Verified answer

Solution :

Find the mean of the following distribution

[tex]\begin{gathered}\qquad\begin{gathered}\boxed{\begin{array}{c|c} \sf Class\:interval & \sf Frequency \: \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 - 2& \sf 1\\ \\ \sf 2 - 4 & \sf 2 \\ \\ \sf 4 - 6& \sf 1 \\ \\ \sf 6 - 8 & \sf 5 \\ \\ \sf 8 - 10 & \sf 6 \\ \\ 10- 12 & \sf 2 \: \\ \\ \:12 - 14 & \sf 3 \\\\Total & \sf 20\end{array}} \\ \end{gathered} \\ \\ \end{gathered}[/tex]

Answer:

[tex]\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \begin{gathered}\sf \boxed{\sf \:{ Mean = 8.1}}\\ \\ \end{gathered} [/tex]

Step-by-step explanation:

[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c|c} \sf Class\:interval & \sf Frequency \: f_i & \sf \: Midvalue \: x_i& \bf \: f_ix_i\\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{}& \frac{\qquad }{}& \frac{\qquad }{} \\ \sf 0 - 2& \sf 1& \sf 1 & \sf 1\\ \\ \sf 2-4 & \sf 2& \sf 3 & \sf 6 \\ \\ \sf 4 - 6 & \sf 1& \sf 5& \sf 5 \\ \\ \sf 6 - 8& \sf 5& \sf 7& \sf 35 \\ \\ \sf 8- 10 & \sf 6& \sf 9 & \sf 54 \: \: \: \\ \\ \sf 10 - 12 & \sf 2& \sf 11 & \sf 22 \: \:\\ \\ \sf 12 - 14& \sf 3& \sf 13 & \sf 39 \: \: \:\\ \\ \sf Toral & \sf 20& & \sf 162 \end{array}} \\ \end{gathered} \\ \\ \end{gathered} [/tex]

So, from above calculations, we concluded

[tex]\begin{gathered}\sf \: \sum \: f_i = 20 \\ \\ \end{gathered} [/tex]

[tex]\begin{gathered}\sf \: \sum \: f_i x_1= 162 \\ \\ \end{gathered} [/tex]

Now,

[tex]\begin{gathered}\sf \: Mean = \dfrac{ \sum \: f_i x_1}{ \sum \: f_i } \\ \\ \end{gathered} [/tex]

[tex]\begin{gathered}\sf \: Mean = \dfrac{ 162}{ 20 } \\ \\ \end{gathered} [/tex]

[tex]\begin{gathered}\sf\implies \boxed{\sf \:{ Mean = 8.1}}\\ \\ \end{gathered} [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

1. Mean using Direct Method

[tex]\begin{gathered}\boxed{ \rm{ \:Mean = \dfrac{ \sum f_i x_i}{ \sum f_i} \: }} \\ \\ \end{gathered} [/tex]

( 2 ) Mode of the continuous series is given by

[tex]\begin{gathered} {{ \boxed{\sf{Mode = l + \bigg(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \bigg) \times h }}}} \\ \\ \end{gathered} [/tex]

where,

l is lower limit of modal class.

[tex]\sf{f_1}[/tex] is frequency of modal class

[tex]\sf{f_0}[/tex] is frequency of class preceding modal class

[tex]\sf{f_2}[/tex] is frequency of class succeeding modal class

h is class height.

Step-by-step explanation:

A survey was conducted by a group of students as a part of their environment awareness programme,in which they collected the following data regarding the number of plants in a locality. find the means number of plants per house

0-2, 2-4,4-6,6-8,8-10,10-12,12-14

1,2,1,5,6,2,3