descriptive statisticsinterquartilemedianr
Take these data:
5, 8, 9, 14
R says the interquartile range is 3:
IQR(c(5, 8, 9, 14))
# 3
…but I make it 5. What am I doing wrong? Here are the steps I've taken:
8.5(5, 8), (9, 14)6.5 and 11.5, respectively6.5 from 11.5, which yields 5The correct answer should be similar, so that's probably correct; by my reckoning the box plot's lower inner-fence is -12.45 so your quartiles are probably fine (sample quartile definitions vary, but are generally close to Tukey's hinges, and - for some quartile definitions at least - sometimes hinges and quartiles do coincide).
Since all your observations are positive, there are no "outliers" (by the present definition) on the low side.
If your data are necessarily positive but sufficiently right-skew and with some values close to zero, negative values for the inner fence are quite common.
The rule has no way to know that negative values are impossible for your data; the rule isn't necessarily entirely appropriate for this kind of data, but it looks to me like you have correctly carried the calculation out on the data you have.
Incidentally, for the boxplot, here's an illustration of essentially the same "outlier" bound calculation:
Personally, I don't shy away from using standard deviation on likert scales, but there are other measures that might make you more comfortable. Like mean absolute deviation or median absolute deviation. Few people are familiar with these statistics, however. Likert scale statistics are often presented in terms of what the questioner cares about, such as "50% in the two top boxes." So you might consider something like that, such as 100% in the three middle boxes for B and 13/16% in the three middle boxes for A, which suggests the split responses in A.
Best Answer
Your difficulty is that in order to find IQR you must first find the two quartiles. And there are many different formulas for quantiles (including quartiles) in common use.
In particular, major statistical software packages disagree on which methods to implement as their default: (a) SAS, (b) Minitab and SPSS, and (c) R (and its parent S) use three different methods. Furthermore, these methods differ from methods found in reputable elementary texts. (Adding to the confusion: Tukey's 'fourths', sometimes used in making boxplots and often considered essentially the same as quartiles, use yet other criteria.)
Generally speaking the differences among these methods become negligible for large sample sizes. However, there can be marked differences for small samples. Fortunately, it is for large samples that quantiles make the most sense. (Roughly, quartiles are intended to divide a sample into four 'chunks' of equal size: how do you do that with a sample of size 10?)
In R, you can type
? quantileto see the nine different types of quantiles supported by R (using an extra argument), mentioned just now by @EdM. The default result from quantile is min, Q1, med, Q3, max, so once you have selected atypeyou could define your own IQR function based on the idea in the @Glen_b Comment and code likeas.numeric(diff(quantile(x, type=5)[c(2,4)])).