Question

A Scientist is weighing each 30 fishes. Their mean weight worked out is 30 gm and standard deviation of 2 gm, Later, it is found that the measuring scale was misaligned and always under reported every fish weight by 2 gm. The correct mean and standard deviation (in gm) of fishes are respectively…...

Answer 1

Hint:
We have two linear expressions for one for mean and other for standard deviation. Mean\[({x_i} + \lambda ) = \overline x + \lambda \], standard deviation \[({x_i} + \lambda )\] remains the same. Think about how change in one quantity, i.e. mean, variance or standard deviation, will affect the others and form your calculations based on that.

Complete step by step solution:
weighted mean of each fish is 30
 Standard deviation is 2 and error in measuring weight is 2 gm
Steps 1 Let’s suppose the weight of fish is given by${x_i}$, where i= 1,2,3,4…
Calculate the mean
Mean of the 30 fishes is given by \[\overline x \]
  \overline x = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_{30}}}}{{30}} = 30 \\
  \overline x = {x_1} + {x_2} + {x_3} + ... + {x_{30}} = 900 \\

Step 2 Calculate the Standard deviation
\[\sigma = \sqrt {\dfrac{{\sum x_i^2}}{n} - \overline x } \]
 Step 3 Calculate the new mean after noted corrected weight
\[\overline x = \dfrac{{{x_1} + 2 + {x_2} + 2 + {x_3} + 2 + ... + {x_{30}} + 2}}{{30}} = \dfrac{{\sum {x_i} + 60}}{{30}} = 30 + 2 = 32\]
Step 4 Calculate the new standard deviation
\[\begin{gathered}
  \sigma = \sqrt {\dfrac{{\sum x_i^2}}{n} - \overline x } \\
   \Rightarrow 2 = \sqrt {\dfrac{{\sum x_i^2}}{{30}} - 30} \\
   \Rightarrow \sum x_i^2 = 904 \\
\end{gathered} \]
But new weight is given by \[{x_i} + 2\]

$\begin{gathered}
  {\sigma _{new}} = \sqrt {\dfrac{{\sum {{\left( {{x_i} + 2} \right)}^2}}}{n} - \overline x } \\
   = \sqrt {\dfrac{{\sum x_i^2 + 4 + 2{x_i}}}{n} - \overline x } \\
   = \sqrt {904 + 4 + 120 - {{32}^2}} \\
   = 2 \\
\end{gathered} $
Hence ${\sigma _{new}} = 2$ and ${\mu _{new}} = 32$

Note:
Note that standard deviation and variation are not changed when each number is either increased or decreased by some constant number. But when each variable is multiplied by constant$\lambda $, the new standard deviation and variation change. Mean, variance and standard deviations are the basic fundamentals of statistics and students need to be familiar with the concepts. They also need to memorize the formulas of mean, variance and standard deviation. Also learn about how changes in one of mean, variance or standard deviation can affect the others.