Ties and Average Ranks

Why Ties Need a Consistent Rule

Ranking replaces measured values with their positions in an ordered list. If every value differs, each observation has an unambiguous rank. But real data often contain repeated values: measurements may have been rounded, or several individuals may genuinely share the same recorded value. A ranking rule must give equal observations equal treatment rather than arbitrarily deciding that one tied observation came first.

The standard solution is to assign each observation in a tied group the average of the rank positions that group occupies. These are called average ranks, or midranks. A midrank need not be a whole number. It is a summary of the tied observations' positions, not a new measurement in the original units.

The ranking set must be defined before assigning ranks. For a pooled comparison, rank all observations in the specified groups together. For a within-subject comparison, rank the measurements separately within each subject if the analysis is based on each subject's relative ordering. These choices answer different questions; midranks resolve ties within the chosen ranking set, but do not determine which set is appropriate.

Calculating Midranks

For a tied group, find its first and last rank positions and average them. For example, if three equal observations occupy positions 4, 5, and 6, each receives rank \((4+6)/2=5\). The three ranks are therefore 5, 5, and 5, not 4, 5, and 6. Their assigned ranks still add to 15, the sum of the occupied positions.

Why the Total Rank Sum Is Preserved

Midranks preserve the total of all ranks because each tied group receives, in total, exactly the sum of the positions it occupies. This fact is useful as an arithmetic check and explains why average ranks are a natural way to handle ties.

To justify the result, consider one tied group occupying positions \(a\) through \(b\), with \(m=b-a+1\) observations. Its midrank is \((a+b)/2\), so the sum of the \(m\) assigned ranks is \[ m\frac{a+b}{2}. \] The integer positions from \(a\) through \(b\) form an arithmetic sequence with \(m\) terms and average \((a+b)/2\). Their sum is also \(m(a+b)/2\). Thus the tied group contributes the same total as its occupied positions would have contributed without averaging. The tie groups and untied observations together partition all positions \(1,\ldots,N\), whose sum is \(N(N+1)/2\). Therefore the overall rank sum is unchanged.

This result concerns the sum of the ranks, not the individual ranks or every statistic calculated from them. Replacing distinct ranks with equal midranks can change a group’s rank sum, a contrast between groups, or another statistic that depends on which observations belong to which group.

Ties in Pooled Groups and Rank Sums

A rank sum is the sum of the ranks belonging to a specified group. In a pooled ranking, tied observations receive the same midrank regardless of group membership. The resulting rank sums can be fractional, and their total across all groups must equal the overall rank total.

Repeated Measurements: Choose the Ranking Set

Repeated measurements make the ranking set especially important. If the aim is to compare measurements within each subject, rank that subject's measurements separately. A tie between two measurements for one subject receives a midrank within that subject's measurements. The ranks then describe relative position within a subject, not the subject's position among every measurement in the dataset.

What Ties Change in a Rank-Based Statistic

A statistic calculated from ranks depends on how ranks are assigned to observations. Midranks preserve the total rank sum, but they can alter a particular group’s rank sum or any statistic that weights ranks by group membership. The pooled example showed this directly: the tied observations contributed \(4+4\) to Group A and 4 to Group B. Which group contains each tied observation matters to the group-specific sums.

The same principle applies to other rank-based summaries. If a statistic adds selected ranks, multiplies ranks by fixed weights, or forms differences between rank sums, use the midranks assigned under the stated ranking rule. Do not replace them with arbitrary consecutive ranks, and do not round them before calculating. For example, rounding a midrank of 2.5 to 3 changes the sum and can change a downstream statistic.

A rank sum by itself is not enough to establish statistical significance. Inference requires a particular procedure, its assumptions, and a suitable reference distribution; methods for those steps are addressed separately. The purpose here is to get the rank assignments and arithmetic right so that any later calculation starts from a consistent set of ranks.

Common Mistake: Breaking Ties Arbitrarily

A tempting approach is to sort tied observations by their row order and give them consecutive ranks. That assigns different ranks to observations with the same recorded value. In a group comparison, it can also make the result depend on which tied observation happened to be listed first.

The correction is to average all the rank positions occupied by the tie and assign that average to each tied observation. If the tie occupies positions 6 and 7, both observations receive rank 6.5. Then retain the decimal ranks through the rank-sum calculation and check the overall total. Remember also that tied values must be identified using the precision of the data as recorded: values recorded as equal receive a tie under that ranking, even if the underlying unrounded quantities might have differed.