Question

16.12 Compute the sample mean and sample median for the dataset 1,2,...,N

in case N is odd and in case N is even. You may use the fact that 1+2+···+N = N(N +1) 2 .

Answer 1

MEAN (Formula for mean is same for the both the values of N i.e when N is even or odd, the formula will be same)

Let's take 1,2,3,4......N is Arithmetic Progression

So here, a=1, n=N and d= 1

Then, S_{​​​​​​n =} n/2{2a + (n - 1)d}

On putting values we get

S_{​​​​​​n} = N/2{2×1 + (N- 1)×1} = N/2{2 + N - 1} = N/2(N + 1)

Now, Mean = Sum of all observations/ Total no of observaobservaobservation

Therefore : Mean = S_{​​​​​​n}​​​​​/ N

N/2(N+1)/N = (N+1)/2

NOW MEDIAN

NOTE : We have to arrange the observations in ascending order

CASE 1 When the number of observations (N) is odd, the median is the value of the {(N+1)/2}^th observation.

For example: if N=9, the value of the {(9+1)/2}^th i.e the median is the 5^th observation.

(ii) When the number of observations (N) is even, the median is the mean of the
(N/2)^th and {(N/2)+ 1}^th​​​ observations

For example: If N= 10, the mean of the values of the (10/2)^th​​​​​th and the {(10/2)+1}th observations i.e the mean of the 5^th and 6^th term.