Actual vs Expected Wins

Lychee University · Lesson 3

A team’s record tells you how often it won.

But does that record match how well it played?

Let’s compare two numbers using baseball.

Actual win percentage

First: actual win percentage.

Games won, divided by games played.

$$\text{Actual win percentage} = \frac{\text{games won}}{\text{games played}}$$

Win sixty out of a hundred games, and that’s sixty percent.

Simple enough.

Tap a game to flip it

100 games played

Win Loss

Actual win %

60%

60–40

Games won ÷ games played

60 ÷ 100 = 0.60

Reset to 60–4050–50

But the record doesn’t tell you how those wins happened.

Win a game ten to zero, and it counts as one win.

Win one to zero, and it still counts as one win.

The margin changes. The record treats them the same.

Two games

Game 1 · final

Your team10

Opponent0

Game 2 · final

Your team1

Opponent0

What the margin says

Game 1

+10 runs

Game 2

+1 run

What the record says

W1 win

=

W1 win

The margin changes. The record treats them the same.

Over a season, a team could win plenty of close games and lose a few by a lot.

Another team could do the opposite.

That’s why we’re adding a second number.

Pythagorean expectation

Pythagorean expectation, developed by baseball analyst Bill James, estimates a team’s win percentage using runs scored and runs allowed.

You’ll also see it called Pythagorean winning percentage.

The classic formula takes runs scored, squared—and divides that by runs scored squared plus runs allowed squared.

$$\text{Expected win percentage} = \frac{\text{runs scored}^2}{\text{runs scored}^2 + \text{runs allowed}^2}$$

Let’s use a made-up team.

Over a hundred games, it scores five hundred runs and allows four hundred.

Five hundred squared is two hundred fifty thousand.

Four hundred squared is one hundred sixty thousand.

Divide two hundred fifty thousand by their combined total, four hundred ten thousand.

That gives us an expected win percentage of about sixty-one percent.

Across a hundred games, that’s about sixty-one expected wins.

$$\text{Expected wins} = \text{expected win percentage} \times \text{games played}$$

Use the percentage as a decimal in that calculation: about $0.61 \times 100$.

1 · Runs

Runs scored

500

Runs allowed

400

Games played

100

→

2 · Expected win %

500² = 250,000

400² = 160,000

250,000410,000

≈ 0.61

≈ 61%

→

3 · Expected wins

0.61 × 100 games

≈ 61 wins

Runs → expected win % → expected wins

Equal runs land at exactly 50%

Score more while allowing less, and the estimate rises.

Score and allow the same number of runs, and it lands at fifty percent.

Score more, allow less

Runs scored500

Runs allowed400

Games played100

Expected win %

61.0%

≈ 61.0 expected wins in 100 games

500² ÷ (500² + 400²) = 0.610

0.610 × 100 = 61.0 wins

Scoring more than allowing: above 50%

Make them equalReset

We’re using the classic squared version here. Other versions adjust the exponent, but the idea stays the same. Lesson 2 compares the exponents.

Comparing actual and expected wins

Now we have two numbers.

Our example team actually won sixty percent of its games.

Its scoring-based estimate was about sixty-one percent.

Close—but not identical.

Now calculate both numbers for each team over the same season.

You could put them in a table.

But a scatter plot lets us see how they relate.

Put the Pythagorean estimate on the horizontal axis, and actual win percentage on the vertical axis.

Each dot is one team’s season.

Our example team goes at about sixty-one percent across and sixty percent up.

Move right, and the scoring-based estimate gets higher.

Move up, and the actual win percentage gets higher.

One team, two numbers

Example team · 100 games

Pythagorean estimateAcross≈ 61%

Actual win %Up60%

Move right and the scoring-based estimate gets higher. Move up and the actual win percentage gets higher.

What the gap tells us

Now draw a diagonal line where the two percentages are equal.

Fifty percent across, fifty percent up.

Sixty percent across, sixty percent up.

Use the same scale on both axes so that comparison is easy to see.

A dot on that line means the team’s record matches its scoring-based estimate.

Above it? The team won more than the formula estimated.

Below it? Fewer.

Our example team sits just below the line.

The vertical gap is about one percentage point—or about one win across those hundred games.

Where is the dot?

Expected (across)

61.0%

Actual (up)

60.0%

Vertical gap (actual − expected)

−1.0 pts

≈ −1.0 winover100162games

Below the line: the team won fewer than the formula estimated.

Example teamOn the lineAboveBelow

A bigger gap gives us more to investigate.

Did the team win a lot of close games?

Did a few blowout losses affect its run totals?

The dot doesn’t tell us the cause.

And the gap isn’t an automatic prediction that the team will improve or decline.

It tells us where the record and the scoring-based estimate differ.

From a table to a relationship

Now step back and look at all the dots.

Do teams with higher expected win percentages also tend to have higher actual win percentages?

Do the dots stay close to the diagonal, or spread far away?

The table

TeamExpectedActual

Summit62.1%59.9%

Bayside59.0%57.4%

Redrock54.1%54.3%

Meadowbrook54.3%51.2%

Northgate52.8%50.6%

Lakeshore50.7%49.4%

Pinewood50.0%53.1%

Ironvale47.9%48.8%

Riverton47.1%44.4%

Highland44.2%46.3%

Coastline41.5%43.2%

Stonebridge37.4%40.7%

Actual = expected Above the line Below the line

We’ve turned a table of numbers into a picture of the relationship.

DiagonalGapsAbove / below

Selected team

Pinewood

86–76

680 scored · 680 allowed

Expected

50.0%

81.0 wins

Actual

53.1%

86 wins

Gap +5.0 wins (above)

Biggest gaps

• Stonebridge+5.5 W
• Meadowbrook−5.0 W
• Pinewood+5.0 W
• Riverton−4.3 W

A gap is something to investigate. It doesn't explain itself, and it isn't a forecast.

One detail matters here.

We drew that diagonal ourselves. It marks where the two percentages are equal.

We haven’t fitted a line to the data yet.

That comes next.

Two kinds of line

The diagonal

We drew it before looking at the dots. It marks where the two percentages are equal.

A fitted line

The data decides its slope and position. Quantifying the relationship this way is linear regression.

Next lesson: linear regression →

Next lesson: Linear regression—how we fit a line to quantify that relationship.

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Source

The classic formula and its mathematical background: A Derivation of the Pythagorean Won-Loss Formula in Baseball.