5 students of a class have an average height 150 cm and variance 18 cm2. A new student, whose height is 156 cm, joined them. The variance (in cm2) of the height of these six students is :
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Official solution
Average height of 5 students,
\[\overline x = {{{x_1} + {x_2} + {x_3} + {x_4} + {x_5}} \over 5} = 150\]
\[ \Rightarrow \,\,\,\sum\limits_{i = 1}^5 {{x_i}} = 750\]
We know,
Variance \[\left( \sigma \right) = {{\sum {x_i^2} } \over 5} - {\left( {\overline x } \right)^2}\]
given that,
\[{{\sum {x_i^2} } \over 5} - {\left( {150} \right)^2} = 18\]
\[ \Rightarrow \,\,\,\sum {x_i^2} = 112590\]
Height of new student, x6 = 156 cm
New average height \[\left( {{{\overline x }_{new}}} \right) = {{750 + 156} \over 6} = 151\]
New variance \[ = {{\,\sum\limits_{i = 1}^6 {x_i^2} } \over 6} - {\left( {{{\overline x }_{new}}} \right)^2}\]
\[ = {{112590 + {{\left( {156} \right)}^2}} \over 6} - {\left( {151} \right)^2}\]
=22821−22801=20