An important use of statistics is to measure variability or the spread ofdata. For example, two measures of variability are the standard deviation andthe range. The standard deviation measures the spread of data from the mean orthe average score. Once the standard deviation is known other lineartransformations may be conducted on the raw data to obtain other scores thatprovide more information about variability such as
The standard deviation can be an effective tool for teachers. The standarddeviation can be useful in analyzing class room test results. A large standarddeviation might tell a teacher the class grades were spead a great distance fromthe mean. A small standard deviation might reflect the opposite of thepreceding. In analyzing test results, a teacher can make an assumption thatwith a small standard deviation the students understood the material uniformly.With a large standard deviation the teacher may assume there is a large amountof variation in regards to students understanding the material tested.
One of the limits of statistics can be found in calculating the range. Sinceoutliers are used to determine the range, they are very influential in theresulting statistic. For example, if one class had test grades of 0%, 50%, and100%, and another class had grades of 50%, 90%, and 100% the range is differsgreatly between the two classes due to the outliers. The ranges would be 100 and 50 respectively.
When analyzing test results it is helpful to group scores intonumber of standard deviatons from the mean. For the data set [3, 5, 7, 7, 7,38] the mean is 11.16 and the standard deviation is 13.24, one standard deviationfrom the mean would be 11.16 plus or minus 13.24 or 24.40 and -2.08 and two standard deviations from the meanwould be 11.16 plus or minus twice 13.24 or 37.64 and -15.72. Imagine that the above data are points scored in abaseball game. Thirty-eight is between two and three standard deviationsabove the mean.