Find Q1 and Q3 from median and IQR

Question

A study gives the following:

$n = 67$

mean = 73

sd = 68

median = 55

IQR = 66

Is it possible from this information to get the actual Q1 and Q3 values? I used the $n$, mean & sd to get 95% CI. Should that be roughly similar?

Answer 1

As @Dave noted in a comment, you would have to make some assumptions about the distribution. Given the mean and the median being so different, it's likely that there is substantial skew - and you confirm this in a comment.

Various assumptions might be reasonable.

With median = 55 and IQR = 66 (and no other info or assumptions), then, with a symmetric distribution, you would have 22 and 88 for the quartiles. But you could have anything from -10 and 56 to 54 and 120. But you have additional info: The mean and sd - these will limit the possibilities. And you probably also can figure out some things from the nature of the variable (e.g. is it always positive?) and try various distributions.

Answer 2

You should have given some context, what (real-life) variable $x$ do your data represent? Some questions you probably know answers for:

• What is the possible range for $x$? That is, is $x$ nonnegative? or a count? ...

• Can we suppose independence?

Nevertheless, some observations:

• the mean is larger than the median, and a 95% confidence interval for the mean based on normal distributions give about $( 56.4, 89.6)$, the observed median is just outside. So the data casts doubt on symmetry, and points to a right-skewed distribution.

• The observed mean and standard deviation are close, pointing to an exponential (or more generally gamma) distribution.

• One can also get a rather close fit with a lognormal distribution, I get that $\mu=4, \sigma=0.778$ is close. One could also try normal or skew-normal distributions. As soon as you decide to try some distributional family as a model, you can use the given descriptive statistics to find moment-type estimators.

• and given those estimators, it is now easy to calculate the quartiles.

Can we say something more? Maybe trying to compare some such models? I doubt normal or skew-normal models can give a good fit, let us try the gamma and lognormal models. We can simulate data from such models, and try abc-methods (approximate bayes computations) to compare them. Some details here: How to do estimation, when only summary statistics are available?

Answer 3

One distribution that fits these parameters is a mixture of:

• 50% a lognormal distribution with $\mu=\ln 55,\ \sigma=.89$
• 25% a point mass at $31.38$
• 25% a point mass at $97.38$

So the quartiles could be at those point masses, though there are many other possibilities also.

I found this by solving some equations; here is the explanation:

• The median of the lognormal is $55$, so the median of the mixture is also $55$.

• The mean of the lognormal is $81.58$, so the mean of the mixture is $$(25\%)31.38+(50\%)81.58+(25\%)97.38=73.$$

• The point masses are at roughly the $26^{th}$ and $74^{th}$ percentiles of the lognormal, so they are at the $13^{th}-38^{th}$ and $62^{nd}-87^{th}$ percentiles of the mixture. In particular, they are the quartiles of the mixture and the IQR is $66$.

• The second moment of the lognormal is $14643$, so the variance of the mixture is $$(25\%)(31.38-73)^2+(50\%)(14643 - 2(73)(81.58)+73^2)+(25\%)(97.38-73)^2=68^2.$$

The final mixture is reasonably easy to understand, and you could tinker with it to get smaller point masses, an $n=67$ dataset with the same properties, or other possibilities.