Answers:
- Mean = 88.5
- Median = 91
- Mode = 91 and 93 are both the mode
- The median is the best measure of center
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Explanation:
Part (a)
To get the mean, first we add up the scores
89+91+94+93+93+67+90+91 = 708
Then we divide over 8 since there are 8 scores total
708/8 = 88.5
The mean is 88.5
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Part (b)
To get the median, we first need to sort the values from smallest to largest.
That sorted set looks like: {67,89,90,91,91,93,93,94}
Now cross off the first and last values to get this smaller subset: {89,90,91,91,93,93}
Repeat the last set of steps to cross off the first and last items to get {90,91,91,93}
Do this one more time to get {91,91}. The midpoint of these two values is 91, so it's the median of the entire original set.
The median is 91.
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Part (c)
The mode is {91, 93}
In other words, we have two modes and they are 91 and 93. It's perfectly possible to have more than one mode. The mode is the most frequent item(s). In this case, both 91 and 93 show up exactly twice which is the most of any other value.
As you can see, the mode isn't particularly useful as a single measure of center since we have two values instead of 1.
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Part (d)
As mentioned in the previous section, the mode isn't useful in this case. So we can rule it out. We can also rule out the mean because the value 67 is far from the group of other values (which is basically like the main cluster). We consider 67 an outlier.
The outlier pulls on the mean to be smaller than it should be. The median is a better measure of center since it's not affected by outliers.
As a real world example, consider home prices. The median price is used because of very expensive multimillion dollar mansions that greatly skew the mean to be larger than it should be.
