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Quantitative Literacy and the Humanities

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Statistical thinking: Variability

Students recognize and interpret measures of variability--that is, how dispersed the data points are from the mean.  Students recognize that two data sets can have the same median and mean, yet differ in their variability. 

Several measures of variability are possible; two are considered here.

Interquartile range

The range between 25% and 75% of a set.  Statistics about incoming students, such as SAT scores and GPA, are often presented in terms of interquartile range.

Standard deviation

The standard deviation tells us how dispersed the data points are from their mean.  It usually takes the units of the thing being measured.  For instance, the standard deviation of human height may be expressed in inches.

The standard deviation is related to the normal distribution. 

By definition, 68.2 percent of observations lie within one standard deviation of the mean (34.1 percent bigger than the mean and 34.1 percent smaller than the mean).  A total of 95.4 percent of observations will lie within two standard deviations of the mean, and 99.7 percent within three standard deviations.

Z-score is another word for the number of standard deviations a data point is from the mean.  The higher the absolute value of the z-score, the further the data point is from the mean.

You can also use the "five number summary": report the minimum value, first quartile value (Q1), median, third quartile (Q3), and maximum value. 

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