The question that i'm being asked it:
the mean of 10 observed scores was 20 and the standard deviation if 6.0. the observed scores are {16,11,20,24,29,24,16,20,,} what are the two missing scores
I have tried to figure this out and i keep ending up needing to use the quadratic formula and getting two numbers that equate to a different standard deviation. I really do not know what more i can do. please help me
Best Answer
Let the missing scores be $a$ and $b$. The non-missing scores have mean $20$, so the missing scores must add up to $40$.
The standard deviation, if we use "$n-1$", not $n$, is equal to $$\sqrt{\frac{1}{9} (226+(a-20)^2+(b-20)^2)}.$$ This is equal to $6$, so some manipulation gives $$324=226+(a-20)^2+(b-20)^2.$$ Let $x=a-20$ and $y=b-20$. We get $x+y=0$ and $x^2+y^2=98$. Substitute. We get $x^2=49$. So $x=7$ and $y=-7$, or the other way around. Our missing values are $13$ and $27$.