Statistics | Exercise 13.2

Question 5

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.

Question diagram 1
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Solution
Understand the Question
  • The mode of grouped data represents the value around which the maximum concentration of data points occurs.
  • To find the mode:
    1. Identify the modal class, which is the class interval with the highest frequency.
    2. Apply 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 where:
      • ll = lower limit of the modal class
      • hh = size of the class interval
      • f1f_1 = frequency of the modal class
      • f0f_0 = frequency of the class preceding the modal class
      • f2f_2 = frequency of the class succeeding the modal class

Step 1 · Identify the Modal Class and Parameters

From the given distribution, the maximum frequency is 1818, which lies in the class interval 4000−50004000 - 5000.Diagram 1

Therefore, the modal class is 4000−50004000 - 5000.

  • Lower limit of modal class, l=4000l = 4000
  • Frequency of modal class, f1=18f_1 = 18
  • Frequency of preceding class, f0=4f_0 = 4
  • Frequency of succeeding class, f2=9f_2 = 9
  • Class size, h=5000−4000=1000h = 5000 - 4000 = 1000

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

Substituting the values:

Mode=4000+[18−42(18)−4−9]×1000=4000+(1436−13)×1000=4000+(1423)×1000=4000+1400023=4000+608.69565…≈4608.7\begin{aligned} \text{Mode} &= 4000 + \left[ \dfrac{18 - 4}{2(18) - 4 - 9} \right] \times 1000 \\[0.6em] &= 4000 + \left( \dfrac{14}{36 - 13} \right) \times 1000 \\[0.6em] &= 4000 + \left( \dfrac{14}{23} \right) \times 1000 \\[0.6em] &= 4000 + \dfrac{14000}{23} \\[0.6em] &= 4000 + 608.69565\dots \\[0.6em] &\approx 4608.7 \end{aligned}
Answer

4608.7 runs4608.7\text{ runs}

Common Mistakes
  • Confusing Frequencies: Misidentifying f0f_0 and f2f_2. Always remember f0f_0 is the frequency before the modal class, f1f_1 is the modal class frequency, and f2f_2 is the frequency after.
  • Denominator Order of Operations: In evaluating 2f1−f0−f22f_1 - f_0 - f_2, subtracting before multiplying 2f12f_1 leads to incorrect results.
  • Rounding Off Errors: Truncating 1400023\dfrac{14000}{23} prematurely instead of dividing to at least two decimal places (608.70608.70) before rounding to one decimal place (608.7608.7).

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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