Range - Concepts such as mean, median, mode and range

[Don Winton]

Found this interesting:

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I am taking my first ever statistics course. However as an elementary teacher, I did teach such concepts as mean, median, mode and range. In this course we learned that range is the highest score/data minus the lowest then add 1 to the difference. I had taught that the way to figure out the range was to simply subtract the lowest score/data from the highest. This is also the way it is done in the statistics program I have. My husband (a statistician) agrees with my method. Which method is correct? If the first method is correct, please explain the reason one adds 1 to the difference. Thanks!

In response to Range: {Not Mine, Don}

Dona, you and your husband is right. The method you used of subtracting the smallest data value from the largest is the correct way of calculating the sample range. The only possible alternative I could think of would be if you wanted to construct an unbiased estimate of the population range.

For example, suppose you have a uniform distribution between A and B, with both limits being unknown. You then calculate from the data, the sample range, r = largest minus smallest. This r necessarily is less than the population range, R=B-A. To obtain an unbiased estimate of R, you need to multiply r by (N+1)/(N-1), where N is the sample size. Note that this is a multiplicative factor, not an additive factor. In short, I can't think of any reason to add one to the sample range.

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In this course we learned that range is the highest score/data minus the lowest then add 1 to the difference. I had taught that the way to figure out the range was to simply subtract the lowest score/data from the highest.

Thoughts,

Regards,
Don

 

[John C]

Don,
My statistical training was always sketchy and is now very rusty, so I offer this only as a possible clue to the problem. I can’t vouch for it’s validity;
For variables, the range is highest minus lowest. That’s the way I was taught. eg measurement in inches; Highest 10, lowest 5. 10 - 5 = 5.
For attributes, it’s the number of consecutive attributes in the sample (or potential attributes, if all possible have not been observed) that gives the range, eg; marks on a screen; Highest sample had 10, lowest had 5. The range can be considered to include; 5, 6, 7, 8, 9 and 10. So that’s 6 levels of attributes. 10-5 = 5. 5+1 = 6.
rgds, John C

 

[Kevin Mader]

Don,

As part of figuring Sigma, finding the estimation and unbiased estimation is merely the result of data from a population or from a sample. As always, the sample always has potential error beyond the population estimate and n-1 is normally used.

Not sure why to add/subtract a 1 for the Range though. JC's explanation seems reasonable, but I can not confirm.

 

[Don Winton]

For variables, the range is highest minus lowest. That’s the way I was taught. eg measurement in inches; Highest 10, lowest 5. 10 - 5 = 5. For attributes, it’s the number of consecutive attributes in the sample (or potential attributes, if all possible have not been observed) that gives the range, eg; marks on a screen; Highest sample had 10, lowest had 5. The range can be considered to include; 5, 6, 7, 8, 9 and 10. So that’s 6 levels of attributes. 10-5 = 5. 5+1 = 6.

Agreed. I believe the response cited was probably considering this example. Or rather, see below.

As part of figuring Sigma, finding the estimation and unbiased estimation is merely the result of data from a population or from a sample. As always, the sample always has potential error beyond the population estimate and n-1 is normally used.

It is also possible that the respondent was thinking sigma rather than range. I posted this in order to see what the responses would be. Many thanks, guys.

Regards,
Don