Question

If 69.5 is the arithmetic mean of 72, 70, x, 62, 50, 71, 90, 64, 58 and 82, find the value of x.

Answer 1

Hint: Use the formula where x1, x2, x3, …, xn are the given set of observations and$A = \dfrac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}$ n is the number of observations to compute the arithmetic mean. Compare the answer thus obtained with the given mean and solve the equation to find the value of x.

Complete step by step solution: We are given a set of values or observations 72, 70, x, 62, 50, 71, 90, 64, 58 and 82 with x being the missing value.
We are also given the arithmetic mean of these numbers which is 69.5.
Our aim is to find the missing value x.
So, let’s solve this by using the formula for arithmetic mean or average
The given data is an ungrouped one.
The formula for arithmetic mean or average of a set of finitely many values or observations say, x1, x2, x3, …, xn is given by
$A = \dfrac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}$where n is the number of observations
In the given set, we have 10 values.
Therefore, n = 10.
According to the formula, the arithmetic mean of the 10 values 72, 70, x, 62, 50, 71, 90, 64, 58 and 82 is
$A = \dfrac{{\sum\limits_{i = 1}^n {{x_i}} }}{n} = \dfrac{{72 + 70 + x + 62 + 50 + 71 + 90 + 64 + 58 + 82}}{{10}}$
Thus, we get the average or arithmetic mean as
$A = \dfrac{{619 + x}}{{10}}$
But the given arithmetic mean is 69.5.
Therefore, we must have
 $\dfrac{{619 + x}}{{10}} = 69.5$
Multiply 10 on both the sides
$ \Rightarrow 619 + x = 695$
Now, subtract 619 from both the sides
$ \Rightarrow x = 695 - 619 = 76$
Therefore, x = 76 is the answer.
That is, 76 is the missing number in the given set of observations whose arithmetic mean is 69.5.

Note: Arithmetic mean is commonly called as simply mean or average. Therefore, when students encounter the word average in the question, they tend to divide the sum of observations by 2.
Please remember that we divide the sum by 2 only when the no. of observations in the data set is 2.