Question

The mean of $10$ observations is $16.3$ By an error one observation is registered as $32$ instead of $23$ . Then the correct mean is-
A.$15.6$
B.$15.4$
C.$15.7$
D.$15.8$

Answer 1

Hint: Mean is the ratio between the sum of all the numbers to the count of the numbers. Always frame the known and unknown terms given in the question statement.
Use, $\overline x = \dfrac{{\sum {{x_i}} }}{n}$

Complete step by step solution:
Given that: Total number of observations $ = 10$
  Mean, $\overline x = 16.3$
  $\sum {{x_i}} = $ Sum of all the $10$ observations where $i = 1,2,........10$
 One observation is registered as $32$ instead of $23$
 Let, sum of $9$ observations is equal to “A”
Therefore, we can write summation of $10$ observations is equal to the sum of all $9$ observations $ + 32$
  $\sum {{x_{10}}} $ $ = A + 32$
Put all the known values in the standard formula of mean, to find the value of “A”
   \[\overline x = \dfrac{{\sum {{x_{10}}} }}{n}\]
   $16.3 = \dfrac{{A + 32}}{{10}}$
   Cross-multiply and make the subject “A”
   $\begin{array}{l}
163 = A + 32\\
 \Rightarrow A = 163 - 32\\
A = 131
\end{array}$
 Now, the corrected mean would be calculated using the value of $9$ observations and the $10th$ observation which was noted $32$ instead of $23$
Therefore, New Mean $\overline {{x_{new}}} = \dfrac{{\sum {{x_{10}}} }}{{10}}$
\[\begin{array}{l}
\overline {{x_{new}}} = \dfrac{{A + 23}}{{10}}\\
\overline {{x_{new}}} = \dfrac{{131 + 23}}{{10}}\\
\overline {{x_{new}}} = \dfrac{{154}}{{10}}\\
\overline {{x_{new}}} = 15.4
\end{array}\]
Then the corrected mean, $\overline {{x_{new}}} = 15.4$ is the required solution.

Note: Mean in mathematics is also known as the average of the given numbers. Mean is the resultant taking all the values under consideration is the most important property whereas, median is the middle value from the list of numbers arranged in either ascending or the descending numbers.