 52)/(122 − 52 − 38)] × 20

= 60 + (9/32) × 20

= 60 + 5.625

= 65.625

≈ 65.63 hours

Modal lifetime = 65.63 hours

Q3. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure.

Mode

Modal Class = 1500-2000

l = 1500, h = 500

f₁ = 40, f₀ = 24, f₂ = 33

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

= 1500 + [(40 − 24)/(80 − 24 − 33)] × 500

= 1500 + (16/23) × 500

= 1500 + 347.83

= 1847.83

Modal Monthly Expenditure ≈ ₹1847.83

Mean

Mean = Σfx / Σf

= 532500 / 200

= 2662.5

Mean Monthly Expenditure = ₹2662.50

Q4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.

Mean

Mean = Σfx / Σf

= 1022.5 / 35

= 29.21

Mean = 29.21 students per teacher

Mode

Modal Class = 30-35

l = 30, h = 5

f₁ = 10, f₀ = 9, f₂ = 3

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

= 30 + [(10 − 9)/(20 − 9 − 3)] × 5

= 30 + (1/8) × 5

= 30.625

≈ 30.63

Mode = 30.63 students per teacher

Interpretation: The most common teacher-student ratio among the states is about 31 students per teacher, while the average ratio is about 29 students per teacher.

Q5. The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches. Find the mode of the data.

Modal Class = 4000-5000

l = 4000, h = 1000

f₁ = 18, f₀ = 4, f₂ = 9

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

= 4000 + [(18 − 4)/(36 − 4 − 9)] × 1000

= 4000 + (14/23) × 1000

= 4000 + 608.70

= 4608.70

Mode ≈ 4608.7 runs

Q6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data.

Modal Class = 40-50

l = 40, h = 10

f₁ = 20, f₀ = 12, f₂ = 11

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

= 40 + [(20 − 12)/(40 − 12 − 11)] × 10

= 40 + (8/17) × 10

= 40 + 4.71

= 44.71

Mode ≈ 44.71 cars