Quartiles and the Interquartile Range

You used positions to locate a median in grouped data. Here the race times are exact: 21, 23, 24, 26, 27, 29, 30, 32, 35, 41 and 58 minutes. Order values from smallest to largest before finding a median or quartile. With eleven values, the median is at position (11 + 1) / 2 = 6: 29 minutes. For an even count, average the two middle values.

Quartile conventions differ. Here, split an ordered list into lower and upper halves. Exclude the central observation when the count n is odd. Each half's median is its quartile. For odd n, their positions are (n + 1) / 4 and 3(n + 1) / 4. Average adjacent values for a halfway position. Eleven runners give positions 3 and 9: 24 and 35 minutes. Subtract lower from upper to find the interquartile range.

Quartiles mark roughly the 25% and 75% ranks. Finite lists and repeated values need not give equal counts in the intervals; count entries for an exact answer. The median need not be halfway between quartile values. IQR cannot exceed range because quartiles lie within the extremes. If the quartiles are equal, every ordered value between their positions equals that value.

Suppose only the last race time increases from 58 to 90 minutes. Positions 3 and 9 still hold 24 and 35, so the IQR stays 11. The range becomes 90 - 21 = 69. This shows resistance to an extreme value, not immunity to every change. Changing a value used for a quartile can change the IQR. Adding a new observation also changes the count, so find the quartiles again.

Adding the same amount to every observation shifts the median and both quartiles by that amount. Their difference stays unchanged, so equal IQRs need not mean equal medians. Sets can also share a median and quartiles while having different extremes. To compare their ranges, calculate each maximum minus minimum, then subtract the two ranges.

For an even list, keep every observation and split it into two equal halves. Find each half's median, averaging its middle pair when needed. You can recover a missing value: if a quartile is (a + b) / 2, then b = 2 × quartile - a. If the lower quartile is missing, use lower quartile = upper quartile - IQR first. Check that any recovered value preserves the list's order.

You can reverse a quartile-position calculation. For an odd-sized list, a lower-quartile position p satisfies p = (n + 1) / 4. Multiply by 4 and subtract 1 to get n = 4p - 1. Check that the resulting count is odd before using this rule.

To find quartiles for combined groups, collect every observation from both groups, keeping repeated values. Sort the combined list, then apply the stated quartile convention to its new count. Do not average the groups' IQRs to obtain the combined IQR. Each group's quartiles describe positions within that group; combining groups changes which observations occupy the quartile positions.