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 - 5& \sf 3\\ \\ \sf 5 - 10 & \sf 5 \\ \\ \sf 10 - 15& \sf 2 \\ \\ \sf 15 - 20 & \sf 6 \\ \\ \sf 20 - 25 & \sf 2 \\ \\ 25 - 30 & \sf 4 \: \\ \\ \:30 - 35 & \sf 7 \\\\35 - 40 & \sf 1\end{array}} \\ \end{gathered} \\ \\ \end{gathered}[/tex]

Answer:

[tex]\begin{gathered}\sf \boxed{\sf \:{ Mean = 21.16}}\\ \\ \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 - 5& \sf 3& \sf 2.5 & \sf 7.5\\ \\ \sf 5 - 10 & \sf 5& \sf 7.5 & \sf 37.5 \\ \\ \sf 10 - 15 & \sf 2& \sf 12.5 & \sf 25 \\ \\ \sf 15 - 20 & \sf 6& \sf 17.5 & \sf 135 \\ \\ \sf 20 - 25 & \sf 2& \sf 22.5 & \sf 55 \: \: \: \\ \\ \sf 25 - 30 & \sf 4& \sf 27.5 & \sf 110 \: \:\\ \\ \sf 25 - 30 & \sf 7& \sf 32.5 & \sf 227.5 \: \: \:\\ \\ \sf 35 - 40 & \sf 1& \sf 37.5 & \sf 37.5 \end{array}} \\ \end{gathered} \\ \\ \end{gathered} [/tex]

So, from above calculations, we concluded

[tex]\begin{gathered}\sf \: \sum \: f_i = 30 \\ \\ \end{gathered} [/tex]

[tex]\begin{gathered}\sf \: \sum \: f_i x_1= 636 \\ \\ \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{ 635}{ 30 } \\ \\ \end{gathered} [/tex]

[tex]\begin{gathered}\sf\implies \boxed{\sf \:{ Mean = 21.16}}\\ \\ \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.