Question

A student wants to report on the number of movies her friends watch each week. The collected data are below: 2 14 1 2 0 1 0 2 Which measure of center is most appropriate for this situation, and what is its value? Median; 1. 5 Median; 3 Mean; 1. 5 Mean; 3.

Answer 1

Answer: Median = 1.5

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Explanation:

Original List = {2,14,1,2,0,1,0,2}

Sorted List = {0, 0, 1, 1, 2, 2, 2, 14}

The main cluster of values is from 0 to 2. The 14 is far from this cluster and is considered an outlier. The large outlier pulls on the mean to make the mean larger than it should be. Think of it like the gravitational pull of a distant planet.

If we were to compute the mean with the outlier, then:

mean = (0+0+1+1+2+2+2+14)/8 = 22/8 = 2.75

Now let's compute the mean without the outlier

mean = (0+0+1+1+2+2+2)/7 = 8/7 = 1.14 approximately

The true center is closer to 1.14 than it is to 2.75; mainly because most if not nearly all of the values are clumped around 0 to 2. Having a mean larger than 2 makes no sense if we were to ignore that 14.

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Because the mean is sensitive to any outliers, it's much better to go for the median. The median stays the same no matter how distant the outlier may be.

Refer back to the sorted list

{0, 0, 1, 1, 2, 2, 2, 14}

There are 8 items in this list. Circle the two middle-most items which are the "1, 2". They are tied for the middle. The midpoint is (1+2)/2 = 3/2 = 1.5

The median of the given list is 1.5 which is the most appropriate center.

Answer 2

The student's friend watch 1.5 movies every week.

Data provided to us {2, 14, 1, 2, 0, 1, 0, 2}.

Number of data values = 8

Sorted or rearranged data {0, 0, 1, 1, 2, 2, 2, 14}

Median of the data provided;

Median is the mid point value of the data provided to us, but as in this case there are 8 data provided (even number of data), we will take the average of the 2 mid values of the data.

Mid Values = 1 and 2

$\bold{Median=\dfrac{1+2}{2} = 1.5}$

Mean of the data provided;

$Mean=\dfrac{Sum\ of\ all\ data\ values}{Number\ of\ data\ values} \\\\Mean= \dfrac{(0+0+1+1+2+2+2+14)}{8}\\\\Mean=2.75$

As we know, The data values provided are 0, 1 and 2, and there is 14 as well in the data which is the maximum value in the data provided to us but the value is far from the other data values provided to us. Therefore, the data value 14 is considered to be an outlier.

An outlier is the maximum value that increases the mean, which makes it larger than it should be.

If we were to compute the mean without the outlier,

$Mean=\dfrac{Sum\ of\ all\ data\ values}{Number\ of\ data\ values} \\\\Mean= \dfrac{(0+0+1+1+2+2+2)}{7}\\\\Mean=1.1428$

Now, we have understood that the mean is sensitive to outlier can be easily manipulated. thus, mean is not the right option to go for in this case.

Therefore, the median value of 1.5 is the right option.

Hence, the student's friend watch 1.5 movies every week.

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