13.3 Ranking Data

The last few sections in this chapter require one to rank a data set. To rank a data set, you first must arrange the data from smallest to largest. The smallest value gets a rank of 1, the next smallest gets a rank of 2, etc. If there are any values that tie, then each of the tied values gets the average of the corresponding ranks.

# rank() averages tied ranks by default - exactly the midrank rule above
x <- c(8, -4, 1, -3, 5, 2, -3, 0, 5, 3, 5)
data.frame(sorted = sort(x), rank = rank(sort(x)))
# -> 1, 2.5, 2.5, 4, 5, 6, 7, 9, 9, 9, 11  (both -3s share 2.5; the three 5s share 9)

# Second sample: what rank does 15 get?
y <- c(10, 25, 15, 8, 20, 15, 10, 9, 8, 22)
rank(y)[y == 15]   # -> 6.5

# Change one value to break a tie and watch the whole rank column shift

Example 1

Rank the following random sample: $8, -4, 1, -3, 5, 2, -3, 0, 5, 3, 5.$

Show solution

First, sort the data from smallest to largest: $-4, -3, -3, 0, 1, 2, 3, 5, 5, 5, 8$

Next, rank the data. The $-4$ gets a rank of 1. There is a tie between the next two values of $-3$ . They would have received the ranks of 2 and 3, but we do not want one of the values to be ranked higher than the other so we give both $-3$ ’s a rank of $\frac{2+3}{2} = 2.5$ . Then the next value of $0$ gets a rank of 4 (we already used the 2nd and 3rd positions). The next set of ties for the three $5$ ’s occurs for the rank of 8th, 9th and 10th place. The average of these ranks is $\frac{8+9+10}{3} = 9$ . The following is a table of the sorted data with the corresponding ranks.

Data $-4$ $-3$ $-3$ $0$ $1$ $2$ $3$ $5$ $5$ $5$ $8$ Rank 1 2.5 2.5 4 5 6 7 9 9 9 11

If there is no tie for the last data point, then your last rank will be the same as your sample size.

Example 3

The Tevis Cup Ride is a 24-hour, 100-mile horse race over the Sierra Nevada mountains from Lake Tahoe to Auburn in a single day. The top 10 completion times for 2019 are shown below. Rank the completion times.

Name Completion Time Sanoma Blakeley $09\text{:}27 \text{ PM}$ Jeremy Reynolds $09\text{:}27 \text{ PM}$ Haley Moquin $09\text{:}36 \text{ PM}$ Richard George $09\text{:}37 \text{ PM}$ Suzanne Huff $09\text{:}54 \text{ PM}$ Karen Donley $09\text{:}54 \text{ PM}$ Nicki Meuten $10\text{:}06 \text{ PM}$ Gwen Hall $10\text{:}20 \text{ PM}$ Lindsay Fisher $10\text{:}28 \text{ PM}$ Suzanne Hayes $10\text{:}29 \text{ PM}$

Show solution

The data are already ordered. There are two ties at $9\text{:}27$ and $9\text{:}54$ .

Time	$9\text{:}27$	$9\text{:}27$	$9\text{:}36$	$9\text{:}37$	$9\text{:}54$	$9\text{:}54$	$10\text{:}06$	$10\text{:}20$	$10\text{:}28$	$10\text{:}29$
Rank	1.5	1.5	3	4	5.5	5.5	7	8	9	10

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.