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I have daily measurements of nitrogen dioxide for one year (365 days) and the interquartile (IQR) is 24 microgram per cubic meter. What does "24" mean in this context, apart from the definition of IQR which is the difference between the 25th and 75th percentile? How would you explain this figure to a journalist, for example?


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what is the mean or median? – mpiktas Jan 12 '11 at 8:04

From definition, this defines the range witch holds 75-25=50 per cent of all measured values.

: (median-24/2,median+24/2). Median should be written somewhere near this IQR.
The above was false of course, it seems I was still sleeping when writing this; sorry for confusion. It is true that IQR is width of a range which holds 50% of data, but it is not centered in median -- one needs to know both Q1 and Q3 to localize this range.

In general IQR can be seen as a nonparametric (=when we don't assume that the distribution is Gaussian) equivalent to standard deviation -- both measure spread of the data. (Equivalent not equal, for SD, (mean-$\sigma$,mean+$\sigma$) holds 68.2% of perfectly normally distributed data).

EDIT: As for example, this is how it looks on normal data; red lines show $\pm 1\sigma$, the range showed by the box on box plot shows IQR, the histogram shows the data itself: alt text
you can see both show spread pretty good; $\pm 1\sigma$ range holds 68.3% of data (as expected). Now for non-normal data
alt text
the SD spread is widened due to long, asymmetric tail and $\pm 1\sigma$ holds 90.5% of data! (IQR holds 50% in both cases by definition)

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Thanks you for the mathematical explanations. – user2742 Jan 12 '11 at 9:41

This is a simple question asking for a simple answer. Here is a list of statements, starting with the most basic, and proceeding with more precise qualifications.

The IQR is the spread of the middle half of the data.

Without making assumptions about how the data are distributed, the IQR quantifies the amount by which individual values typically vary.

The IQR is related to the well-known "standard deviation" (SD): when the data follow a "bell curve," the IQR is about 35% greater than the SD. (Equivalently, the SD is about three-quarters of the IQR.)

As a rule of thumb, data values that deviate from the middle value by more than twice the IQR deserve individual attention. They are called "outliers." Data values that deviate from the middle value by more than 3.5 times the IQR are usually scrutinized closely. They are sometimes called "far outliers."

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The interquartile range is an interval, not a scalar. You should always report both numbers, not just the difference between them. You can then explain it by saying that half the sample readings were between these two values, a quarter were smaller than the lower quartile, and a quarter higher than the upper quartile.

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@onestop This prompts me to think whether IQR should be an interval or a scalar. From Wikipedia, it is defined as a scalar and is consistent with what I have learnt. – user2742 Jan 12 '11 at 9:37
I've always known that the IQR is the difference between the 3rd and the 1st quartile, hence a scalar. – nico Jan 12 '11 at 10:50
Though IQR might be defined as a scalar, if it is reported as an interval it carries much more information. For example [-1,1] and [499,501] will have the same range, but implications are different. – mpiktas Jan 12 '11 at 13:23
(IMHO) IQR is a scalar: it is the lenght of an interval. Obviously, since it is always possible to pass from the second (lying in a subset of R^2) to the first (lying in R+) but not viceversa, the latter is more informative than the former. – glassy Jan 13 '11 at 22:05
I expected this to be controversial and realise that it's not the usual definition, but i think it's a more useful one. In my experience of the medical literature, it's far more common to report both quartiles as the IQR rather than the difference between them, and I'm glad that's the case. – onestop Jan 16 '11 at 13:54

Roughly speaking, I would say to a journalist that I could declare the daily level of nitrogen dioxide being sure, after discarding the highest values and the lowest values, that in each one of one-half of the days in that year the observed value is not beyond a distance of IQR/2 from the declared level.

For example, if your first quartile and third quartile are 100 and 124, you could say that daily level is 112 (average of 100 and 124) and assure your interlocutor that in half of the days the error you make isn't greater than 12.

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Could you explain the distinction you are making between "observed" and "declared" level? (I also suspect a good journalist would be bothered by the implication that you are "discarding" values and would follow up with some difficult questions :-).) – whuber Jan 13 '11 at 16:08
@whuber: I add an example to my answer. About the difficult questions, I suppose that they arise even if you answer refers to the normal distribution (as other answers to this question do), or some other concept. Could be simpler to explain a 50%-trimmed mean? If it is so, then say that the IQR is the length of the range of its observations – glassy Jan 16 '11 at 19:00

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