1. **Problem Statement:** Calculate the variance of profit data given in class intervals with frequencies. 2. **Understanding the Data:** We have profit intervals (0-10, 10-20, etc.) and the number of companies (frequencies, $f_i$) in each interval. 3. **Mid Values ($x_i$):** For each class interval, the mid value is the average of the lower and upper limits. For example, for 0-10, mid value $x_1 = \frac{0+10}{2} = 5$. 4. **Coding Variable ($u_i$):** To simplify calculations, we use a coding variable $u_i = \frac{x_i - A}{h}$ where $A$ is the assumed mean and $h$ is the class width. - Here, $A = 25$ (mid value of the middle class 20-30), $h = 10$. - For $x_1 = 5$, $u_1 = \frac{5 - 25}{10} = -2$. 5. **Calculate $f_i u_i$ and $f_i u_i^2$:** Multiply frequency by coding variable and its square. - For first class: $f_1 u_1 = 6 \times (-2) = -12$, $f_1 u_1^2 = 6 \times 4 = 24$. 6. **Sum values:** - $\sum f_i u_i = 9$ - $\sum f_i u_i^2 = 121$ 7. **Variance formula:** $$\sigma^2 = \left( \frac{\sum f_i u_i^2}{N} - \left( \frac{\sum f_i u_i}{N} \right)^2 \right) \times h^2$$ - $N = 100$ total companies. 8. **Plug in values:** $$\sigma^2 = \left( \frac{121}{100} - \left( \frac{9}{100} \right)^2 \right) \times 10^2 = (1.21 - 0.0081) \times 100 = 1.2019 \times 100 = 120.19$$ 9. **Interpretation:** The variance of profit is $120.19$ (lakhs)$^2$, which measures the spread of profit values around the mean. This step-by-step shows how mid values, coding variables, and frequency products help compute variance efficiently.