Does the mean describe the data well?

[eddy2017]

Hello, dear friends: I have come across this question as I study the central tendency (mean, median, mode and range), but I am having difficulty understanding the question. Could anybody give me a lead here?. The only thing I can think of is that the data set given is not arranged from the least to greatest, but I'm still uncertain, because the question asks me to tell if the mean describe the data well???. I do not know if that is what is being asked of me. I want you to confirm if I am drawing the right conclusion here.
Thanks for what you do everyday for people who struggle with math. I can't thank you enough.

Here's the question:

The heights of 5 starting players on the basketball team are 74 in,. 74 in., 73in., 70 and 68 in. Decide whether the mean describes the data well.

Thank you in advance for any help you can offer.
eddy.

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[lev888]

Please read https://en.wikipedia.org/wiki/Central_tendency or your textbook. Do you understand what mean is? It does not depend on whether the set of values is ordered, so not sure what you mean.

Let's say all 5 players have the height of 70". The mean is 70". Does it describe the data well? Sure.
What about 60, 65, 70, 75, 80? I guess.
What about 50, 50, 80, 84, 86? The same mean - 70. If you knew the mean, but not the individual heights, would you be surprised the first time you saw the players? I don't know the answer.
Read about the topic, maybe there are some formal rules regarding which measure is a better fit for a particular set of values.

 

[eddy2017]

I got it, thanks. Now, reflecting upon it, I think the answer is a yes. the mean describes the data well. There is no outlier, so the central tendency to be used is mean. That is what I can think of. I am not being asked to find the mean. I know how to do that. it is pretty easy. It is asking if the mean describes the data well, I think so. The answer is yes.
What do you think?

 

[JeffM]

First of all the question as posed is subjective. There probably are criteria for which of the common measures of central tendency is best for some specific purpose. But the question does not ask that. It asks a general question.

One way to go is to calculate all three of the common measures and see which one strikes you as most typical (understanding that this is a subjective assessment). The mode is 74, which does not recognize that most of the data are below that figure and none are below it. So I personally would say that the mode is not typical. The median is 73, and the mean is 71.8. The sum of the absolute deviations from the median relative to the median is (1+1+0+3+5)/73 = 13.7%. The sum of the absolute deviations from the mean relative to the mean is (2.2+2.2+1.2+0.8+3.8)/71.8 = 14.2%. That says the median is slightly better than the mean. The square root of the sum of the squared deviations from the median relative to the median is 8.2%. The square root of the sum of the squared deviations from the mean is 7.13%. That says the mean is slightly better than the median. So I'd conclude that the mean and the median are both decent measures of central tendency.

If I were giving a report to a board of directors (which I used to do), I'd give both the mean and the median. The fact that the mean is below the median provides extra information. If I were forced to choose, however, I'd go with the mean because more people understand it.

The point is that a measure of central tendency is a single number that summarizes many numbers. It almost always loses information. The question always is whether it loses too much information for the number to be highly informative. Ultimately, that is a subjective decision, but one that can be informed by careful thought.

 

[lev888]

Intuitively it would appear to be correct. Don't know enough about the subject to be sure.