ThreeWave Ranking Data In Lists

This page describes worksheet formulas you can use to work with data ranking in Excel worksheets.
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Introduction

Excel is often used to keep track of data that needs to be ranked. This might be anything from sports scores to sales data. Excel provides a worksheet function named RANK that you can use to do rudimentary ranking, but this function is rather limited. However, we can use some worksheet formulas to provide a much deeper and meaningful anlaysis of data using the RANK function.

You can an example workbook with the formulas described on this page.

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Simple Ranking

The simplest use of ranking is to get the rank of one or more values in a list. The rank of a number in a list is the position at which the value would be placed if the data list were sorted. The RANK function does exactly this and it supports ranking unsorted data, assigning ranks in either ascending order or descending order. Descending order is the default for the RANK function. In descending order, the higher score is given the numerically lower rank. For example, if you have bowling scores of {150, 125, 180, 175}, the value 180, the highest value in the list, is given a rank of 1 and the value 125, the lowest value in the list, is given the rank of 4. Even though in common language usage, the highest score is said to be the "highest ranked" score, the actual numeric value of the rank is the lowest. (E.g., "the highest ranked bowling score was 180", may sound correct, but the rank of 180 is actually the lowest rank value, 1). When you rank in ascending order, the lower values are given the numerically lower rank. Golf scoring is an example in which you would use ascending ranking. The lower the score the better, so lower values are assigned lower numerical rankings. With the scores {72, 88, 75, 82}, the value 72, the lower score in the list, is given a rank of 1 and 88, the highest score in the list, is given a rank of 4.

The RANK function is essentailly the inverse of the LARGE (and SMALL) function. While RANK returns the rank of a value in a list, LARGE returns a value based on its rank. If cells B4:B8 contain the values {5,2,4,3,1}, the formula =LARGE($B$4:$B$8,RANK(B4,$B$4:$B$8,0)) in cells C4:C8 returns the original data in the original order, confirming the relationship between RANK and LARGE. A similar formula can be created with the SMALL function: =SMALL($B$4:$B$8,RANK(B4,$B$4:$B$8,1)).

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Unique Ranks In Descending Order

One of the features (or failures, depending on your perspective) of the RANK function is that it returns the same rank value for items of equal value. For example, with the data {33, 22, 44, 22, 11} the RANK function returns a rank of 1 to the value 44, the value 2 for 33, the value 3 for both occurrences of 22, and then a rank of 5 for 11. No value is ranked as 4. You can use a formula to prevent this duplication of rank value. For example, the following formula entered in cell B1 and filled down to B5 will return unique ranks of the values in A1:A5

=RANK(A1,$A$1:$A$5,0)+COUNTIF($A$1:A1,A1)-1

To see how this formula works, enter the values 33, 22, 44, 22, 11 in cells A1:A5 and enter the formula =RANK(A1,$A$1:$A$5,0)+COUNTIF($A$1:A1,A1) in cell B1. Fill the formula down from B1 through B5. This will return the values 2, 3, 1, 4, 5 to cells B1:B5. Examine the formulas in cells B2 and B4, the cells that are adjacent to the duplicated value 22. In B2, the RANK returns 3 (as it should), and the COUNTIF function is COUNTIF($A$1:A2,A2) which returns 1. Subtract 1 from the result of COUNTIF and you'll get 0, which when added to the value 3 from RANK gives us 3, the correct rank of the first occurrence of 22. Now, look at the formula in cell B4, which is adjacent to the second occurrence of the value 22. As before, the RANK function returns 3, but the COUNTIF function is now COUNTIF($A$1:A4,A4) which returns 2, the number of values 22 in range A1:A4. Subtract 1 from that result and add that to the result of RANK, and you'll get 4. This formula will work with any number of duplicated values. The reason that it works is that for any value V that occurs N times in a list, with a rank of R, there are N rank values R, and the next N-1 occurrences of V have the rank R. This means that N-1 rank values are omitted and the next rank value is R-N. The COUNTIF()-1 piece of the formula adds back N-1, and R+N-1 is the rank of the Nth occurrence of the value V. For as many occurrences N of V exist in the list, the RANK function omits N-1 rank value. The COUNTIF()-1 simply adds those missing rank values back into the ranking. In the following screen shot, the original data values to be ranked are shown in blue, the intermediate calculated results in black, and the final results shown in .

RankSimple

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Unique Ranks In Ascending Order

The logic for unique ascending ranks is a bit different than that for descending unique ranks. With your data in the range A1:A5, enter the following formula in cell B1 and fill down to cell B5.
=COUNT($A$1:$A$5)-(RANK(A1,$A$1:$A$5)+COUNTIF($A$1:A1,A1)-1)+1

The formula works in the following manner. The total number of elements minus the descending rank of a value is that value's ascending rank as if the rankings were 0-based rather than 1-based. For example, the unique descending rank of the value 33 is 2. Subtracting this from the total number of elements, 5, returns the zero based rank of 3. Adding 1 brings the rank to a 1-based system and therefore 33 gets the rank of 4. It is important to note that even though this formula calculates the ascending ranks, it uses the descending format of the RANK function. In the screen show below, the data to be ranked is shown in blue, the intermediate calculations in black, and the final results in red.

UniqueAscending

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Descending Player Rankings

While the formulas illustrated above are useful for the direct ranking of data, it is very often the case that you want to return a ranked list of the players or other entities that have ranked scores or values. For example, a ranking of bowling scores is useful really only when you can retrieve the players in scored rank. The actual scores are less useful than are the player names themselves. This section will create a list of players according to the ranked scores, in descending rank order.

In the following example, the player names are in B4:B13 and each player's score is in C4:C13. We use the following formula to calculate the unique descending rankings of the players' scores. This is the same formula descibed earlier in this article.

=RANK(C4,$C$4:$C$13,0)+COUNTIF($C$4:C4,C4)-1

Enter this formula in cell E4 and fill down to E13. This gives us the unique rankings of the player's scores that we will use to retrieve the player names in scored order. Enter the following formula in cell G4 and fill down to G13. This will list the names of the players, listed in B4:B13 in the order of the unique ranks in E4:E13.

=OFFSET(B$4,MATCH(SMALL(E$4:E$13,ROW()-ROW(G$4)+1),E$4:E$13,0)-1,0)

In the screen image below, the input data is shown in blue, the unique rankings in black, and the final results in .

RankPlayers

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Ascending Player Rankings

The previous section display players ordered by a descending unique rank. The same players may be ordered by ascending rank order. With the player names in cells B18:B27 and their scores in cells C18:C27, enter the following formula in E18:E27

=COUNT($C$18:$C$27)-(RANK(C18,$C$18:$C$27,0)+COUNTIF($C$18:C18,C18))+2

This formula will create unique player ranks in ascending order. These ranks will be used to create the order list of players. In cell G18, enter the following formula and fill it down to cell G27.

=OFFSET(B$18,MATCH(SMALL(E$18:E$27,ROW()-ROW(G$18)+1),E$18:E$27,0)-1,0)

This will list the players in ascending rank order. The screenshot below illustrates this.

PlayersAscending

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Double Ranking And Tie Breaking

In the examples above, we had tied scores, two players having the same score. In those scenarios, there was no way to break a tie. The tied elements would appear in the result list according to their position in the input list. This section of the article describes how to use multiple tables to resolve ties. We first have a primary table that contains the players name and their scores. Next, we have a secondary table that is used to resolve ties. If two player have the same score in the primary table, those players' scores in the secondary table are used to break the tie. (If those same player have a tied score in the secondary table, the results are returned in the order in which they appear in the secondary list.

The core of working with the two lists and breaking the ties in to create a third table, called the Composite table. This table lists all of the player and calcualtes a composite rank that is calculated from the player's rank in the primary table and her rank in the secondardy table. Assuming you have the primary table with player names and scores in the range B6:C15, called Table1, the secondary table, also wih player names and scores, in cells E6:F7, named Table2, and the composite table in cell H6:J16, with Player Names in column H, the composite score (to be calcuated later) on column I, and the Composite Rank (also discussed later) in column J, you can list the players by score in descending rank in column L and the players by score in acsendning rank in column N. The basic layout of the tables is shown below. The rest of this section will discuss how write the formula and what those formulas actually do.

The formulas on this page are Array Formula, so you must press CTRL SHIFT ENTER rather than ENTER when you first enter the formula and whenever you edit it later. If you do this correctly, Excel will display the formulas enclosed in curly braces { }. See the Array Formulas page for more information about working with array formulas.

Rank2Players

In the cells of column I in Table1 enter the following formula:

=(C6/MAX(ABS($C$6:$C$15)))+(F6/(10^(MAX(LEN(C$6:C$15)+1))))

This formula calculates the Composite Score. This score is calculated with the following formula:

=(C6/MAX(ABS($C$6:$C$15)))+(F6/(10^(MAX(LEN(C$6:C$15)+1))))

For each row in the Composite Score, this calcualtes the data value divided the maximum data element. This number, N is subject to constraint 0 <= N <= 1. Added to this is is the value of the secondary (tie breaking) rank divided by (10^(MAX(LEN($C$6:$C$15)+1). This calculated value is 10 to power of Z where Z is the longtest (in terms of character length, not numeric value) of the numbers in cells C6:C15. If the longest number has six characters (e.g., 123.456), the formula (10^(MAX(LEN($C$6:$C$15)+1) reutrns 10^7. When we divide that into the cell value from the score from the secondary table, we get a fractional number that is 0's until the number of digits surpass the digit to right of the decimal point from the score value from Table2. Finally, to break ties between scores in Table2, we divide the current Row number by ((ROW()/(10^MAX(LEN($F$6:$F$15)+1)))). This ensure that the least significant protion of the number is scaled past the end of the decimal number to which it is added.

One we have composite score, which combines the scores in the primary and second tables to generate a unique, properly ordered key, we must rank the composite scores. We do this with the same formula we used earlier in this to determine unique ranks. In cell J6, enter the following formula and then fill down to J15.

=RANK(I6,$I$6:$I$15,0)+COUNTIF($I$6:I6,I6)-1
This is just the unique rank of the composite scores. Finally, after all this, we can use the composite ranks to return the names of the players in either Descending or Ascending order. To return the list in Descending order, enter the formula in cell L6 and fill down to L15.

=OFFSET(B$6,MATCH(SMALL(J$6:J$15,ROW()-ROW(J$6)+1),J$6:J$15,0)-1,0)

To return the names in Ascending order, enter the following formula in cell N6 and fill down to N15,

=OFFSET(B$6,MATCH(LARGE(J$6:J$15,ROW()-ROW(N$6)+1),J$6:J$15,0)-1,0)

As a practical matter, you don't need to include the player names in any table other than the primary tab. As longs as everything is in the same rows, you can simply the display to the following: RankNoNames

You can an example workbook with the formulas described on this page.

This page last updated: 19-Sept-2007