Question

I bought two packets of apples, 25 in each packet. The mean and standard deviation of weights
of apples in the first packet are 235 and 3; and the mean and standard deviation for the second
packet are 237.5 and 4. Write down the mean and standard deviation formulae for all the fifty
apples and compute them.

Answer 1

The formulas for the mean and standard deviation of a population are:
$\mu=\frac{1}{n}\Sigma_i\ x_i\\\\\sigma=\sqrt{\frac{1}{n}\Sigma_i\ (x_i-\mu)^2}$

The formulas for the mean and standard deviation of the total combined population, of  two populations  n1  and n2 with means  μ1,  μ2  and standard deviations σ1  and σ2  are as follows:

$\mu=\frac{\mu_1\ n_1 + \mu_2\ n_2}{n_1+n_2}=\frac{25*235+25*237.5}{50}=236.25\\\\\sigma=\sqrt{\frac{n_1\ \sigma_1^2+n_2\ \sigma_2^2}{n_1+n_2}}=\sqrt{\frac{25*3^2+25*4^2}{50}}=\sqrt{12.5}=3.53$

I hope that is easy enough to follow.