what percentage confidence interval for the observation which is within 2 standard deviations from the mean? The answer should be greater than 95% because 95% confidence interval is within 1.96 standard deviation from the mean. My question is how different the confidence interval is from what I said in the subject: 75% observations lie within 2 standard deviation?

If u use the Chebyshev’s inequality at least 1-1/4 = 75% of observations must lie within the +/- 2 std from the mean. And it states at least… states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a quantitative description of “nearly all” and “close to”. In particular, No more than 1/4 of the values are more than 2 standard deviations away from the mean; No more than 1/9 are more than 3 standard deviations away; No more than 1/25 are more than 5 standard deviations away; and so on. In general: No more than 1/k2 of the values are more than k standard deviations away from the mean. Your 1.96 and 95% comes from a Std Normal Distribution and that also DOES NOT VIOLATE the CHEBYSHEV’s inequality.

thanks a lot. I really appreciate it