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the question states that no data set with only 5 terms has an outlier, and I'm stumped

you're given a set of 5 numbers in ascending order $$ x_1, x_2, x_3, x_4, x_5 $$

I started by finding the IQR, so

$$ \frac {x_4+x_5-x_1-x_2}{5} $$ then I used the outlier formula

(UB = Upper Boundary)

$UB = UQ + 1.5 * IQR $

to get

$$ 0.5 x_4+ 0.5 x_5 +0.75 x_4 +0.75 x_5 - 0.75 x_1 - 0.75 x_2 $$

then did the same for the lower boundary to get

$$ 0.5 x_1+ 0.5 x_2 +0.75 x_1 +0.75 x_2 - 0.75 x_4 - 0.75 x_5 $$

from there I don't know what to do, there must be something related to the numbers being in ascending order but I can't figure it out

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  • $\begingroup$ Please add the self-study tag & read its wiki. $\endgroup$
    – Stephan Kolassa
    Commented Mar 18 at 14:14
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    $\begingroup$ The entire concept of "outliers" is very dubious. I believe I am not saying too much if I state that the consensus here is that "outliers", as defined by this formula, should not, e.g., be blindly removed. The formula you are looking for is a tool to determine which data points should be plotted how in a boxplot - nothing more. $\endgroup$
    – Stephan Kolassa
    Commented Mar 18 at 14:15
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    $\begingroup$ That said, look at the formula for "upper outliers" and simplify it. Which of your data points would be a candidate for an "upper outlier"? What relation would it need to satisfy, given the (simplified) formula? Can it satisfy this relation? $\endgroup$
    – Stephan Kolassa
    Commented Mar 18 at 14:16
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    $\begingroup$ Welcome to CV, Bongo. There's actually no statistical content to this question (not your fault) and its answer will be of little value, because it's just an algebra problem in disguise. Moreover, this definition of "outlier" is not only questionable, it's idiosyncratic: the original formula for the upper boundary was $x_4 + 1.5(x_4-x_2),$ which will indeed highlight a sufficiently large value of $x_5$ for further examination: that is, as an "outlier." I would therefore suggest finding a different resource for learning this material. $\endgroup$
    – whuber ♦
    Commented Mar 18 at 15:00
  • $\begingroup$ Although I agree with @whuber that this is really an algebra problem, I think it should stay open (and I don't see any close votes) because the comments and answers have useful statistical content. $\endgroup$
    – Peter Flom
    Commented Mar 18 at 15:34

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As whuber said in a comment, this is really an algebra problem. And, as he and several others said, the whole question is a bit dubious (not your fault). But, to show this, just imagine that you sample 5 people and take their height. Just my chance, basketball player Victor Wembanyama is in your data set. He is 7 feet 4 inches (2.24 meters) tall. The other four people are of relatively normal height.

Any definition that says that Victor isn't an outlier is ludicrous. An outlier is a surprising point and Victor is surprising.

Rather than say "no data set with only 5 items has an outlier" it would be better to say "taking the IQR of a data set with only 5 items is silly, and any definition based on the IQR will be equally silly."

Good luck on the algebra.

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