The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
Number of observations (n) = 100
Incorrect mean (x̅)= 20
Incorrect standard deviation (σ) = 3
Incorrect sum of observations = 2000
Correct sum of observations = 2000-21-21-18 = 2000 - 60 = 1940
correct mean = \( \dfrac{Correct\;sum}{100-3}\)
= \( \dfrac{1940}{97}\) = 20
\( (σ)= \sqrt{\dfrac{1}{n} \displaystyle\sum_{i=1}^{n} X_1-\dfrac{1}{n^2} (\displaystyle\sum_{i=1}^{n}X)^2}\)
\( 3= \sqrt{\dfrac{1}{n} \displaystyle\sum_{i=1}^{n} X_1^2-\overline(X)^2}\)
\(3= \sqrt{\dfrac{1}{100}\times Incorrect \displaystyle\sum X_1^2-(20)^2}\)
\( Incorrect \displaystyle\sum X_1^2=100(9+400)\)
\( Incorrect \displaystyle\sum X_1^2=40900\)
\(correct \displaystyle\sum_{i=1}^{n}X_1^2 \) =
\( Incorrect \displaystyle\sum X_1^2-(21)^2-(21)^2-(18)^2\)
= 40900-441-441-324
= 40900 - 1206
= 39694
Hence, correct standard deviation
\(3= \sqrt{\dfrac{correct \displaystyle\sum X_1^2}{n}-(correct\;mean)^2}\)
\( =3= \sqrt{\dfrac{39694}{97}-(20)^2}\)
= \( \sqrt{409.216-400}\)
= 3.036
Answered by Pragya Singh | 1 year agoFind the mean deviation from the mean and from a median of the following distribution:
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
| No. of students | 5 | 8 | 15 | 16 | 6 |
The age distribution of 100 life-insurance policy holders is as follows
| Age (on nearest birthday) | 17-19.5 | 20-25.5 | 26-35.5 | 36-40.5 | 41-50.5 | 51-55.5 | 56-60.5 | 61-70.5 |
| No. of persons | 5 | 16 | 12 | 26 | 14 | 12 | 6 | 5 |
Compute mean deviation from mean of the following distribution:
| Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |
| No. of students | 8 | 10 | 15 | 25 | 20 | 18 | 9 | 5 |
Find the mean deviation from the mean for the following data:
| Classes | 95-105 | 105-115 | 115-125 | 125-135 | 135-145 | 145-155 |
| Frequencies | 9 | 13 | 16 | 26 | 30 | 12 |
Find the mean deviation from the mean for the following data:
| Classes | 0-100 | 100-200 | 200-300 | 300-400 | 400-500 | 500-600 | 600-700 | 700-800 |
| Frequencies | 4 | 8 | 9 | 10 | 7 | 5 | 4 | 3 |