Standard deviation (SD), Standard error (SE) and Confidence Interval (CI)

Hi all,

I had a tough time understanding these 3 statistics and let me share what I understood with you guys.

• The first thing we should know is that we will always be dealing with samples from a large population. For example, say we want to know the average height of a Japanese woman. We will measure the height of n=100 Japanese women and we will draw some conclusions about the entire Japanese women population.
• Now, standard deviation (SD) of the 100 measurements will give us how our measurements vary around mean of the measurements.
• the standard error (SE) (of the mean) will tell us the variability of our computed mean. I repeat; SD gave the variability of data around the mean and SE gives the variability of the computed mean, not the data around it.
• finally, the confidence interval (CI) give us where the real mean of the large population might lie.


• Now, how do we compute these 3 values?
• SD is computed directly from our measurements.

• SE is computed from SD.

                    This is how you get it.

• if xi are n independent observations from a population that has a mean u, and standard deviation s then the variance of the total

is ns^2 (because for two it will be s1^2 + s2^2)

• The variance of T/n (mean) must be 1/n^2 * (ns^2) = s^2 / n
• Thus, the standard deviation of T/n = s/(sqrt(n))

• CIs are computed from the SE.
• Say, we need a 95% CI, then
• upper 95% limit =

• lower 95% limit =

1.96 comes from 0.975quantile of the normal distribution -- (100-95)/2%=0.975