Question

Mean of 25 observations was found to be 78.4. But later on, it was discovered that one observation 96 was misread as 69. Find the correct meaning.

Answer 1

Hint: It is given that we have 25 observations. Assume that the observations other than 96 are \[{{x}_{1}},{{x}_{2}},...........{{x}_{23}}\] , and \[{{x}_{24}}\] . When one observation was misread as 69, then our observations are \[{{x}_{1}},{{x}_{2}},...........,{{x}_{24}}\] and 69. We know the formula of mean \[Mean=\dfrac{sum\,of\,all\,observations}{total\,number\,of\,observations}\] . It is given that the mean when one observation was misread as 69 is 78.4 Now, calculate the mean using the formula and make it equal to 78.4. Solve it further and get the value of \[\left( {{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}} \right)\] . When one observation is read correctly as 96, then our observations are \[{{x}_{1}},{{x}_{2}},...........,{{x}_{24}}\] and 96. Now, use the formula of mean and then put the value of \[\left( {{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}} \right)\] . Simplify it further and get the value of mean.

Complete step by step answer:
According to the question it is given that,
The mean of 25 observations when one observation 96 was misread as 69 = 78.4 …………………………………(1)
The total number of the observations = 25 ………………………………………..(2)
First of all, let us assume that the observations other than 96 are \[{{x}_{1}},{{x}_{2}},...........{{x}_{23}}\] , and \[{{x}_{24}}\] .
We know the formula of mean, \[Mean=\dfrac{sum\,of\,all\,observations}{total\,number\,of\,observations}\] …………………………………………….(3)
When one observation was misread as 69, then our observations are \[{{x}_{1}},{{x}_{2}},...........,{{x}_{24}}\] and 69 ………………………………………….(4)
Now, from equation (2), equation (3), and equation (4), we get
\[Mean=\dfrac{{{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}}+69}{25}\] ……………………………………………(5)
From equation (1), we have the mean of 25 observations when one observation was misread as 69.
Now, on comparing equation (1) and equation (5), we get
\[\Rightarrow 78.4=\dfrac{{{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}}+69}{25}\]
\[\begin{align}
  & \Rightarrow 1960={{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}}+69 \\
 & \Rightarrow 1960-69={{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}} \\
\end{align}\]
\[\Rightarrow 1891={{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}}\] ………………………………………………(6)
Here, we are calculating the correct mean when one observation 96 is read correctly as 96.
When one observation is read correctly as 96, then our observations are \[{{x}_{1}},{{x}_{2}},...........,{{x}_{24}}\] and 96 …………………………………………(7)
Now, calculating the mean using equation (2), equation (3), and equation (7), we get
\[Mean=\dfrac{{{x}_{1}}+{{x}_{2}}+...........+{{x}_{24}}+96}{25}\] ……………………………………………(8)
From equation (6) and equation (8), we have
\[\begin{align}
  & \Rightarrow Mean=\dfrac{1891+96}{25} \\
 & \Rightarrow Mean=\dfrac{1987}{25} \\
 & Mean=79.48 \\
\end{align}\]

Therefore, the mean of all 25 observations when one observation is read correctly as 96 is 79.48.

Note:
In this question, one might make a silly mistake while understanding the language of the question. One might think that one of the observations is 69 that was misread as 96. This is wrong because it is given that one observation 96 was misread as 69.