Question

Find the value of p, if the mean of the following distribution is 7.68.
3, 5, 7, 9, 11, 13, 6, 8, 15, p, 8, 4

Answer 1

Hint: To solve this question, we will use the formula of mean given as \[\text{Mean}=\dfrac{\text{Sum of all observations}}{\text{number of observations}}.\] From the given data, we will calculate the sum of all the observations having p in it and by using the above formula we will calculate the value of p.

Complete step-by-step answer:
The data is given as
3, 5, 7, 9, 11, 13, 6, 8, 15, p, 8, 4
The sum of all the given observations is given by
\[3+5+7+9+11+13+6+8+15+p+8+4\]
Let it be S. Therefore, we get,
\[S=89+p\]
Then the sum of all the observations is given as,
\[S=89+p\]
The number of observations is given as 12. Finally, we will calculate the mean by using the formula \[\text{Mean}=\dfrac{\text{Sum of all observations}}{\text{number of observations}}.\] Substituting the value of the sum of all observations as 89 + p and that of the number of observations as 12. Also, the mean of the given distribution is given as 7.68. So, substituting all these values in the above formula, we get,
\[\text{Mean}=7.68=\dfrac{89+p}{12}\]
\[\Rightarrow 7.68=\dfrac{89+p}{12}\]
\[\Rightarrow 7.68\times 12=89+p\]
\[\Rightarrow 92.16=89+p\]
\[\Rightarrow p+89=92.16\]
\[\Rightarrow p=92.16-89\]
\[\Rightarrow p=3.16\]
Hence, we have determined the value of p as 3.16

Note: Observe that as all the values of the data are given as positive, hence the mean would also be positive. So, here this means 7.68. Also, if any value like p is obtained to be negative then it would definitely hamper the mean. Even if some values of the data are positive and some negative the mean can be positive or negative both.