13.2 The Arithmetic Mean: Finding the Balance Point

First, identify the Class Size ($h$). Here, $h = 25 - 15 = 10$. Next, calculate the Class Marks ($x_i$) for each interval. $x_1 = (15+25)/2 = 20$. The subsequent marks increase by 10: 20, 30, 40, 50, 60, 70, 80.

Select the Assumed Mean ($a$). The class marks are 20, 30, 40, 50, 60, 70, 80. The middle value is 50. Let $a = 50$.

Calculate $u_i$ for each row using $u_i = (x_i - a)/h = (x_i - 50)/10$.

• For 20: $(20-50)/10 = -3$
• For 30: $(30-50)/10 = -2$
• For 40: $(40-50)/10 = -1$
• For 50: 0
• For 60: 1
• For 70: 2
• For 80: 3

Multiply $f_i \times u_i$:

• $6 \times (-3) = -18$
• $11 \times (-2) = -22$
• $7 \times (-1) = -7$
• $4 \times 0 = 0$
• $4 \times 1 = 4$
• $2 \times 2 = 4$
• $1 \times 3 = 3$

Calculate Sums: $\sum f_i = 6+11+7+4+4+2+1 = 35$ $\sum f_i u_i = (-18 -22 -7) + (4 + 4 + 3) = -47 + 11 = -36$

Apply the Formula: $\bar{x} = a + (\frac{\sum f_i u_i}{\sum f_i}) \times h$ $\bar{x} = 50 + (\frac{-36}{35}) \times 10$ $\bar{x} = 50 + (\frac{-360}{35}) = 50 - 10.28$ $\bar{x} = 39.72$

Interpretation: The mean percentage of female teachers in primary schools across these states is approximately 39.72%.