Question

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

No. of heart beats per minute	65 - 68	68 - 71	71 – 74	74 – 77	77 – 80	80 – 83	83 – 86
No. of women	2	4	3	8	7	4	2

Answer 1

Hint: We will first create a new table using the data given above to us with the mid value of the intervals, assumed mean and then the product of assumed mean and frequency. After this, we will use the assumed mean method to find the answer.

Complete step-by-step solution:
Let us first find the mid – points of all the intervals.
65 – 68: the mid – point will be \[\dfrac{{65 + 68}}{2} = \dfrac{{133}}{2} = 66.5\]
68 – 71: the mid – point will be \[\dfrac{{68 + 71}}{2} = \dfrac{{139}}{2} = 69.5\]
71 – 74: the mid – point will be \[\dfrac{{71 + 74}}{2} = \dfrac{{145}}{2} = 72.5\]
74 – 77: the mid – point will be \[\dfrac{{74 + 77}}{2} = \dfrac{{151}}{2} = 75.5\]
77 – 80: the mid – point will be \[\dfrac{{77 + 80}}{2} = \dfrac{{157}}{2} = 78.5\]
80 – 83: the mid – point will be \[\dfrac{{80 + 83}}{2} = \dfrac{{163}}{2} = 81.5\]
83 – 86: the mid – point will be \[\dfrac{{83 + 86}}{2} = \dfrac{{169}}{2} = 84.5\]
Now, let us assume that ${u_i} = \dfrac{{{x_i} - 75.5}}{3}$
Now, let us create a new table:

Class - Interval	Mid – value $({x_i})$	Frequency $({f_i})$	${u_i} = \dfrac{{{x_i} - 75.5}}{3}$	${f_i}{u_i}$
65 - 68	66.5	2	-3	-6
68 - 71	69.5	4	-2	-8
71 – 74	72.5	3	-1	-3
74 – 77	75.5	8	0	0
77 – 80	78.5	7	1	7
80 – 83	81.5	4	2	8
83 - 86	84.5	2	3	6
	Total	30	0	4

Now, we also have formula of mean in assumed mean method which is given by:
$\bar x = A + h\left( {\dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}} \right)$, where A is the assumed value which is 75.5 here and h is the class interval which is 3 here.
Now, putting in the values as per our question, we will get:-
$ \Rightarrow \bar x = 75.5 + 3\left( {\dfrac{4}{{30}}} \right)$
Simplifying the RHS of the above expression will lead us to:-
$ \Rightarrow \bar x = 75.5 + 0.4$
Simplifying the RHS further, we will get as follows:-
$ \Rightarrow \bar x = 75.9$
Hence, the mean heart beats per minute for these women is 75.9.

$\therefore $ The required answer is 75.9.

Note: The students must know that mean refers to the average. Like if we take the example of the given question, if we pick out a random woman, the average heartbeat of her will be 75.9.
We can find mean by normal methods as well in which we have to calculate the sum of product of the mid - value and the frequency and divide it by total number (frequency) but that will involve a lot of difficult calculation which eventually may lead to mistakes, therefore, we here used the Assumed mean method to calculate the same.