SOLUTION: What proportion of the following sample of ten measurements lies within 1 standard deviation of the mean? Within 2 standard deviations? Within 2 standard deviations? 3 4 1 3

Hi

3 4 1 3 2 4 3 5 2 3 mean = 3, sd = 1.15 (n-1) Used

Proportion which lies within 1 standard deviation (2.85, 4.15) 6/10 = 60%

Proportion which lies within 2 standard deviation  (1.7, 5.30) 9/10 = 90%

Proportion which lies within 3 standard deviations (.2, 6.45)  10/10 = 100%

Note: For the normal distribution: 

one  standard deviation from the mean accounts for about 68% of the set 

two standard deviations from the mean account for about 95%

and three standard deviations from the mean account for about 99.7%.

Below:  z = 0, z = ± 1, z= ±2 , z= ±3 are plotted.  

Note: z = 0 (x value the mean) 50% of the area under the curve is to the left and %50 to the right