Statistics | Exercise 13.1

Question 4

Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.

Question diagram 1
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Solution
Understand the Question
  • To find the mean of grouped data efficiently when class marks and frequencies involve decimals or large numbers, we can use the step-deviation method.
  • The class mark xix_i for each class interval is the midpoint:
    xi=Upper limit+Lower limit2x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}
  • We choose an assumed mean a=75.5a = 75.5 and determine the class width h=3h = 3.
  • Using the step deviation ui=xi−ahu_i = \dfrac{x_i - a}{h}, the mean is calculated as: xˉ=a+(∑fiui∑fi)×h\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h

Step 1 · Prepare the Frequency Distribution Table

Class mark formula: xi=Upper limit+Lower limit2x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}

Let assumed mean a=75.5a = 75.5 and class size h=3h = 3.

Number of heartbeats per minuteNumber of women (fi)Class mark (xi)di=xi−75.5ui=di3fiui65−68266.5−9−3−668−71469.5−6−2−871−74372.5−3−1−374−77875.500077−80778.531780−83481.562883−86284.5936Total∑fi=30∑fiui=4\begin{array}{|c|c|c|c|c|c|} \hline \text{Number of heartbeats per minute} & \text{Number of women } (f_i) & \text{Class mark } (x_i) & d_i = x_i - 75.5 & u_i = \dfrac{d_i}{3} & f_i u_i \\ \hline 65 - 68 & 2 & 66.5 & -9 & -3 & -6 \\ \hline 68 - 71 & 4 & 69.5 & -6 & -2 & -8 \\ \hline 71 - 74 & 3 & 72.5 & -3 & -1 & -3 \\ \hline 74 - 77 & 8 & 75.5 & 0 & 0 & 0 \\ \hline 77 - 80 & 7 & 78.5 & 3 & 1 & 7 \\ \hline 80 - 83 & 4 & 81.5 & 6 & 2 & 8 \\ \hline 83 - 86 & 2 & 84.5 & 9 & 3 & 6 \\ \hline \text{Total} & \sum f_i = 30 & & & & \sum f_i u_i = 4 \\ \hline \end{array}

From the table: ∑fi=30\sum f_i = 30 ∑fiui=4\sum f_i u_i = 4

Step 2 · Calculate the Mean

Using the step-deviation formula: xˉ=a+(∑fiui∑fi)×h\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h

Substitute the values:

xˉ=75.5+(430)×3=75.5+1230=75.5+0.4=75.9\begin{aligned} \bar{x} &= 75.5 + \left( \dfrac{4}{30} \right) \times 3 \\[0.6em] &= 75.5 + \dfrac{12}{30} \\[0.6em] &= 75.5 + 0.4 \\[0.6em] &= 75.9 \end{aligned}
Answer

75.9 beats per minute75.9\text{ beats per minute}

Common Mistakes
  • Forgetting to multiply by hh: A common error in the step-deviation formula is omitting the multiplication by class size h=3h = 3.
  • Sign errors in uiu_i and fiuif_i u_i: Forgetting negative signs for values of xi<ax_i < a will give an incorrect value for ∑fiui\sum f_i u_i.

More questions in Exercise 13.1

Q1

A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Which method did you use for finding the mean, and why?

Q2

Consider the following distribution of daily wages of 50 workers of a factory.

Find the mean daily wages of the workers of the factory by using an appropriate method.

Q3

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is ₹18. Find the missing frequency ff.

Q4

Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.

Q5

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Q6

The table below shows the daily expenditure on food of 25 households in a locality.

Find the mean daily expenditure on food by a suitable method.

Q7

To find out the concentration of SO2\text{SO}_2 in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:

Find the mean concentration of SO2\text{SO}_2 in the air.

Q8
  1. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
Q9
  1. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
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