The mean and standard deviation of a sample may change if dataare rescaled (for

Question

The mean and standard deviation of a sample may change if dataare rescaled (for instance, temperature changed from Fahrengeit toCelsius). For a sample with mean xbar, adding a constant c to eachobservation changes the mean to xbar + c, and the standarddeviation s is unchanged. Multiplying each observaiton byc>0 changes the mean to cxbar and the standard deviation tocs. a) Scores on a difficult exam have a mean of 57 and astandard deviation of 20.   The teacher boosts all thescores by 20 points before awarding grades. Report the meanand standard deviation of the boosted scores. Explain whichrule above you used, and identify c. b) Suppose that annual income for some group has a mean of39,000 and a standard deviation of 15,000. Values areconverted to British pounds for presentation to a Britishaudience. If one British pound equals 2.00, report the meanand standard deviation in British currency. Explain whichrule above you used, and identify c. c). Adding a constant and/or multiplying by a constant iscalled a linear transformation of the data. Do lineartransformations change the shape of the distribution? Explainyour reasoning. The mean and standard deviation of a sample may change if dataare rescaled (for instance, temperature changed from Fahrengeit toCelsius). For a sample with mean xbar, adding a constant c to eachobservation changes the mean to xbar + c, and the standarddeviation s is unchanged. Multiplying each observaiton byc>0 changes the mean to cxbar and the standard deviation tocs. a) Scores on a difficult exam have a mean of 57 and astandard deviation of 20.   The teacher boosts all thescores by 20 points before awarding grades. Report the meanand standard deviation of the boosted scores. Explain whichrule above you used, and identify c. b) Suppose that annual income for some group has a mean of39,000 and a standard deviation of 15,000. Values areconverted to British pounds for presentation to a Britishaudience. If one British pound equals 2.00, report the meanand standard deviation in British currency. Explain whichrule above you used, and identify c. c). Adding a constant and/or multiplying by a constant iscalled a linear transformation of the data. Do lineartransformations change the shape of the distribution? Explainyour reasoning.

Explanation / Answer

Basically, what you need to know in intro data transformationis that adding to the values affects only the Mean, not the SD.Multiplying data affects both the Mean and SD. With that it mind,I'll go through the problems. a. c = 20. Because we only added, the newMean and SD are: = + 20 = 57 + 20 = 77 = 20 We used the first rule. b. c = 1/2. Because we multiplyed, the newMean and SD are: = * 1/2 = 19500 = * 1/2 = 7500 We used the second rule. c. No. A linear transformation changes onlythe scale of the distribution, it does not affect theshape of the distribution. Because linear transformations changeonly the scale of a distribution, the relative distances andcomparisons among data remain exactly the same. If you measured theheights of two males, A and B in inches, and A was taller than B,if you changed the data to cm, A would still be taller than B. Thisis an example of a linear transformation.

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