September 23, 2025

Estimating rare proportions

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There is a statistic circulating on social media claiming that the average person in the USA thinks 21% of the population is transgender.  Obviously this isn’t true (both obviously it isn’t 21% and obviously that isn’t what the average person believes). It’s similar in some ways to the claim that some Americans think Iran is in the middle of the Atlantic Ocean, which I’ve dealt with before, except that estimating small proportions is an extensively studied problem in psychology, so a lot is known about the biases. In fact, if you look at the original source for the claim, demonstrating this phenomenon was the actual point of the story.

As Danielle Navarro explains, all small proportions are overestimated and all large proportions underestimated when people aren’t certain of the true value. This is an extremely consistent phenomenon, to the extent that we can actually say Americans are better informed about the proportion of transgender people than they are about other comparably extreme proportions.

[Update: Andrew Gelman writes about a slightly different, but related phenomenon, in the context of people reporting having been present for mass shootings.  It’s slightly different because people are reporting their own experience, which they presumptively do know, rather than their estimates of some proportion they have no way of knowing. We’d expect the bias to be smaller in this setting, but to still be present — it’s like the estimate of the frequency of virgin birth from the National Longitudinal Study of Youth]

United Rugby Championship Predictions for Week 1

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Team Ratings for Week 1

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Leinster 13.41 13.41 0.00
Bulls 8.86 8.86 -0.00
Glasgow 6.18 6.18 -0.00
Stormers 4.17 4.17 0.00
Munster 3.65 3.65 -0.00
Edinburgh 2.67 2.67 -0.00
Sharks 1.29 1.29 0.00
Scarlets -0.54 -0.54 -0.00
Lions -1.19 -1.19 0.00
Connacht -1.39 -1.39 -0.00
Ospreys -2.15 -2.15 -0.00
Benetton -2.32 -2.32 -0.00
Cardiff Rugby -2.74 -2.74 -0.00
Ulster -3.24 -3.24 -0.00
Zebre -11.02 -11.02 -0.00
Dragons -15.66 -15.66 -0.00

 

Predictions for Week 1

Here are the predictions for Week 1. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Stormers vs. Leinster Sept 27 Leinster -2.20
2 Glasgow vs. Sharks Sept 27 Glasgow 11.90
3 Ulster vs. Dragons Sept 27 Ulster 19.40
4 Bulls vs. Ospreys Sept 28 Bulls 18.00
5 Zebre vs. Edinburgh Sept 28 Edinburgh -6.70
6 Scarlets vs. Munster Sept 28 Scarlets 2.80
7 Cardiff Rugby vs. Lions Sept 28 Cardiff Rugby 5.50
8 Connacht vs. Benetton Sept 28 Connacht 7.90

 

Top 14 Predictions for Round 4

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Team Ratings for Round 4

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Stade Toulousain 10.67 11.56 -0.90
Bordeaux Begles 4.76 4.78 -0.00
Toulon 3.58 3.49 0.10
Section Paloise 2.47 2.21 0.30
Racing 92 2.23 1.88 0.30
Stade Rochelais 1.95 1.22 0.70
Clermont 1.67 1.88 -0.20
Castres Olympique 1.36 0.59 0.80
Bayonne 0.82 1.48 -0.70
Montpellier 0.74 -0.21 1.00
Lyon 0.16 -0.45 0.60
Stade Francais -1.88 -2.17 0.30
USA Perpignan -4.13 -3.37 -0.80
Montauban -11.49 -10.00 -1.50

 

Performance So Far

So far there have been 21 matches played, 15 of which were correctly predicted, a success rate of 71.4%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Clermont vs. Section Paloise Sept 21 50 – 27 4.50 TRUE
2 Bordeaux Begles vs. Montauban Sept 21 71 – 24 21.10 TRUE
3 Castres Olympique vs. Bayonne Sept 21 48 – 17 5.40 TRUE
4 Lyon vs. Stade Francais Sept 21 42 – 37 9.00 TRUE
5 USA Perpignan vs. Racing 92 Sept 21 15 – 28 1.10 FALSE
6 Montpellier vs. Stade Toulousain Sept 21 44 – 14 -5.50 FALSE

 

Predictions for Round 4

Here are the predictions for Round 4. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Stade Francais vs. Bordeaux Begles Sept 28 Bordeaux Begles -0.10
2 Montauban vs. Montpellier Sept 28 Montpellier -5.70
3 Racing 92 vs. Clermont Sept 28 Racing 92 7.10
4 Section Paloise vs. Lyon Sept 28 Section Paloise 8.80
5 Stade Rochelais vs. USA Perpignan Sept 28 Stade Rochelais 12.60
6 Stade Toulousain vs. Castres Olympique Sept 28 Stade Toulousain 15.80
7 Bayonne vs. Toulon Sept 29 Bayonne 3.70

 

Rugby Premiership Predictions for Round 1

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Team Ratings for Round 1

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Bath 10.30 10.30 0.00
Sale Sharks 6.70 6.70 -0.00
Leicester Tigers 5.55 5.55 -0.00
Saracens 5.03 5.03 -0.00
Gloucester 4.13 4.13 0.00
Bristol 3.66 3.66 -0.00
Northampton Saints -1.47 -1.47 -0.00
Harlequins -3.02 -3.02 -0.00
Exeter Chiefs -4.58 -4.58 0.00
Newcastle Falcons -18.45 -18.45 -0.00

 

Predictions for Round 1

Here are the predictions for Round 1. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Sale Sharks vs. Gloucester Sept 26 Sale Sharks 9.60
2 Harlequins vs. Bath Sept 27 Bath -6.30
3 Newcastle Falcons vs. Saracens Sept 27 Saracens -16.50
4 Northampton Saints vs. Exeter Chiefs Sept 29 Northampton Saints 10.10
5 Bristol vs. Leicester Tigers Sept 29 Bristol 5.10

 

NRL Predictions for the Preliminary Finals

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Team Ratings for the Preliminary Finals

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Storm 8.07 9.29 -1.20
Panthers 8.05 8.50 -0.50
Roosters 8.04 7.44 0.60
Sharks 6.70 5.10 1.60
Broncos 4.33 -1.82 6.10
Bulldogs 2.21 0.07 2.10
Dolphins 1.52 -1.96 3.50
Sea Eagles 1.02 2.97 -2.00
Raiders 0.92 -3.61 4.50
Warriors -1.33 -1.68 0.30
Cowboys -1.69 4.11 -5.80
Eels -1.79 -3.02 1.20
Rabbitohs -5.19 -4.35 -0.80
Dragons -5.45 -4.55 -0.90
Titans -7.67 -5.50 -2.20
Wests Tigers -7.73 -10.97 3.20
Knights -10.01 -0.05 -10.00

 

Performance So Far

So far there have been 210 matches played, 133 of which were correctly predicted, a success rate of 63.3%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Raiders vs. Sharks Sept 20 12 – 32 -1.10 TRUE
2 Bulldogs vs. Panthers Sept 21 26 – 46 -1.20 TRUE

 

Predictions for the Preliminary Finals

Here are the predictions for the Preliminary Finals. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Storm vs. Sharks Sept 26 Storm 4.40
2 Broncos vs. Panthers Sept 28 Panthers -0.70

 

Bunnings NPC Predictions for Week 9

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Team Ratings for Week 9

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Taranaki 9.03 5.30 3.70
Bay of Plenty 8.70 6.48 2.20
Canterbury 8.68 4.02 4.70
Wellington 3.84 6.60 -2.80
Hawke’s Bay 3.74 0.68 3.10
Counties Manukau 2.14 -0.26 2.40
Otago 2.04 -1.15 3.20
Tasman 1.50 3.49 -2.00
Waikato 0.15 5.46 -5.30
Auckland 0.13 -0.04 0.20
North Harbour -2.52 1.50 -4.00
Northland -7.09 -6.54 -0.60
Southland -10.66 -5.90 -4.80
Manawatu -13.51 -13.45 -0.10

Performance So Far

So far there have been 56 matches played, 35 of which were correctly predicted, a success rate of 62.5%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Taranaki vs. Hawke’s Bay Sept 19 38 – 24 6.50 TRUE
2 Counties Manukau vs. Auckland Sept 20 49 – 28 2.50 TRUE
3 Wellington vs. Southland Sept 20 75 – 19 12.60 TRUE
4 Canterbury vs. Otago Sept 20 36 – 38 11.60 FALSE
5 Tasman vs. Waikato Sept 20 24 – 29 6.00 FALSE
6 North Harbour vs. Northland Sept 21 21 – 22 9.10 FALSE
7 Manawatu vs. Bay of Plenty Sept 21 19 – 55 -16.60 TRUE

Predictions for Week 9

Here are the predictions for Week 9. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Hawke’s Bay vs. Auckland Sept 25 Hawke’s Bay 6.60
2 Southland vs. Tasman Sept 26 Tasman -9.20
3 Otago vs. North Harbour Sept 27 Otago 7.60
4 Taranaki vs. Wellington Sept 27 Taranaki 8.20
5 Bay of Plenty vs. Waikato Sept 28 Bay of Plenty 11.60
6 Counties Manukau vs. Manawatu Sept 28 Counties Manukau 18.70
7 Northland vs. Canterbury Sept 28 Canterbury -12.80

AFL Predictions for the Grand Final

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Team Ratings for the Grand Final

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Geelong Cats 30.57 15.04 15.50
Western Bulldogs 25.54 18.20 7.30
Hawthorn Hawks 23.30 21.95 1.30
Brisbane Lions 21.89 22.65 -0.80
Adelaide Crows 14.77 2.69 12.10
Collingwood 11.15 5.39 5.80
Gold Coast Suns 10.16 -6.41 16.60
GWS Giants 9.94 9.08 0.90
Fremantle Dockers 6.56 5.99 0.60
Sydney Swans 1.16 12.60 -11.40
Melbourne Demons 0.76 -0.21 1.00
Carlton Blues -4.46 5.01 -9.50
St Kilda Saints -6.98 0.89 -7.90
Port Adelaide Power -12.97 7.63 -20.60
North Melbourne -23.62 -37.08 13.50
Essendon Bombers -27.28 -10.15 -17.10
Richmond Tigers -31.46 -31.00 -0.50
West Coast Eagles -41.44 -34.67 -6.80

Performance So Far

So far there have been 215 matches played, 147 of which were correctly predicted, a success rate of 68.4%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Geelong Cats vs. Hawthorn Hawks Sept 19 115 – 85 -9.20 FALSE
2 Collingwood vs. Brisbane Lions Sept 20 71 – 100 5.10 FALSE

Predictions for the Grand Final

Here are the predictions for the Grand Final. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Geelong Cats vs. Brisbane Lions Sept 27 Geelong Cats 8.70
September 16, 2025

Top 14 Predictions for Round 3

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Team Ratings for Round 3

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Stade Toulousain 11.71 11.56 0.10
Bordeaux Begles 3.95 4.78 -0.80
Toulon 3.58 3.49 0.10
Section Paloise 3.09 2.21 0.90
Stade Rochelais 1.95 1.22 0.70
Racing 92 1.73 1.88 -0.20
Bayonne 1.63 1.48 0.20
Clermont 1.05 1.88 -0.80
Castres Olympique 0.55 0.59 -0.00
Lyon 0.40 -0.45 0.80
Montpellier -0.31 -0.21 -0.10
Stade Francais -2.12 -2.17 0.10
USA Perpignan -3.63 -3.37 -0.30
Montauban -10.69 -10.00 -0.70

 

Performance So Far

So far there have been 14 matches played, 11 of which were correctly predicted, a success rate of 78.6%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Montauban vs. Lyon Sept 14 18 – 25 -4.30 TRUE
2 Bayonne vs. Montpellier Sept 14 26 – 23 9.20 TRUE
3 Section Paloise vs. Stade Francais Sept 14 34 – 10 10.80 TRUE
4 Stade Rochelais vs. Clermont Sept 14 34 – 16 6.50 TRUE
5 Stade Toulousain vs. USA Perpignan Sept 14 31 – 13 22.40 TRUE
6 Toulon vs. Castres Olympique Sept 14 16 – 12 10.30 TRUE
7 Racing 92 vs. Bordeaux Begles Sept 15 44 – 32 3.20 TRUE

 

Predictions for Round 3

Here are the predictions for Round 3. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Clermont vs. Section Paloise Sept 21 Clermont 4.50
2 Bordeaux Begles vs. Montauban Sept 21 Bordeaux Begles 21.10
3 Castres Olympique vs. Bayonne Sept 21 Castres Olympique 5.40
4 Lyon vs. Stade Francais Sept 21 Lyon 9.00
5 USA Perpignan vs. Racing 92 Sept 21 USA Perpignan 1.10
6 Montpellier vs. Stade Toulousain Sept 21 Stade Toulousain -5.50
7 Toulon vs. Stade Rochelais Sept 22 Toulon 8.10

 

NRL Predictions for Finals Week 2

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Team Ratings for Finals Week 2

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Storm 8.07 9.29 -1.20
Roosters 8.04 7.44 0.60
Panthers 7.21 8.50 -1.30
Sharks 5.86 5.10 0.80
Broncos 4.33 -1.82 6.10
Bulldogs 3.05 0.07 3.00
Raiders 1.76 -3.61 5.40
Dolphins 1.52 -1.96 3.50
Sea Eagles 1.02 2.97 -2.00
Warriors -1.33 -1.68 0.30
Cowboys -1.69 4.11 -5.80
Eels -1.79 -3.02 1.20
Rabbitohs -5.19 -4.35 -0.80
Dragons -5.45 -4.55 -0.90
Titans -7.67 -5.50 -2.20
Wests Tigers -7.73 -10.97 3.20
Knights -10.01 -0.05 -10.00

 

Performance So Far

So far there have been 208 matches played, 131 of which were correctly predicted, a success rate of 63%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Storm vs. Bulldogs Sept 12 26 – 18 8.00 TRUE
2 Warriors vs. Panthers Sept 13 8 – 24 -3.90 TRUE
3 Sharks vs. Roosters Sept 13 20 – 10 -0.20 FALSE
4 Raiders vs. Broncos Sept 14 28 – 29 0.70 FALSE

 

Predictions for Finals Week 2

Here are the predictions for Finals Week 2. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Raiders vs. Sharks Sept 20 Sharks -1.10
2 Bulldogs vs. Panthers Sept 21 Panthers -1.20

 

Bunnings NPC Predictions for Week 8

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Team Ratings for Week 8

The basic method is described on my Department home page.
Here are the team ratings prior to this week’s games, along with the ratings at the start of the season.

Current Rating Rating at Season Start Difference
Canterbury 9.65 4.02 5.60
Taranaki 8.13 5.30 2.80
Bay of Plenty 7.41 6.48 0.90
Hawke’s Bay 4.64 0.68 4.00
Tasman 2.32 3.49 -1.20
Wellington 1.39 6.60 -5.20
Auckland 1.37 -0.04 1.40
Otago 1.08 -1.15 2.20
Counties Manukau 0.90 -0.26 1.20
Waikato -0.67 5.46 -6.10
North Harbour -1.76 1.50 -3.30
Northland -7.85 -6.54 -1.30
Southland -8.21 -5.90 -2.30
Manawatu -12.23 -13.45 1.20

 

Performance So Far

So far there have been 49 matches played, 31 of which were correctly predicted, a success rate of 63.3%.
Here are the predictions for last week’s games.

Game Date Score Prediction Correct
1 Northland vs. Wellington Sept 11 43 – 26 -9.60 FALSE
2 Waikato vs. North Harbour Sept 12 29 – 28 5.10 TRUE
3 Bay of Plenty vs. Hawke’s Bay Sept 13 21 – 24 7.30 FALSE
4 Otago vs. Taranaki Sept 13 28 – 26 -6.00 FALSE
5 Auckland vs. Manawatu Sept 14 52 – 29 14.60 TRUE
6 Canterbury vs. Tasman Sept 14 31 – 25 11.70 TRUE
7 Southland vs. Counties Manukau Sept 14 10 – 64 -0.30 TRUE

 

Predictions for Week 8

Here are the predictions for Week 8. The prediction is my estimated expected points difference with a positive margin being a win to the home team, and a negative margin a win to the away team.

Game Date Winner Prediction
1 Taranaki vs. Hawke’s Bay Sept 19 Taranaki 6.50
2 Counties Manukau vs. Auckland Sept 20 Counties Manukau 2.50
3 Wellington vs. Southland Sept 20 Wellington 12.60
4 Canterbury vs. Otago Sept 20 Canterbury 11.60
5 Tasman vs. Waikato Sept 20 Tasman 6.00
6 North Harbour vs. Northland Sept 21 North Harbour 9.10
7 Manawatu vs. Bay of Plenty Sept 21 Bay of Plenty -16.60

 

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