Statistics

Using step deviation method, calculate the mean for the following data:

Height (in cm) Number of boys
$135 - 140$ $4$
$140 - 145$ $9$
$145 - 150$ $18$
$150 - 155$ $28$
$155 - 160$ $24$
$160 - 165$ $10$
$165 - 170$ $5$
$170 - 175$ $2$


Class size $= $ Upper class limit $-$ Lower class limit $= h = 5$

Let assumed mean $= A = 152.5$

Height (in cm) Number of boys (frequency) $\left(f_i\right)$ Mid-value $\left(x_i\right)$ deviation $= x_i - A $ $ t_i = \dfrac{x_i - A}{h} $ $ f_i \times t_i $
$135 - 140$ $4$ $137.5$ $-15$ $-3$ $-12$
$140 - 145$ $9$ $142.5$ $-10$ $-2$ $-18$
$145 - 150$ $18$ $147.5$ $-5$ $-1$ $-18$
$150 - 155$ $28$ $152.5$ $0$ $0$ $0$
$155 - 160$ $24$ $157.5$ $5$ $1$ $24$
$160 - 165$ $10$ $162.5$ $10$ $2$ $20$
$165 - 170$ $5$ $167.5$ $15$ $3$ $15$
$170 - 175$ $2$ $172.5$ $20$ $4$ $8$


$\Sigma f_i = 100$, $\;$ $\Sigma f_i \times t_i = 19$

Mean $= A + \dfrac{\Sigma f_i \times t_i}{\Sigma f_i} \times h$

i.e. $\;$ Mean $= 152.5 + \dfrac{19}{100} \times 5 = 153.45$