If a question says
You have a data set with six numbers: $4,5,7,10,x,y$. The mean is 7, and the square of the standard deviation is $\frac{13}3$. What are the two missing points?
How do I solve that? I began by using the formula for the average and standard deviation to try and solve for the missing values, but I am stuck. Can someone please help me with the steps I should follow to solve problems like these?
Best Answer
We know the mean is 7 and that there are six numbers, so the sum of all numbers is 42. It follows that the sum of the two unknowns is $$x+y=42-(4+5+7+10)=16\tag1$$ Now use the formula for variance (which is the square of the standard deviation): $$\sigma^2=\frac{13}3=\frac1N\sum_i(x_i-\mu)^2$$ Substituting what we know into this equation (sample size, mean, data points) yields $$\frac{(4-7)^2+(5-7)^2+(7-7)^2+(10-7)^2+(x-7)^2+(y-7)^2}6=\frac{13}3$$ which simplifies to $$\frac{22+(x-7)^2+(y-7)^2}6=\frac{13}3$$ $$22+(x-7)^2+(y-7)^2=26$$ $$(x-7)^2+(y-7)^2=4\tag2$$ We now have two unknowns in two equations $(1)$ and $(2)$, which means we can solve for $x$ and $y$. In this case we are lucky, since we can just guess and find that $x=7$, $y=9$ (or the other way around) forms the solution. Hence the completed data set is $$4,5,7,7,9,10$$