Statistics | Exercise 13.2

Question 6

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 :

Question diagram 1
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Solution
Understand the Question
  • The mode of grouped data is the value that occurs most frequently and lies within the modal class (the class interval with the highest frequency).
  • Formula for Mode: Mode=l+(f1−f02f1−f0−f2)×h\text{Mode} = l + \left( \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h where:
    • l=l = lower limit of the modal class
    • h=h = size of the class interval
    • f1=f_1 = frequency of the modal class
    • f0=f_0 = frequency of the class preceding the modal class
    • f2=f_2 = frequency of the class succeeding the modal class

Step 1 · Identify the Modal Class and Key Values

From the given frequency distribution table, the maximum frequency is 2020, which corresponds to the class interval 40−5040 - 50.

Therefore, the modal class is 40−5040 - 50.Diagram 1

From the modal class, we identify:

  • Lower limit of modal class (ll) =40= 40
  • Frequency of modal class (f1f_1) =20= 20
  • Frequency of preceding class (f0f_0) =12= 12
  • Frequency of succeeding class (f2f_2) =11= 11
  • Class size (hh) =50−40=10= 50 - 40 = 10

Step 2 · Calculate the Mode

Using the mode formula: Mode=l+[f1−f02f1−f0−f2]×h\text{Mode} = l + \left[ \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right] \times h

Substitute the values:

Mode=40+[20−122(20)−12−11]×10=40+[840−12−11]×10=40+[828−11]×10=40+[817]×10=40+8017=40+4.7058…≈44.7\begin{aligned} \text{Mode} &= 40 + \left[ \dfrac{20 - 12}{2(20) - 12 - 11} \right] \times 10 \\[0.6em] &= 40 + \left[ \dfrac{8}{40 - 12 - 11} \right] \times 10 \\[0.6em] &= 40 + \left[ \dfrac{8}{28 - 11} \right] \times 10 \\[0.6em] &= 40 + \left[ \dfrac{8}{17} \right] \times 10 \\[0.6em] &= 40 + \dfrac{80}{17} \\[0.6em] &= 40 + 4.7058\dots \\[0.6em] &\approx 44.7 \end{aligned}
Answer

44.7 cars44.7\text{ cars}

Common Mistakes
  • Subscript Confusion: Confusing f0f_0, f1f_1, and f2f_2. Remember: f1f_1 is the modal class frequency, f0f_0 is the previous (preceding) class frequency, and f2f_2 is the next (succeeding) class frequency.
  • Denominator Calculation Error: Forgetting that 2f12f_1 means 2×f1=402 \times f_1 = 40 before subtracting f0f_0 and f2f_2.
  • Mode Range Check: The calculated mode (44.744.7) must always fall inside the modal class interval (40−5040 - 50). If it does not, recheck your calculations.

More questions in Exercise 13.2

Q1
  1. 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 given above. Compare and interpret the two measures of central tendency.

Q2
  1. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components :

Determine the modal lifetimes of the components.

Q3
  1. 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 :
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.

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.

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 :

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