Question

If $\sum {{f_i}{x_i} = 75}$ and $\sum {{f_i} = 15}$, then find the mean $\overline x$.

Answer 1

Here we have to find the mean value. Also, all the data is given to us therefore we will apply the formula of mean to get the correct answer. After doing some simplification we get the required answer.

Formula used: Mean $= \overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$

Complete step by step solution:
From the question we know that $\sum {{f_i}{x_i} = 75}$ and$\sum {{f_i} = 15}$.
The formula of mean $\overline x$ is:
$\overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$
On substituting the values, we get:
$\overline x = \dfrac{{75}}{{15}}$
On simplifying we get:
$\overline x = 5$
Therefore, the mean is $5$

Hence the mean value for the given question stated as the given data is $5$

Note: We know the formula for the mean of ungrouped data it is: ${\text{Mean = }}\dfrac{{{\text{sum of terms}}}}{{{\text{number of terms}}}}$
Therefore, in grouped data if we add all the frequencies of all the individual data values, we will find the total number of terms in the distribution therefore, the total number of terms are ${f_i}$.
And to get the sum of all the terms in the distribution we have to add all the data values multiplied by the sum of its frequencies; therefore, it can write as $\sum {{f_i}} \times \sum {{x_i}}$,
Which could be simplified and written as: $\sum {{f_i}{x_i}}$.
Since we have both the values of the sum of terms and the total number of terms the formula can be hence deduced.
To find the median from the frequency table we first calculate $\sum {{f_i}}$ is the sum of the term and if the number is even the median is the average of the two terms in the middle and if it is odd then it is the middle term.
To find the mode from the frequency table we look at the value which has the most frequency, which means the data value ${x_i}$ which has the highest ${f_i}$ is the mode for the data.