Question

please find proportionthat lies within

a) 1% of std deviation

b) 2% of stf dev

c) 3% of std dev

What proportion of the following sample of ten measurements lies within 1 standard deviation of the mean? Within 2 standard deviations? Within 3 standard deviations? 2 2 3 4 3 5 Proportion which lies within 1 standard deviation - %

Answer 1

X 3 (X-2.9)^2 0.01 1.21 1.21 4 4 3 0.01 1 3.61 5 4.41 2 0.81 2 0.81 2 0.81 3 0.01 12.9 Total 29

Mean :

$\bar{x}=\frac{ \sum x}{n}=\frac{29}{10}=2.9$

standard deviation :

$S=\sqrt{}\frac{ \sum (x-\bar{x})^2}{n-1}$

$S=\sqrt{}\frac{ \sum (x-2.9)^2}{10-1}$

$S=\sqrt{}\frac{ 12.9}{9}$

S = 1.197

For 1 standard deviation :

$(\bar{x}-1*S,\bar{x}+1*S)= (2.9-1*1.197,2.9+1*1.197)=(1.703,4.097)$

8 value lies between 1.703 to 4.097 hence proportion 8/10 = 0.8 = 80%

Proportion lies within 1 standard deviation is 80%

For 2 standard deviation :

$(\bar{x}-2*S,\bar{x}+2*S)= (2.9-2*2.197,2.9+2*1.197)=(0.506,5.294)$

all 10 value lies between 0.506 to 5.294 hence proportion 10/10 = 0.1 = 100%

Proportion lies within 2 standard deviation is 100%

For 3 standard deviation :

$(\bar{x}-3*S,\bar{x}+3*S)= (2.9-3*2.197,2.9+3*1.197)=(-0.692,6.492)$

all 10 value lies between -0.692 to 6.492 hence proportion 10/10 = 0.1 = 100%

Proportion lies within 3 standard deviation is 100%