Question

If ${u_i} = \dfrac{{{x_i} - 45}}{{10}},\sum\limits_{} {{u_i}{f_i} = 30} $ and$\sum\limits_{} {{f_i} = 100} $, where the symbols have their usual meanings, then $\overline x $ is equal to

Answer 1

Hint: Using the short cut method of finding the mean of the data and substituting given values in a short cut formula to get the value of$\overline x $.
Formula used: Mean of the data =\[A+\dfrac{{\sum\limits_{}^{} {{f_i}{u_i}} }}{{\sum\limits_{}^{} {{f_i}} }} \times h\], where ${u_i}$ is deviation of the observation from any term (A = $45$) with difference of the interval is $10$.

Complete step by step solution:

We have various methods to find the meaning of the data. But according to given terms in the statement we know that method using is a short cut method. This method is applied for data given in interval form.

In the short cut method we first find the mid value $({x_i})$ of the interval given in the data.

After calculating $({x_i})$we assume or choose any number from $({x_i})$called assumed number (A).

Now, we calculate the difference of each value of $({x_i})$ with assumed number (A) and divide the difference with length of the interval (h). Naming the result so, obtain is ${u_i}$
i.e. ${u_i} = \dfrac{{{x_i} - A}}{h}$
Comparing \[{u_i} = \dfrac{{{x_i} - A}}{h}\] with given ${u_i} = \dfrac{{{x_i} - 45}}{{10}}$ we have A = $45$ and h =$10$.

Mean of the data by short cut method is given by
$Mean(\overline {x)} = A + \dfrac{{\sum\limits_{}^{} {{f_i}{u_i}} }}{{\sum\limits_{}^{} {{f_i}} }} \times h$

Substituting values in above formula we have
$Mean(\overline x ) = 45 + \dfrac{{30}}{{100}} \times 10$
\[ \Rightarrow Mean(\overline x )\, = \,45\, + \,\dfrac{{300}}{{100}}\]
$ \Rightarrow Mean(\overline x ) = 45 + 3$
$ \Rightarrow Mean(\overline x ) = 48$
Hence, from above we see that mean $\overline x $of the given data is $48$.

Note: To find mean of the data we have different methods, but we chose the method according to given data to calculate mean of the data.