Find Midpoints ($x_i$) for Each Class

Calculate the midpoint for each class interval using the formula $x_i = \frac{\text{Lower limit} + \text{Upper limit}}{2}$. For 10⁻²⁰ it is 15, for 20-30 it is 25, for 30-40 it is 35, and for 40-50 it is 45.

$x_i = \{15, 25, 35, 45\}$

Calculate Product of Frequency and Midpoint ($f_i \cdot x_i$)

Multiply each frequency $f_i$ by its corresponding midpoint $x_i$: $5 \times 15 = 75$, $10 \times 25 = 250$, $15 \times 35 = 525$, and $10 \times 45 = 450$.

$\sum f_i \cdot x_i = 75 + 250 + 525 + 450 = 1300$

Find Total Frequency ($\sum f$_i)

Sum up all the given frequencies to find the total number of observations.

$\sum f_i = 5 + 10 + 15 + 10 = 40$

Compute Arithmetic Mean

Apply the arithmetic mean formula for grouped data by dividing the sum of $f_i x_i$ by the total frequency $\sum f_i$.

$\overline{x} = \frac{1300}{40} = 32.50$