# Teaching range as a measure of spread

This was originally part of a question on averages but it was suggested to me that it should be a question in itself. If people think the question is too similar it can be deleted.

I'll link the questions when I've worked how to do it. How do you teach range to students so that they get a good understanding of range as a measure of spread. To be clear computing highest take lowest is not really the issue as almost all can do that. I'm trying to combat statements such as "the highest range is the best" in inappropriate situations.

• present students with multiple situations, some where a large spread is good and some where a small spread is good – celeriko Mar 23 '15 at 18:44
• You can also look at quartiles to help subdivide and find outliers. – Chris C Mar 23 '15 at 18:50

This may be too complicated, but you could find more straightforward plots of data with vertical ranges that make the same point. (Image from AMS Positron Excess.)

• I like the idea of comparing vertical ranges. I'll look at the site. – Karl Mar 24 '15 at 8:20

In our high-school curriculum, range and especially inter-quartile range (IQR) have replaced standard deviation as the main measures of spread. It has the following benefits:

1. Student are able to quickly ascertain them manually, without tons of mindless calculations.

2. It is obvious what they mean, unlike the "black box" of standard deviation calculations, so students can intuitively discuss samples in real language based on these statistics.

3. Simple box-plots based on these range measures simultaneously display range, median, IQR, and skew. The shape of the distribution is not hidden as is the case in mean/SD summaries.

4. The median, range, and IQR are often more useful for quick, informal population inferences especially in the presence of outliers and non-normal distributions.

5. Traditional confidence intervals are still easy to calculate. For example: median +/- 1.5 * IQR is equivalent to 2 standard deviations either side of mean as the basis of a 95% confidence interval on a normal distribution. On non-normal distributions mean and SD are problematic anyway.

6. For more advanced calculations, such as difference between medians/means, the curriculum specifies that students use bootstrapping/Montecarlo techniques to give accurate confidence intervals. Montecarlo is easy for students to understand and intuitive influences are more easily drawn. Though the CLT says the distributions tend towards normality, in reality it is often far from normal and automated Montecarlo techniques can be more accurate. The resulting Montecarlo distributions can again be analysed using the median/IQR distribution summaries.

7. As a bonus, Montecarlo techniques can easily be extended to create confidence intervals for arbitrarily complicated hypotheses, such as regression analysis which is impossible with mean/SD. Advanced Montecarlo is not part of the high-school curriculum.

This methodology is not intended to replace traditional techniques to be learned at university, but they give a good grounding in statistical concepts of centre, range, comparing samples, and making inferences. The concepts learned are not tied to traditional mean/SD and are easily extended to modern statistical techniques.

• This is all good as insight into why range and IQR are prominent in the curriculum, but does not really address the OP's question, which was about how to teach it to students so that they will understand it. – mweiss Mar 25 '15 at 2:27