(a) If the wrong item was omitted:
When the wrong item 8 is omitted, the correct sum of the observations would be:
Correct sum = (Mean) * (Number of observations)
Correct sum = 10 * 20 = 200
Now, since the incorrect item of 8 was omitted, the correct sum of the remaining 19 observations would be:
Correct sum of 19 observations = 200 - 8 = 192
The correct mean is then:
Correct mean = Correct sum / Number of observations
Correct mean = 192 / 19 = 10.1053
(b) If the wrong item is replaced by 12:
When the wrong item 8 is replaced by 12, the correct sum of the 20 observations would be:
Correct sum = 10 * 20 + 12 - 8 = 200 + 4 = 204
The correct mean is then:
Correct mean = Correct sum / Number of observations
Correct mean = 204 / 20 = 10.2
To calculate the correct standard deviation in each case, we need to use the formula for standard deviation:
Standard deviation = sqrt [(Σ(x - mean)^2) / n]
For case (a) when the wrong item was omitted:
Correct standard deviation = sqrt [(Σ(x - mean)^2) / n]
Correct standard deviation = sqrt [(16/19)] = sqrt(0.8421) ≈ 0.9176
For case (b) when the wrong item was replaced by 12:
Correct standard deviation = sqrt [(Σ(x - mean)^2) / n]
Correct standard deviation = sqrt [(16/20)] = sqrt(0.8) ≈ 0.8944
Therefore, the correct mean and standard deviation would be:
(a) Correct mean = 10.1053, Correct standard deviation ≈ 0.9176
(b) Correct mean = 10.2, Correct standard deviation ≈ 0.8944
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,
(a) If the wrong item was omitted.
(b) If it is replaced by 12.
1 answer