Question

If you only had the SD from the sample of 5 people, how would you know it is large or small? What does a large SD indicate regarding the data? What does it mean if the SD is small?

Answer 1

Let X be the variable under consideration. Then it’s standard deviation is

i.e. the square root of average of the squared deviation of each observation from their mean

SD is always positive. it equal to zero when all observations are same. That is all observations are concentrated at one value. So there is no spread in the data. So SD close to zero means smaller the spread. And the spread of the data increases as SD increases.

I will show 2 different frequency polygon of normal distribution with same mean with two different standard deviation

in the first graph standard deviation =1 and mean is 0. the probability(area under the curve) of observations go above and 2 and below -2 is very low.

But look at second graph. mean is still same. and just one unit increase in the standard deviation made the graph more wide. compare the probability of points go above 2 and below -2