Descriptive Statistics Test 5

Answer:- one $$\Uparrow$$ represents $$5$$ houses.

Given that one man symbol represents $$12$$ men.

$$73,77,81...113$$ are in A.P

Answer:- Converting the whole pictograph into frequency distribution table-

A symbol is used to represent $$10$$ flowers

Answer:- Converting the whole pictograph into frequency distribution table-

In the graph, 1 unit on y-axis $$=Rs.100$$

$$\textbf{Step - 1: Finding the mean}$$

                   $$\text{The } n \text{ natural numbers are } 1, 2, 3,..., n$$

                   $$\text{The Sum of } n \text{ natural number is } \dfrac{{n(n+1)}}{{2}}$$

                   $$\text{Mean} = \dfrac{ \dfrac{{n(n+1)}}{{2}}}{n} = \dfrac{{n+1}}{{2}}$$

$$\textbf{Step - 2: Finding the Variance}$$

                   $$\text{Variance} = \dfrac{{\sum(x_i)^2}}{{n}} - (\text{Mean})^2$$

                   $$ \sum(x_i)^2 = \dfrac{{1^2 +2^2 + ...+ n^2}}{{n}}$$

                   $$\text{Since } 1^2 +2^2 + ...+ n^2 = \dfrac{{n(n+1)(2n+1)}}{{6}}$$

                   $$\dfrac{{\sum(x_i)^2}}{{n}} = \dfrac{{n(n+1)(2n+1)}}{{6n}}$$

                   $$\text{Variance} =  \dfrac{{n(n+1)(2n+1)}}{{6n}} - (\dfrac{{n+1}}{{2}})^2$$

                   $$=\dfrac{{(n+1)}}{{2}}\times(\dfrac{{2n+1}}{{3}}-\dfrac{{n+1}}{{2}})$$

                   $$= \dfrac{{(n+1)}}{{2}}(\dfrac{{4n+2-3n-3}}{{6}})$$

                   $$=\dfrac{{n+1}}{{2}}\times\dfrac{{n-1}}{{6}}$$

                   $$= \dfrac{{n^2 - 1}}{{12}}$$

$${\textbf{Hence, the correct answer is Option B}.}$$

Formula to calculate Standard deviation of a grouped frequency distribution is $$ \sqrt { \frac { \sum { { (x }_{ i }-\bar { x } )^{ 2 } } { f }_{ i } }{ N }  } $$
where $$ f_1 $$ is the frequency of observation $$ x_i $$ ;
$$ bar {x} $$ is the mean of the frequency distribution
$$ N $$ is the sum of all frequencies