Khan Academy: Summarizing Quantitative Data

Question 1

Why do we use √ (sum( x-sample mean)^2/(n-1)) to estimate the population standard deviation using the sample?

Answer 1

Normally we can find the variance by averaging the squared deviations from the mean. But remember we don’t know the real population mean—we had to estimate it by using the sample mean.
Any particular item in our sample is likely to be closer to the average of the sample. This is comparable to the mean of population. But, the squared deviations from the mean we calculated will probably underestimate the actual deviations from the population mean.
To compensate for this underestimation, rather than simply averaging the squared deviations from the mean, we total them and divide by n - 1.

Ref

Question 2

How can we calculate the IQR?

Answer 2

IQR=Q3-Q1, to calculate Q3 and Q1, we first find the median(Q2), then without considering the median, we calculate the median of the upper half and lower half of the sample.
81, 82, 83, 83,84, 84, 84 ,85
Q2= mean(83,84)= 83.5
upper half: 84,84,84,85
Q2 of upper half: mean (84,84)=84
lower half: 81,82,83,83
Q2 of lower half: mean (83,84)= 83.5
so IQR= 1.5