Question 274180: If each score in a set of test scores is divided by 10, describe the effect of the division on the mean, median, mode, range, and standard deviation of the set.
Answer by solver91311(24713)   (Show Source):
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The mean:
The mean of every data element divided by 10:
Factor out 
\left(\frac{\sum_{i=1}^n{x_i}}{n}\right))
Hence 
The median is either an actual data element, in the case of an odd number of data elements, or it is the average of two data elements, in the case of an even number of data elements. In the first case, if all of the data elements have been divided by 10, the one selected for the median would also be divided by 10. In the second case, the mean of two data elements divided by 10 would also be divided by 10 (see discussion of the mean, above).
The mode is, in all cases, one of the data elements. If all of the data elements have been divided by 10, then the mode must be divided by 10 as compared to the original.
The range:
With the data elements divided by 10:
Factor out
\left(x_{max}\ -\ x_{min}\right))
Hence 
Standard Deviation:
\sum_{i=1}^n{\left(x_i\,-\,\overline{x}\right)^2})
The mean of every data element divided by 10 and every instance of the mean divided by 10 as demonstrated earlier:
\sum_{i=1}^n{\left(\frac{x_i}{10}\,-\,\frac{\overline{x}}{10}\right)^2})
Factor out
\sum_{i=1}^n{\left(\left(\frac{1}{10}\right)\left(x_i\,-\,\overline{x}\right)\right)^2})
\left(\frac{1}{100}\right)\sum_{i=1}^n{\left(\right)\left(x_i\,-\,\overline{x}\right)^2})
\sqrt{\left(\frac{1}{n}\right)\sum_{i=1}^n{\left(\right)\left(x_i\,-\,\overline{x}\right)^2})
Hence 
John
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