# Statistics

I need help with this problem. I think I have some of the answers right but I'm not sure.

There is a data set consisting of 33 whole-number observations. Its five number summary is (16, 20, 22, 30, 46)

1. How many observations are stricty less than 22?

I think it is 16 because the median is 33 so if you think of the numbers on the "less than median" side ther should be 16 right?

2. How many observations are strictly less than 20?

How would I figure this out? 20 is a quartile so does that mean it is 33 divided by 4?

3. Is it possible that there is no observation equal to 20?

Well its a quartile so does it have to be part of the data? How do I figure this out?

4. Approximately where is the mean?

How dop I figure this out is I don't have the data? I only have the minimum, max, quartiles, and median.

Thank you!

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1. *-------1-------M-------3-------*
With 33 observations, the median and quartiles fall exactly on real observations, as shown in the above figure.

1. The number could vary from 9 to 16, since observations 10 to 16 could be all 22's, and consequently are not strictly less than 22.

2. Similar response to question 1.

3. See figure above. It is impossible that there is no observation of 20, since the quartiles and the median are all actual observations (#9,#17,#25)

4. There are probably better estimates of the mean given quartiles, median and range.

Here's one[see ref.] that relies solely on the numbe of observations, n, the median m, and the extreme valus a,b.
μ = (a+2m+b)/4 + (a-2m+b)/4n
The second term is usually negligible for large n.
The estimate of the mean according to this formula is therefore
mean = (16+2*22+46)/4 + (16-2*22+46)/(4*33)
= 26.6

Ref:
http://www.biomedcentral.com/1471-2288/5/13

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