Which measure will be affected by an outlier the most?

a) mean
b) median
c) range
d) mode

Answer 1

Range

An outlier is a data point that is distant from the other observations. For instance, in a data set of #{1,2,2,3,26}#, 26 is an outlier. There is a formula to determine the range of what isn't an outlier, but just because a number doesn't fall in that range doesnt necessarily make it an outlier, as there may be other factors to consider.
The #color(red)(median)# is the middle number of a set of numerically ordered numbers. If the number of values in the set is odd, then the #color(red)(median)# is the central number, with equal amounts of data on both its left and its right. If the set has an even number of values, then the #color(red)(median)# is the average of the two central numbers. For example, in the set of #{1,2,3,4,5,6,7,8}#, there is an even amount of numbers, therefore we must find the mean of the two central numbers, which results in #(5+4)/2=4.5#, the #color(red)(median)# .
The #color(green)("range")# #r# is the distance from the highest value to the lowest value, and is calculated as #r=h-l#, where #h# is the highest value, and #l# is the lowest value. So if we have a set of #{52,54,56,58,60}#, we get #r=60-52=8#, so the #color(green)("range")# is 8.
Given what we now know, it is correct to say that an outlier will affect the #color(green)(ran)##color(green)(g)##color(green)(e)# the most. This is because the #color(red)(median)# is always in the centre of the data and the #color(green)(ran)##color(green)(g)##color(green)(e)# is always at the ends of the data, and since the outlier is always an extreme, it will always be closer to the #color(green)(ran)##color(green)(g)##color(green)(e)# then the #color(red)(median)#.
For example, take the set #{1,2,3,4,100}#, with 100 as the outlier. The #color(green)(ran)##color(green)(g)##color(green)(e)# of this set is #r=100-1=99#, while the #color(red)(median)# is 3. If we take the outlier 100 out, so the set is now #{1,2,3,4}#, the #color(green)(ran)##color(green)(g)##color(green)(e)# becomes #4-1=3#, while the #color(red)(median)# becomes #(3+2)/2=2.5#. Evidently, it was the #color(green)(ran)##color(green)(g)##color(green)(e)# which was affected the most.

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Answer 2

The measure most affected by an outlier is the mean. An outlier is a data point that significantly differs from other data points in a dataset. When calculating the mean, the outlier's value is included in the calculation, which can skew the average. As a result, the mean can be pulled towards the outlier, making it an unreliable representation of the central tendency of the data. Other measures such as the median and mode are less affected by outliers because they are not influenced by extreme values to the same extent as the mean. The median represents the middle value when the data is ordered, and the mode represents the most frequently occurring value, both of which can remain relatively stable even in the presence of outliers. Therefore, the mean is the measure most sensitive to the presence of outliers.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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