The number of observations in a survey is $40$. If the mean of the first $10$ observations is $4.5$ and that of remaining is $3.5$, then the mean of the whole survey is

Let's use the properties of weighted averages to find the total combined mean:Total observations ($N$): $40$ First group ($n_1$): $10$ observations with mean (${\bar{x}}_1$) = $4.5$ $
$\text{Sum of first group}=n_1\times {\bar{x}}_1=10\times 4.5=45$$Secondremaininggroup($ n_2$):$ 40 - 10 = 30$observationswithmean($ \bar{x}_2$)=$ 3.5
$\text{Sum of second group} = n_2 \times \bar{x}_2 = 30 \times 3.5 = 105
Total mean ($\bar{X}$):
$$\text{Total Sum}=45+105=150$$
$$\bar{X}=\frac{\text{Total Sum}}{\text{Total Observations}}=\frac{150}{40}=\frac{15}{4}=3.75$$