Mean and mode of grouped frequency distribution

Calculation of Mean:

• Mid-values $x_i$: 110, 130, 150, 170, 190, 210, 230.
• Let assumed mean $A = 170$, $h = 20$. $u_i = (x_i - 170)/20 = -3, -2, -1, 0, 1, 2, 3$.
• $f_i u_i$: $10(-3) = -30$, $15(-2) = -30$, $20(-1) = -20$, $22(0) = 0$, $18(1) = 18$, $12(2) = 24$, $13(3) = 39$.
• $\sum f_i u_i = -80 + 81 = 1$.
• $\sum f_i = 110$.
• $\text{Mean} = A + h \left(\frac{\sum f_i u_i}{\sum f_i}\right) = 170 + 20 \left(\frac{1}{110}\right) = 170 + 0.18 = 170.18\text{ ₹}$.

Calculation of Mode:

• Maximum frequency is 22 $\implies$ Modal class is 160 – 180.
• $l = 160$, $f_1 = 22$, $f_0 = 20$, $f_2 = 18$, $h = 20$.
• $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h = 160 + \left(\frac{22 - 20}{44 - 20 - 18}\right) \times 20 = 160 + \frac{2}{6} \times 20 = 160 + 6.67 = 166.67\text{ ₹}$.