Spread, the IQR and the Standard Deviation

The big idea

Spread is how far apart the values are, and two sets with the same center can differ completely in spread. The range, maximum minus minimum, is quick but depends on two values. The interquartile range, Q3 minus Q1, is the spread of the middle half, the width of the box, and like the median it resists extreme values; the quartiles are the medians of the two halves, found the way a graphing calculator finds them. The standard deviation grows out of the mean absolute deviation: instead of dropping the sign of each distance from the mean, square it, average the squares, and take the square root at the end to return to the original units, the same square, add, square root move as the distance formula, with the square root undoing the square. Dividing by n gives sigma, for a whole group; dividing by n minus 1 gives s, for a sample, slightly larger because a sample tends to underestimate the spread. A bigger standard deviation means the values sit farther from the mean, and 0 means every value is the same.