# Teaching the difference between standard deviation and interquartile range

Are there any good examples for high school studends where:

• Interquartile range is "better" to describe "spread" in an (empirical) statistical distribution of data
• standard deviation is a "better" measure and why it is used instead of interquartile range in many applications
• No particular examples to give you, however a comment about SD: There are two good reasons to use standard deviation. One of them is that it matches the vision of stats as geometry: the distance between a point $(x_1, \dots, x_n)$ and the one where they're all the mean $(\bar{x}, \dots, \bar{x})$ is close to the standard deviation. The other is that there is a well-known distribution describing the possible standard deviations in a sample of size $n$, so probability calculations work better. – DavidButlerUofA Jul 6 '14 at 5:44

A simple example for the IQR is to consider the following two data sets:

A = {1,1,1,1,1,1,1} and B = {1,1,1,1,1,1,100000000}.

IRQ for both is 0, but SD is very different. You can argue about which is really better, but this example very nicely illustrates that the IQR tells you where the middle 50% of the data is located while the SD tells you about the spread of the data. It also shows that the IQR is very resistant to outliers (and to some degree skew) while the SD is not.

Although maybe not directly relevant to the beginning student, the standard deviation plays a central role in statistics for two reasons: it is a key factor in the central limit theorem (which explains to students why increasing the sample size of a data set should give us a "better" estimate for the mean) and also because it gives us the variance (which is fundamental for such methods as ANOVA, linear regression, etc). However, one use of the standard deviation that is very important for beginning students to understand is that for the normal distribution, we actually think of the SD as a measuring stick that helps us tell how likely an even is to occur (i.e. the all important "68–95–99.7 rule"). An example I give to my students wrt SD is to ask them to consider 2 hypothetical job offers: one is in city A, located in rural area where the salary ~ N(30K,5K) and the other is in city B, a large city where salary ~N(60K,10K). Suppose that salary offer from city A is 45K and the salary offer from city B is 60K, which should you choose? Here simply looking at how far the salary offers from the means in units of SD, tells us that city A is the better offer.

• If it's not a normal distribution you have also a weaker rule by chebycheff, i.e. that 75 percent of all points are in the 2sigma interval around the mean value – Julia Jul 6 '14 at 18:43
• The main point that I wanted to emphasize however which is often overlooked is the SD is like a "measuring stick". The point being that when we work with a normal distribution, what is of principal interest in many calculations a beginning statistics student performs is how far a value is from the mean in units of SD. This is a key point that is easy for a student to miss. – Matt Brenneman Jul 6 '14 at 19:36
• I'd advise students to take the big-city offer instead, since their real income will be higher there, even with a higher cost of living. They might have higher status in their locality with the \$45 K job in the small town, but most people will have a higher quality of life with the \$60 K job in the big city. – user173 Jul 7 '14 at 0:03

Heights and weights are roughly normal, so standard deviation is more standard for them. In those cases it's easy to translate from IQR to standard deviation by a factor of 1.35, so it's better to use the more standard number. By contrast:

Economic data is rarely normal, so interquartile range is often more useful in that area. This is why interquartile ranges are more commonly quoted for both incomes and real estate values. Consider:

What is the interquartile range of New York City incomes? We can calculate it reasonably well from a reasonable sample.

What is the standard deviation of New York City incomes? We'd need to know details about the highest-earning people like:

• Do we count David Koch as a resident, even if he spends less than half the year in the city?
• Did Michael Bloomberg claim the NYC mayoral mansion as his primary residence for tax purposes, and how did that affect the deducibility of mortgage interest on his other properties?

The standard deviation of income depends so much on these details that it won't be as useful as the interquartile range.

What makes one better or worse may depend on many things.

Ultimately (though this is not especially enlightening) - if you want to look at the kind of thing IQR measures, IQR is good at that, while if you want to look at the kind of thing standard deviation measures, standard deviation is good at that. The different ways they respond to data helps determine where they're more useful (in particular, sd is much more impacted by large outliers, so tends to be more valuable in situations where the data doesn't have big tails)

I want to give a slightly different take that may or may not be of some help.

Both IQR and standard deviation can be thought of as measures of a kind of "typical distance between data points".

For example, the IQR is effectively the distance between the median of the top half of the data and the median of the bottom half of the data, and in that sense is a kind of 'typical distance'.

The variance (up to a Bessel correction factor in the sample case) is half the average squared distance between pairs of points; the standard deviation is thereby a root-mean-square distance between pairs of points, divided by $\sqrt{2}$ (or times $\sqrt{\frac{n}{2(n-1)}}$ in samples when applying a Bessel correction in the sample case.

The different way those 'typical distance' measures are affected by observations in different parts of the data is useful; it may be worth exploring an empirical influence function (without necessarily naming it, the concept is intuitive enough).

These lines of discussion may not be suitable for every level of student, but it's a take on two common measures of spread that may be easier to motivate that some others, and has at least one advantage: it reduces the size of that 1.35 "asymptotic conversion factor" at the normal to the perhaps less surprising 0.954 (or 1.048, depending on which way you go) - that's a good deal closer to 1, and it's smaller for the measure less affected by very large observations.