Question

Calculate the mean for the following distribution:

x	$5$	$6$	$7$	$8$	$9$
f	$4$	$8$	$14$	$11$	$3$

Answer 1

Hint: Remember that ‘x’ represents the observation/numbers and the ‘f’ represents the repetitions/frequency. According to the definition, mean is the sum of observations divided by the number of observations. But here mean will be the addition of the product of numbers and their frequency divided by the sum of frequencies. Substitute the values given in the table and find the mean.

Complete step-by-step answer:
Here in this problem, we are given a table of ‘x’ and ‘f’, where ‘x’ represents the observations or numbers and ‘f’ represents their frequency or repetitions in taking the observation. And with this given data we need to find the mean of the distribution.
Before starting with the solution, we need to first understand the concept of mean in a distribution table. According to the fundamental definition of the arithmetic mean of a set of data, it is the average value for the observation and can be calculated by dividing the sum of all the observations by the number of observations.
So by the same definition, the sum of the observations of the distribution can be calculated by multiplying each observation with the number of times it is repeated, i.e. product of each observation(x) and their respective frequency (f). This can be written as:
$ \Rightarrow Mean = \dfrac{{{\text{Sum of all the observations}}}}{{{\text{Total number of observations}}}} = \dfrac{{{\text{Sum of observations times their frequencies}}}}{{{\text{Sum of the frequencies}}}} = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$
Let’s now substitute the value in the above relation:
$ \Rightarrow Mean = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }} = \dfrac{{\left( {5 \times 4} \right) + \left( {6 \times 8} \right) + \left( {7 \times 14} \right) + \left( {8 \times 11} \right) + \left( {9 \times 3} \right)}}{{4 + 8 + 14 + 11 + 3}}$
Now we can numerator and denominator to find the mean:
$ \Rightarrow Mean = \dfrac{{\left( {5 \times 4} \right) + \left( {6 \times 8} \right) + \left( {7 \times 14} \right) + \left( {8 \times 11} \right) + \left( {9 \times 3} \right)}}{{4 + 8 + 14 + 11 + 3}} = \dfrac{{20 + 48 + 98 + 88 + 27}}{{40}} = \dfrac{{281}}{{40}}$
Therefore with this, we get the value for mean as $7.025$

Note: Be careful while doing products and then adding them. Give priority to multiplication and then addition. The symbol of summation used above $\sum {{x_i}{f_i}} {\text{ or }}\sum {{f_i}} $ represents the sum of the values followed by the sigma $\sum {} $ symbol for all possible values of ‘i’. Remember that $\dfrac{{\sum {{a_i}} }}{{\sum {{b_i}} }} \ne \sum {\dfrac{{{a_i}}}{{{b_i}}}} $ . The sum of frequencies represents the total number of observations/quantities including all the repetition.