Question

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted. (ii) If it is replaced by 12.

Answer 1

(i)

The given total number of observations is 20 and the given incorrect mean and standard deviation is 10 and 2 respectively.

The incorrect sum of observations is,

x ¯ = 1 n ∑ i=1 n x i 10= 1 20 ∑ i=1 20 x i ∑ i=1 20 x i =200

So, the incorrect sum of observations is 200.

The correct sum of observation will be,

200−8=192

The correct mean is,

correct mean= correct sum 19 = 192 19 =10.1

The standard deviation will be,

σ= 1 n ∑ i=1 n x i 2 − 1 n 2 ( ∑ i=1 n x i ) 2 2= 1 n ∑ i=1 n x i 2 − ( x ¯ ) 2 2= 1 20 ∑ i=1 n x i 2 − ( 10 ) 2

Squaring both sides,

4= 1 20 ∑ i=1 n x i 2 −100 1 20 ∑ i=1 n x i 2 =104 ∑ i=1 n x i 2 =2080

So, incorrect ∑ i=1 n x i 2 is 2080.

The correct ∑ i=1 n x i 2 is,

correct ∑ i=1 n x i 2 =incorrect ∑ i=1 n x i 2 − ( 8 ) 2 =2080−64 =2016

The correct standard deviation will be,

correct standard deviation= correct ∑ i=1 n x i 2 n − ( correct mean ) 2 = 2016 19 − ( 10.1 ) 2 = 4.09 =2.02

Thus, the correct mean is 10.1 and the correct standard deviation is 2.02.

(ii)

The incorrect sum of observations is 200.

When 8 is replaced by 12, then the correct sum of observations will be,

correct sum of observations=200−8+12 =204

The correct mean is,

correct mean= correct sum 20 = 204 20 =10.2

The standard deviation is,

σ= 1 n ∑ i=1 n x i 2 − 1 n 2 ( ∑ i=1 n x i ) 2 2= 1 n ∑ i=1 n x i 2 − ( x ¯ ) 2 2= 1 20 ∑ i=1 n x i 2 − ( 10 ) 2

Squaring both sides,

4= 1 20 ∑ i=1 n x i 2 −100 1 20 ∑ i=1 n x i 2 =104 ∑ i=1 n x i 2 =2080

So, incorrect ∑ i=1 n x i 2 is 2080.

The correct ∑ i=1 n x i 2 is,

correct ∑ i=1 n x i 2 =incorrect ∑ i=1 n x i 2 − ( 8 ) 2 + ( 12 ) 2 =2080−64+144 =2160

The correct standard deviation will be,

correct standard deviation= correct ∑ i=1 n x i 2 n − ( correct mean ) 2 = 2160 20 − ( 10.2 ) 2 = 3.96 =1.98

Thus, the correct mean is 10.2 and the correct standard deviation is 1.98.