Question

Use step deviation method to find the mean weight of the elephant.

Weight (kg)	100	200	300	400
Elephant	5	10	15	20

(a) 400 kg
(b) 300 kg
(c) 200 kg
(d) 100 kg

Answer 1

Hint: First of all, we are going to find the assumed mean of the following distribution by taking any mid weight. Then make a table in which write the weight, number of elephants corresponding to that weight, make a column which is the deviation from mean in which write the subtraction of assumed mean from the weight of elephant, make another column in which divide the deviation from mean by 100 (which is the successive difference of the two consecutive weights of the elephants) and then make another column in which multiply the new deviation from mean by the number of elephants corresponding to each weight. Now, add all the elements of the last column that we have made and let us call it d. Now, to find the mean weight multiply this d by 100 and then add it to the assumed mean.

Complete step-by-step answer:
We have given a table of the weight of different numbers of elements.

Weight (kg)	100	200	300	400
Elephant	5	10	15	20

Let us represent the first weight of 5 elements as ${{w}_{1}}$ and the 5 elements as ${{n}_{1}}$. Similarly, represent the second weight and its corresponding number of elephants as ${{w}_{2}},{{n}_{2}}$ likewise write for all the weights.

Weight (kg) $\left( {{w}_{i}} \right)$	100 $\left( {{w}_{1}} \right)$	200 $\left( {{w}_{2}} \right)$	300 $\left( {{w}_{3}} \right)$	400 $\left( {{w}_{4}} \right)$
Elephant $\left( {{n}_{i}} \right)$	5 $\left( {{n}_{1}} \right)$	10 $\left( {{n}_{2}} \right)$	15 $\left( {{n}_{3}} \right)$	20 $\left( {{n}_{4}} \right)$

Let us take the assumed mean as 300 and which we call as (A).
As you can see the successive difference between the weights is equal to 100 so let us mark it as “h”.

Weight $\left( {{w}_{i}} \right)$	Elephants $\left( {{n}_{i}} \right)$	${{d}_{i}}={{w}_{i}}-A$	${{d}_{i}}'=\dfrac{{{d}_{i}}}{h}$	${{d}_{i}}''={{n}_{i}}{{d}_{i}}'$
100	5	-200	-2	-10
200	10	-100	-1	-10
300	15	0	0	0
400	20	100	1	20

Now, to find the mean of the weight of the elephants we are going to use the following formula:
$Mean=A+h\left( \dfrac{d}{N} \right)$
Substituting the above values in the equation that we have calculated above we get,
$\begin{align}
  & Mean=300+100\left( \dfrac{0}{50} \right) \\
 & \Rightarrow Mean=300+0=300 \\
\end{align}$
Hence, we have calculated the mean weight of the elephants using step deviation method as 300kg.
So, the correct answer is “Option b”.

Note: You might be thinking as to why we use this step deviation method for mean. The answer is it will reduce the rigorous calculations that the direct method of mean has.
In the below, we have shown the direct method to show you the rigorous calculations that are performed.
 $\begin{align}
  & \dfrac{\sum\limits_{i=1}^{4}{{{w}_{i}}{{n}_{i}}}}{\sum\limits_{i=1}^{4}{{{n}_{i}}}} \\
 & =\dfrac{100\left( 5 \right)+200\left( 10 \right)+300\left( 15 \right)+400\left( 20 \right)}{5+10+15+20} \\
\end{align}$
$\begin{align}
  & =\dfrac{500+2000+4500+8000}{50} \\
 & =\dfrac{15000}{50}=300 \\
\end{align}$
As you can see, we have to do the long additions and divisions in this direct method of finding mean.
One more thing, you might be thinking that, how we can choose the assumed mean. The answer is you can take any near mid value of the observations because we don’t know the exact mean.