Question 18

The mean of 25 observations is 36. If the mean of the first 13 observations is 32 and that of the last 13 observations is 39, then the 13th observation is:

Solution

Let all the 25 observations be {$$X_1,X_2,X_3,.......X_25$$}.

Mean of first 13 observations = $$\dfrac{\ X_1+X_2+X_3+.....X_{13}}{13}=\ 32$$ ------Equation 1

Mean of last 13 observations = $$\dfrac{\ X_{13}+X_{14}+X_{15}+.....X_{25}}{13}=\ 39$$.......Equation 2

Adding Equation 1 & 2 : {$$X_1+X_2+X_3+......X_{25}$$} + ($$X_{13}$$) = 13(32 + 39) = 13(71)

Mean of all 25 observations = $$\dfrac{\ X_1+X_2+X_3+.....X_{25}}{25}=\ 36$$
this implies : {$$X_1+X_2+X_3+......X_{25}$$} = 25(36)

Therefore, $$X_{13}=13(71)-25(36)$$ = $$923-900$$ = 23.

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