Question

What is the value of $x$ if 100, 120, 103, 133, 109, $x$ have a mean of 110?

Answer 1

We need to solve this question by using the concept of mean in statistics. Mean for a set of numbers is nothing but the average for the given set of data which can be calculated by adding all the terms in the set of data and dividing it by the total number of terms in the data set. Using this concept, we calculate the value of x by using the mean formula.

Complete step by step solution:
In order to solve this question, let us first give the formula to calculate the average of a set of numbers. Average or mean of a set of numbers is nothing but the sum of the numbers divided by the total number of terms in. The formula for mean can be given as,
$\Rightarrow \text{Mean=}\dfrac{{{x}_{1}}+{{x}_{2}}+\ldots +{{x}_{n}}}{n}$
Given the data as 100, 120, 103, 133, 109, $x$ , we know the number of terms here is 6. We calculate the mean by substituting in the above formula.
$\Rightarrow \text{Mean=}\dfrac{100+120+103+133+109+x}{6}$
We also know the mean for the given data is already calculated as 110. Substituting this in the above equation too,
$\Rightarrow \text{110=}\dfrac{100+120+103+133+109+x}{6}$
Multiplying both sides by 6 and adding all the terms in the numerator on the right-hand side,
$\Rightarrow \text{110}\times \text{6=565+x}$
Subtracting both sides by 565,
$\Rightarrow 660-565\text{=565+x-565}$
Subtracting the terms on the left-hand side, we get the value of x as,
$\Rightarrow x=95$
Hence, the value of $x$ if 100, 120, 103, 133, 109, $x$ have a mean of 110 is 95.

Note: We need to note that the average of a set of numbers is the same as the mean. It is a very basic statistical concept in mathematics. We need to take care while considering the mean for negative numbers. The numbers are to be considered with the sign for calculating the mean of negative numbers.