[Math] Calculating Quartiles Dilemma

descriptive statisticsstatistics

When I am calculating the lower quartile and upper quartile, why am I taking (n+1)/4 and 3(n+1)/4, instead of n/4 and 3n/4?

According to me, if total number of observations are n, then that should be counted figuring out the quartiles, isn't it?

Best Answer

You need to look at the definition of quartile you are using. For large $n$ it will not matter much, but it appears the definition you are using is the datum that exceeds $25\%$ or $75\%$ of the data. The $+1$ is there for the "exceed". Wikipedia gives three different definitions of quartile, with a data set that shows the differences.

Related Solutions

[Math] What’s the proper way for calculating first quartile

In theory it means the point a quarter of the way up the cumulative distribution. In practice there are various definitions used, particularly in statistical computing. For example in R there are nine different definitions, the first three for a discrete interpretation and the rest for a variety of continuous interpolations. Wikipedia has a tenth.

R types 1, 2, 3 and 5 give $3$ as the first quartile, as you calculated, and if you want to regard this observation as a discrete population that is indeed the obvious answer. The other interpolations range from $2$ to $4$.

Your figure of $1.75$ looks wrong and it is unclear how you calculated it: if it supposed to be a weighted average of $1$ and $3$ then perhaps you were trying to get $1 \times \frac14 + 3 \times \frac34 = 2.5$, which is what R type 6 would produce. Or perhaps you were trying something else.

[Math] Is it possible to calculate the mean and standard deviation from a median and quartiles

It's mathematically impossible to deduce mean or standard deviation from median/quartiles, because medians and quartiles discard most of the data on which the mean and standard deviation are based.

Example:

data   frequency  
   0       50      
 1.4        4     
   2       50

That has a mean of 1.0 and standard deviation of 0.9. (I'm using 2 significant figures so I don't have to go into population versus sample standard deviation.)

data     frequency    
   0       30        
 1.4       44        
   2       30

That data also has the median and quartiles the same as in your example, but now the mean is 1.2 and the standard deviation is 0.8.

data     frequency        
   0       30        
 1.4        3        
   2       70        
10000000    1

Now I've changed my maximum without changing the median or quartiles, you can see even more clearly how the median and quartiles exclude extreme data, because the mean is now 96000 and the standard deviation is 98000 (still 2 sig.fig.).

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Related Solutions

[Math] What’s the proper way for calculating first quartile

[Math] Is it possible to calculate the mean and standard deviation from a median and quartiles

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