Question 15 of 21intermediate🔧 ApplyNumerical3 marks

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:

Number of cars0 - 1010 - 2020 - 3030 - 4040 - 5050 - 6060 - 7070 - 80
Frequency71413122011158
Correct Answer

Modal Class: The maximum frequency is 20, which lies in the class interval 40-50. Therefore, the modal class is 40-50.

Values: Lower limit (ll) = 40 Class size (hh) = 10 Frequency of modal class (f1f_1) = 20 Frequency of preceding class (f0f_0) = 12 Frequency of succeeding class (f2f_2) = 11

Calculation: Mode =l+(f1−f02f1−f0−f2)×h= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h Mode =40+(20−122(20)−12−11)×10= 40 + \left( \frac{20 - 12}{2(20) - 12 - 11} \right) \times 10 Mode =40+(840−23)×10= 40 + \left( \frac{8}{40 - 23} \right) \times 10 Mode =40+(817)×10= 40 + \left( \frac{8}{17} \right) \times 10 Mode =40+8017= 40 + \frac{80}{17} Mode =40+4.71= 40 + 4.71 Mode =44.71= 44.71

Therefore, the mode of the data is 44.71.

Exercise: EXERCISE 13.2 | Q: 6 | (Page 17)
For More Understanding

Explanation

To find the mode of grouped data, we first identify the modal class, which is the class interval with the highest frequency. In this table, the highest frequency is 20, corresponding to the class 40-50. We then apply the formula: Mode =l+[(f1−f0)/(2f1−f0−f2)]×h= l + [(f_1 - f_0) / (2f_1 - f_0 - f_2)] \times h. Here, ll is the lower limit of the modal class, hh is the class width, f1f_1 is the frequency of the modal class, f0f_0 is the frequency of the class preceding the modal class, and f2f_2 is the frequency of the class succeeding the modal class.

Solution Steps

  1. Step 1: Identify the modal class. The highest frequency is 20, so the modal class is 40-50.

  2. Step 2: Identify the parameters: Lower limit (ll) = 40, class size (hh) = 10, frequency of modal class (f1f_1) = 20, frequency of preceding class (f0f_0) = 12, frequency of succeeding class (f2f_2) = 11.

  3. Step 3: Substitute the values into the mode formula: Mode =40+[(20−12)/(2(20)−12−11)]×10= 40 + [(20 - 12) / (2(20) - 12 - 11)] \times 10.

  4. Step 4: Simplify the expression: Mode =40+(8/17)×10=40+80/17= 40 + (8 / 17) \times 10 = 40 + 80/17.

  5. Step 5: Calculate the final value: Mode =40+4.71=44.71= 40 + 4.71 = 44.71.

PreviousNext