# How to calculate the 4th quartile from median and IQR?

How can I calculate the 4th quartile from median and IQR. In a scientific article, I have those values:

• The median is 2.8 ng/ml of bisphenol A and
• The interquartile range, they wrote that 1.5-5.6.

Can I conclude that

• the first quartile is 1.5
• the second quartile 2.8
• and the third quartile 5.6 ?

If it is ok I understand, but I need to recalculate in order to have four quartiles. Can you help me?

• see Ferdi's answer, but are you sure you mean the 4th quartile as a number? It would essentially be the maximum value. – IWS Oct 9 '17 at 10:04
• Can you clarify what you mean by the fourth quartile? There are normally only $q − 1$ different $q$-quantiles (three quartiles, four quintiles, nine deciles etc) unless you're referring to the intervals that the quartiles separate. (If you count the largest value as the fourth quartile you'd also count the smallest observation as the zero-th, and there'd be $q+1$ then, not $1$.) See the second sentence of the second paragraph here and this article. – Glen_b Oct 9 '17 at 12:12
• Values in the third quartile as a set of numbers (rather than a point) might be said to be between $2.8$ to $5.6$. So, in the same way, values in the fourth quartile might be said to go from $5.6$ upwards – Henry Oct 10 '17 at 7:47

Note: In the following answer I assume that you only know the quantiles you mentioned and you do not know anything else about the distribution, for instance you do not know whether the distribution is symmetric or what its pdf or its (centralized) moments are.

It is not possible to calculate the 4th quartile, if you have only the median and the IQR.

Let us look at the following definitions:

median = second quartile.

IQR = third quartile $-$ first quartile.

The 4th quartile is in neither of these two equations. Therefore, it is impossible to calculate it with the information given.

Here is one example:

   x <- c(1,2,3,4,5,6,7,8,9,10)
y <- c(1,2,3,4,5,6,7,8,9,20)

summary(x)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.00    3.25    5.50    5.50    7.75   10.00

summary(y)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
1.00    3.25    5.50    6.50    7.75   20.00


The first quartile is for both "x" and "y" 3,25. Also the median is 5.5 for both. The third quartile is 7.75 for both and the IQR is 7.75 $-$ 3.25 = 4.5 for both. However, the 4th quartile, which is also the maximum, is different, namely 10 and 20.

You can also look at boxplots of x and y and you will see that the first quartile, the second quartile (median) and the third quartile are equal. Therefore, you cannot conclude anything about the rest of the distribution of the datapoints.

df <- data.frame(x,y)
p <- ggplot(stack(df), aes(x = ind, y = values)) + geom_boxplot()
p


• An exception would be if the distribution is known to be symmetric. In that case the quartiles are IQR/2 on either side of the median. – pjs Oct 9 '17 at 16:01
• Good point. I included it in my answer. – Ferdi Oct 9 '17 at 16:07
• All right !! I understand now !! I have been confused actually – Doris TAN Oct 10 '17 at 11:25
• Feel free to accept one of the answers. – Ferdi Oct 11 '17 at 19:40

@Ferdi is correct, but I think that you are asking the wrong question. I think you are confused because "quartile" seems to mean "4 of something". There are, indeed, 4 groups. But that means there are 3 divisions and, at least in what I've read, the term 4th quartile (as a number) is not used at all. If you do calculate the 4th quartile as a number, then you'd also want the 0th quartile, which would be the minimum. But I don't think that's what you want.

In case that isn't clear, picture cutting a rectangle into 4 rectangles. You need three cuts to make four rectangles.

If I have wrongly accused you of being confused, I apologize, but I've seen this confusion more than once.

• That's right, i am surely confused – Doris TAN Oct 10 '17 at 11:26

The first quartile has 25% of the data below it, 2nd quartile = median has 50% of data below it, third quartile has 75% data below and 25% above. IQR = 3rd quartile - 1st quartile. A fourth quartile would be the maximum, which you can't get from the median and IQR. IQR and median tell you very little about the shape of the distribution. You might be able to make an estimate if you know the shape of the distribution, but for many distributions the answer will be infinity. I suspect that the third quartile is what you really want. If you have the IQR and median and know the shape of the distribution you may be able to estimate the third quartile: e.g. median plus half the IQR for a symmetric distribution. However many distributions are not symmetric. Also, be careful in case you have been given the semi interquartile range rather than the IQR.