The mean of a grouped distribution of 40 observations is 45. A further class 80-100 with frequency 10 is now added to the table. What is the mean of all 50 observations?

The old total is $\sum f_i{x}_i=45\times 40=1800$. The new class contributes $10\times 90=900$ (90 is its class mark), so the new total is 2700 over $\sum f_i=50$, giving $\frac{2700}{50}=54$. "52" uses the lower limit 80 instead of the class mark 90; "67.5" averages 45 and 90 as if the two groups were equal in size; "45" wrongly assumes adding a class leaves the mean untouched.