The Semi-Interquartile Range

Dear Students,

Because of its name, the semi-interquartile range is thought to be something horrid by many, many students who are taking their first course in statistics. Actually, the SIQR is something very simple. By the time you finish reading this e-mail message, I guarantee that you'll know just about everything you need to know about the semi-interquartile range.

First, it's important to know that the SIQR is a measure of variability. Thus, it is like a range, a standard deviation, or a variance in that it measures the degree of "spread" among a group of scores. Like other measures of variability, the SIQR will be 0 if all the scores are the same; it will be larger than zero to the extent that the scores are spread out from each other. And like other measures of variability, the SIQR cannot ever be negative.

Next, you need to know that there are 3 quartile points within any group of scores. These 3 points divide the total group into 4 equally large sections (with each section containing 25% of the scores). For example, in a group of 20 scores, the 3 quartile points would divide the full data set into 4 groups of 5 scores. The lowest quartile, Q1, would be a numerical value having 5 scores below it and 15 scores above it. The middle quartile, Q2, would have 10 scores on each side of it. And the upper quartile, Q3, would have 15 scores below it and 5 scores above it. As you have probably guessed, Q1, Q2, and Q3 correspond exactly to the 25th percentile, the 50% percentile, and the 75th percentile, respectively. The middle quartile, Q2, being the 50% percentile, is, of course, equivalent to the median.

The SIQR is simply half the numerical distance between Q1 and Q3. It's somewhat like the range. However, R is based on the highest and lowest scores in the group whereas SIQR is based on the upper and lower quartile points.

A little numerical example will now make everything perfectly clear. Consider these 12 scores: 15, 14, 10, 4, 8, 9, 7, 20, 1, 3, 17, and 6. Now, since N = 12, one-fourth of the group is 3 scores. To determine what the upper and lower quartile points are, you now need to rearrange the scores from low-to-high or high-to-low. Here's the low-to-high reordering: 1, 3, 4, 6, 7, 8, 9, 10, 14, 15, 17, 20. The lowest quartile is positioned between the 4 and the 6, because this point will have 3 scores below it and 9 scores above it. This point, Q1, is the midpoint between the 4 and the 6. Thus, Q1 is equal to 5. The upper quartile is positioned between the 14 and the 15, because this point will have 9 scores below it and 3 scores above it. This point, Q3, is the midpoint between the 14 and the 15. Thus, Q3 is equal to 14.5. Now that we have the numerical values for Q1 and Q3, all we do is figure out half the distance between these 2 quartiles. Since these upper and lower quartile points are 9.5 points apart, SIQR is equal to 4.75.

There you have it! You now are an expert at thinking about and computing the semi-interquartile range. Way to go!!!!

Sky Huck

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Schuyler W. Huck
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