Putting the Data Skills Together

Suppose a company surveys 5 out of 400 employees, its full populationThe entire group being studied, from which a smaller sample may be collected., about commute time in minutes: 10, 15, 15, 20, 60. Step one is always to check the sample sizeThe number of data values collected from a population to represent it in a data set.: is 5 out of 400 enough to trust? That's a real judgment call, and probably not on its own, an analyst would want a larger or more carefully chosen sample before reporting conclusions to the whole company.

Step two is to build a box plotA diagram that displays a numerical data set using five key values: the lowest number, first quartile, median, third quartile, and highest number.. Order the data (it already is), find the median, 15, then the first quartileOne of the values that divides an ordered data set into four equal-sized groups. of the lower half {10, 15}, which is 12.5, and the third quartile of the upper half {20, 60}, which is 40. The lower extremeThe least value in a data set, shown at the left end of a box plot. is 10 and the upper extremeThe greatest value in a data set, shown at the right end of a box plot. is 60. Step three is to describe the shape: this data set is skewA distribution shape where most of the data bunches on one side and a tail stretches out toward the other side.ed, with a long tail toward 60, which is also an outlierA data value that is much greater or much less than the rest of the values in a data set., very different from the rest of the commute times. Because of that outlier, median is the more appropriate measure of center here, not mode, since no value repeats often enough to be meaningful, and this data set isn't symmetricA distribution shape where the data is evenly balanced on both sides of the center., uniformA distribution shape where the data values are spread out fairly evenly across the range, with no clear peak., or bimodalA distribution shape with two separate peaks, meaning the data clusters around two different values..

Step four is to calculate all four statistics precisely: the medianThe middle value of a data set when the values are ordered from least to greatest. is 15, there's no clear modeThe value or values that occur most often in a data set. since 15 is the only repeated value, the rangeThe difference between the greatest and least values in a data set. is 60 minus 10, or 50, and the interquartile rangeThe distance between the first quartile and the third quartile, showing the spread of the middle half of a data set. is 40 minus 12.5, or 27.5. Notice how the range, 50, is much larger than the interquartile range, 27.5, exactly because that one outlier stretches the range without affecting the middle half of the data nearly as much. That's the whole toolkit, working together on one honest data set.

Adapted from Studio Aletheia's The Center for Quantitative Studies curriculum library, drawing on mathematical resources and the SC CCR Mathematics Standards.