# In statistics, what would happen to the variance and standard deviation if the highest and lowest values were taken out? Explain why.I think they'd decrease, but I'm not totally sure why.

Asked on by lizziesed

embizze | High School Teacher | (Level 1) Educator Emeritus

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In statistics, what would happen to the variance and the standard deviation if the highest and lowest values were taken out?

In general they will both decrease, except in the case where all data entries are equal, in which case the variance and standard deviation were zero and remain zero after removing the two data points.

Consider the formula for the variance; in this case I would look at the "shortcut" formula:

`v=(n(sum x^2)-(sum x)^2)/(n(n-1))`

After throwing out the two data points, the denominator will decrease by 4n-6. [Take n(n-1) - (n-2)(n-3)]. A reduced denominator increases the value of a fraction, but the numerator is decreasing also. As long as the decrease in the numerator is more than the decrease in the denominator, the variance will decrease.

This is not universally true -- consider the data 1,1,1,1,1,9,9,9,9,9. If you calculate the variance for this data you get `s(x)~~ 4.216`

while the reduced set (eliminating highest and lowest values) yields `s(x) ~~ 4.276` .

The standard deviation is the square root of the variance, and since the square root function is increasing, if the variance shrinks so does the standard deviation.

So the answer is -- it depends on the data set. Some unusual sets may show an increase, but generally the variance (and hence the standard deviation) will decrease.

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