Question

Can you please tell me Example - 4​

Answer 1

★ Given :-

$\begin{tabular}{|c|c|c|c|c|c|c|c|}\cline{1-8}\sf Marks &\sf 0-10&\sf 10-20&\sf 20-30&\sf 30-40&\sf 40-50&\sf 50-60&\sf 60-70\\\cline{1-8}\sf No.\ of \ students (f_i)&\sf 12&\sf 18&\sf 27&\sf 20&\sf 17&\sf 6&\sf 10\\\cline{1-8} \end{tabular}$

To Find :-

We have to find out the mean using all three 3 methods

• Direct method
• Assumed-Mean Method
• Step Deviation Method

★Solution :-

$\underline{\bigstar{\textit{\textbf{\ (1)\ Direct\ Method:-}}}}$

$\boxed{\sf Mean= \dfrac{\sum f_i x_i}{\sum f_i}}$

$\bullet\sf (x_i)= \dfrac{upper\ limit + lower\ limit}{2}$

$\begin{tabular}{|c|c|c|c|}\cline{1-4}\it Class\ interval&\it frequency (f_i)&\it class\ mark (x_i)&\it (f_i\times x_i)\\\cline{1-4}\sf 0-10&\sf 12 &\sf 5&\sf 60\\\sf 10-20 &\sf 18&\sf 15&\sf 270\\\sf 20-30&\sf 27 &\sf 25&\sf 675\\\sf 30-40&\sf 20&\sf 35&\sf 700\\\sf 40-50&\sf 17&\sf 45&\sf 765\\\sf 50-60&\sf 6&\sf 55&\sf 330\\\sf 60-70&\sf 10&\sf 65&\sf 650\\\cline{1-4}\cline{1-4}\sf &\it\sum f_i=110&\sf &\it \sum(f_i\times x_i)=3450\\\cline{1-4}\end{tabular}$

$\sf \therefore \ Mean = \dfrac{\sum(f_i x_i)}{\sum f_i}= \cancel{\dfrac{3450}{110}}= 31.36$

$\underline{\bigstar{\textsf{\textbf{\ Mean : 31.36}}}}$

$\underline{\bigstar{\textit{\textbf{ (2) \ Assumed\ Mean \ Method:-}}}}$

$\boxed{\sf Mean= A+ \bigg\lgroup\dfrac{\sum f_i d_i}{n}\bigg\rgroup}$

$\bullet\sf\ A= Middle \ value \ of \ x_i\\ \\ \bullet\sf (d_i)= ( x_i-A) \ for \ each \ i\ \\ \\\bullet\sf n=\sum f_i$

Now ,

For First interval ,(0-10)

$\bullet\sf \ x_i= \dfrac{10+0}{2}= 5 \\ \\ \bullet\sf \ A= 35\\ \\\bullet\sf \ d_i= (x_i-A)\\ \\\dashrightarrow\sf 5-35= (-30)\\ \\\bullet\sf \ (f_i\times d_i)= 12\times (-30)= (-360)$

• Similarly find out the all values !

$\begin{tabular}{|c|c|c|c|c|}\cline{1-5}\it Class\ interval&\it frequency (f_i)&\it class\ mark (d_i)&\it deviation (d_i)&\it (f_i\times d_i)\\\cline{1-5}\sf 0-10&\sf 12 &\sf 5&\sf -30&\sf -360\\\sf 10-20&\sf 18&\sf 15&\sf -20&\sf -360\\\sf 20-30&\sf 27&\sf 25&\sf -10&\sf -270\\\sf 30-40&\sf 20&\sf 35\ \rightarrow A&\sf 0&\sf 0\\\sf 40-50&\sf 17&\sf 45&\sf 10&\sf 170\\\sf 50-60&\sf 6&\sf 55&\sf 20&\sf 120\\\sf 60-70&\sf 10&\sf 67&\sf 30&\sf 300\\\cline{1-5}\cline{1-5}\sf &\sf \sum f_i=110&\ &&\sf \sum( f_i d_i)= -400\\\cline{1-5}\end{tabular}$

$\sf \therefore \ Mean = A+\dfrac{\sum( f_i d_i)}{\sum f_i}\\ \\ \\ \dashrightarrow\sf 35+\dfrac{-400}{110}\\ \\ \\\dashrightarrow\sf \dfrac{3850-400}{110}= \cancel{\dfrac{3450}{110}}= 31.36$

$\underline{\bigstar{\textsf{\textbf{\ Mean : 31.36}}}}$

$\underline{\bigstar{\textit{\textbf{\ (3)\ Step-Deviation \ Method:-}}}}$

$\boxed{\sf Mean= A+h\bigg\lgroup \dfrac{\sum(f_i u_i)}{\sum f_i}\bigg\rgroup}$

$\bullet\sf \ h= (upper\ limit - lower \ limit)\\ \\\bullet\sf u_i= \dfrac{x_i-A}{h}\\ \\ \bullet\sf \ x_i= \dfrac{upper\ limit+ lower\ limit }{2}$

• Now, For 1st interval (0-10)

$\bullet\sf \ x_i= \dfrac{10-0}{2}= \cancel{\dfrac{10}{2}}=5\\ \\ \bullet\sf\ h= 10-0=10 \\ \\ \bullet\sf\ A=35\\ \\\bullet\sf\ u_i= \dfrac{x_i-A}{h}\\ \\ \dashrightarrow\sf u_i= \dfrac{5-35}{10}= \cancel{\dfrac{-30}{10}}= (-3)$

• Similarly find all

$\begin{tabular}{|c|c|c|c|c|}\cline{1-5}\it Class\ interval&\it frequency (f_i)&\it Midvalue (x_i)&\it u_i=(x_i-A)/h&\it (f_i\times u_i)\\\cline{1-5}\sf 0-10&\sf 12&\sf 5&\sf -3&\sf -36\\\sf 10-20&\sf 18&\sf 15&\sf -2&\sf -36\\\sf 20-30&\sf 27&\sf 25&\sf -1&\sf -27\\\sf 30-40&\sf 20&\sf 35\ \rightarrow A&\sf 0&\sf 0\\\sf 40-50&\sf 17&\sf 45&\sf 1&\sf 17\\\sf 50-60&\sf 6&\sf 55&\sf 2&\sf 12\\\sf 60-70&\sf 10&\sf 65&\sf 3&\sf 30\\\cline{1-5}\cline{1-5}\sf &\sf \sum f_i=110&\ &&\sf \sum( f_i\times u_i)= -40\\\cline{1-5}\end{tabular}$

$\sf \therefore \ Mean= A+h\bigg\lgroup \dfrac{\sum(f_i u_i)}{\sum f_i}\bigg\rgroup}\\ \\ \\\bullet\sf A= 35 \ \ ;\ \bullet\sf\ h=10\\ \\ \\\dashrightarrow\sf Mean= 35+ \bigg\lgroup 10\times \dfrac{-40}{110}\bigg\rgroup\\ \\ \\\dashrightarrow\sf Mean = 35+ \bigg\lgroup \dfrac{-400}{110}\bigg\rgroup\\ \\ \\\dashrightarrow\sf \dfrac{3850-400}{110} = \cancel{\dfrac{3450}{110}}\\ \\ \\\dashrightarrow\sf Mean= 31.36$

$\underline{\bigstar{\textsf{\textbf{\ Mean : 31.36}}}}$