Question

Find the arithmetic mean of the following distribution by step deviation method and verify using the assumed mean.

x	0-10	10-20	20-30	30-40	40-50
y	5	13	20	15	7

Answer 1

Hint:In the step deviation method, we take an assumed mean from the frequencies, and by using the formula of step deviation method, we can find the arithmetic mean. The values which are given here as y are the frequencies. We also need to find class marks while finding the mean in the assumed mean method.

Complete step by step answer:
We know that the formula used to find the arithmetic mean for a data using step deviation is
\[\bar x\, = A + \dfrac{{{{\sum f }_i}{d_i}}}{{{{\sum f }_i}}}\]
where A is assumed mean,
 and \[{d_i}\] is (${y_i}$ -A) according to the above question,
and \[{\sum f _i}\]is sum of all the frequencies

Classes	Class mark(${y_i}$)	\[{d_i}\]=(${y_i}$ -A)	${f_i}$	${f_i}{d_i}$
0-10	5	-20	5	-100
10-20	15	-10	13	-130
20-30	25(A)	0	20	0
30-40	35	10	15	150
40-50	45	20	7	140

By substituting the above information in mean formula that is mentioned above we get
 \[\bar x\, = A + \dfrac{{{{\sum f }_i}{d_i}}}{{{{\sum f }_i}}}\]
=> \[\begin{gathered}
  \bar x\, = 25 + \dfrac{{ - 100 - 130 + 0 + 150 + 140}}{{5 + 13 + 20 + 15 + 7}} \\
  \,\,\,\,\, = \,25 + \dfrac{{60}}{{60}} = 25 + 1 = 26 \\
\end{gathered} \]
which implies the mean is 26 using step deviation assumed mean method.

Note:
Learn all the formulae of all the methods to find mean, median, mode, and all the information. Do not make calculation mistakes while solving these type of questions. you can also try this problem by direct method but not all problems can be solved by direct so use this method mostly. Read the question properly. In the step deviation method, there are two methods in which we can solve the problem, the first is the direct method in which we do all the calculations directly, and the second is the assumed mean method in which we lessen all the values by taking assumed mean to make the calculation easy.