The image shows the top eight nations in the FIFA women's world rankings as of March 24, 2023. Clicking on the image will take you to the current world rankings on the FIFA website.

The rank of each national team is determined by a rating number. In first place (rank 1) is the USA with a rating number of 2091.38 and in last place (rank 188) is Mauritius with a rating number of 375.59.

This rating number changes with each international match of a nation - not only in tournament matches, but also in friendlies. With the online calculator at the top of the page you can try different cases to get a feeling for how much the rating number increases or decreases in individual cases.

### How is the rating number calculated?

The FIFA Women's World Ranking has existed since 2003. The calculation of the rating numbers is based retrospectively on all matches since 1971. The playing strength of a team is indicated by a so-called ELO number, similar to chess. (Arpad Elo developed this method for calculating playing strength for the American Chess Association in 1960.) This ELO number now determines the position in the FIFA world rankings.

From August 2018, the rating numbers in the men's world rankings were also switched to ELO numbers. However, there are serious differences in the calculation of the rating numbers for women and men. For example, the size of a win or a loss plays no role in the men's ratings. A distinction is only made between won, drawn and lost. On the other hand, the goal difference, the number of goals scored and, if applicable, a home advantage are also taken into account when calculating the rating number for the women. For women, however, there are no special rules for penalties and defeat in a KO phase.

Let's consider an example:

Team A plays against Team B during UEFA European Championship and wins the match by 5 - 2 goals.

In the following, I use the same letters for the variables in the formulas as in the ▸description of FIFA to facilitate comparison with this description.

First, a number S_{act} is created for both teams, which expresses a victory in numbers. The number ranges from 0 to 1, with values close to 1 representing victory and values close to 0 representing defeat. The value for a tie is close to 0.5.

First the number S_{act} for the loser is determined according to the following table.

In our example, the loser, i.e. Team B, scored two goals. The goal difference is 3 goals. In the table, under "Goals scored", we need to go to the number 2, which is the third line. Under "Goal Difference" we go to the number 3, which is the fourth column. In the second row and the fourth column we find the match value S_{act} for the loser in percent. The match value is 5.6 percent or 0.056 in decimal notation.

The rule now states that the winner gets the remaining percent, so 100% - 5.6% = 94.4%, or 1 - 0.056 = 0.944 in decimal notation.

Result:

Team A: S_{act} = 0.944

Team B: S_{act} = 0.056

It's different when there's no loser. For example, if the match ended in a draw with 2 - 2 goals, we would find the match value 51% in the table in the third row (Goals scored = 2) and the first column (Goal Difference = 0). In the event of a tie, both teams are awarded the same match value, in this case 51% or 0.51. As a result, both teams' ratings in the world rankings would improve slightly. If the score was 1 - 1, the rating numbers would not change and if the score was 0 - 0, they would even deteriorate slightly.

In the second step, a number S_{exp} is calculated for each team, which is also between 0 and 1 and is intended to express the expected match result. Team A is expected to have a current match value greater than 0.5 in this match because their current FIFA World Ranking rating is higher than Team B's.

The formula for S_{exp} according to Arpad Elo is:

Or in linear notation: S_{exp} = 1 / (1 + 10^((R_{B} - R_{A} ± H)/400))

R_{A}: rating number Team A (the team whose rating number is being recalculated)

R_{B}: rating number Team B (the opposing team)

H: Fix for home advantage (H = 100 oder H = 0)

If a team has home advantage, their rating number is increased by 100 in the formula above. This calculates a larger value for S_{exp}. In the event of a win, the team thus climbs less up the rankings. If it loses, it slides further down.

Annotation:

In the ▸FIFA description, the two formulas

x = [ r1 - r2 ] / scaling factor

and

P(x) = 1/[ 1 + 10^(x/2) ]

are mentioned. No numerical value is given for the "scaling factor". However, the four numerical examples listed show that the scaling factor 200 is used.

Combining the two formulas and adding the correction for home advantage yields the above formula for S_{exp}.

The denominator in the exponent is 400 (scaling factor times two). This is the same scaling factor as in chess. The value 600 is used for the men's world rankings. A larger value for this number results in larger changes in rating numbers when recalculated.

For Team A, the calculation of S_{exp} looks like this:

And that's S_{exp} = 0.62187.

The value is slightly larger than 0.5. This is to be expected given that Team A's rating number (1811.73) is slightly larger than Team B's (1725.31). If Team A had home advantage, you would have to use the number 1811.73 + 100 = 1911.73 in the formula instead of 1811.73. If Team B had home advantage, you would have to use the number 1725.31 + 100 = 1825.31 in the formula instead of 1725.31.

For Team B, the following term would have to be calculated:

And that's S_{exp} = 0.37813.

The fact that the two values for S_{exp} add up to exactly 1 is a matter of mathematical law.

Calculation of the new rating number R_{aft} for Team A according to the formula:

_{aft}= R

_{bef}+ K * (S

_{act}- S

_{exp})

The following weightings are specified by FIFA:

60 | FIFA Women's World Cup match |

60 | Women's Olympic football tournament |

45 | FIFA Women's World Cup qualifier |

45 | Women's Olympic football qualifier |

45 | Women's Continental finals match |

30 | Women's Continental qualifier |

30 | Women's friendly match between two Top 10 teams |

15 | Women's friendly match |

Since the match Team A - Team B took place in an European Championship and was therefore a Continental finals match, the weighting factor 45 is used.

The calculation for the new rating number of Team A then looks like this:

R_{aft} = 1811.73 + 45 * (0.944 - 0.62187) = 1826.23

The calculation for the Team B's new rating number looks like this:

R_{aft} = 1725.31 + 45 * (0.056 - 0.37813) = 1710.81

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If you would like to calculate the FIFA rating number (women) in Excel, you can download a corresponding Excel solution here:

▸FIFA Rating Calculator with Excel

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