Weighted quadratic, delta method

When do KBO players peak?

Birth dates from the record centre against wRC+ and FIP — once as a snapshot, once as the same player a year apart, which is the only honest way to do it.

Computed on request from the record book (Oct 13, 2026) and 720 finals in the game log.

The finding

Plot this season's 155 hitters with 100+ plate appearances against their age and the curve is close to flat: a weighted quadratic explains 0.3% of the spread in wRC+, and in 39% of resamples it has no peak at all. That is the survivor problem in one picture: a 37-year-old is on a KBO roster only because he can still hit, so the old end of the chart is a hand-picked sample.

Comparing the same 109 hitters a year apart removes most of that selection. Their change in wRC+ falls by 1.0 points for every year of age (95% −2.2–+0.2), crossing zero at 26.0 (95% 20–33): younger than that, hitters tend to improve; older, to decline. Hitters 36 and over moved −19.2 points on average (n = 22), those 23 and under −6.2 (n = 14).

For pitchers, the snapshot of 150 arms with 30+ innings has FIP at its best around 33.6 (95% 30–40) — with the same survivor caveat, and more of it: young pitchers are often given innings to develop, not because they are good.

Growth stops at (same hitter)

26.0

20–33

Change per year of age

−1.0

wRC+ points, p = 0.09

Snapshot r² (hitters)

0.003

155 hitters

Pitcher FIP best at

33.6

30–40

Hitters: wRC+ by age, this season

Each faint dot is a hitter, sized by plate appearances. The line is the PA-weighted quadratic with its 95% band; white dots are two-year age-group means with 95% intervals (grey when fewer than 10 hitters).

Weighted quadratic, 95% bandAge-group mean

18–22: n 20 · 23–24: n 14 · 25–26: n 26 · 27–28: n 16 · 29–30: n 19 · 31–32: n 17 · 33–34: n 11 · 35–36: n 20 · 37–45: n 12

The same hitter, one year later

Change in wRC+ from last season to this for hitters with enough plate appearances in both, grouped by age, with 95% intervals and the fitted line. Above zero, the group improved.

18–23: n 14 · 24–26: n 19 · 27–29: n 21 · 30–32: n 18 · 33–35: n 15 · 36–45: n 22

Pitchers: FIP by age, this season

As above, weighted by innings. Lower is better, so the curve's low point is the best age.

18–22: n 24 · 23–24: n 19 · 25–26: n 24 · 27–28: n 19 · 29–30: n 21 · 31–32: n 18 · 33–34: n 14 · 35–36: n 3 · 37–45: n 8

Method

Age is taken on 1 July of the season from the record centre's birth dates (305 players matched). The snapshot fit is weighted least squares, with plate appearances (or innings) as weights:

value = β₀ + β₁(age − 29) + β₂(age − 29)² · peak = 29 − β₁ ÷ (2β₂)

The peak's interval resamples players 800 times; draws where the curve opens the wrong way (no interior peak) are counted and reported rather than hidden. The delta method takes each hitter with enough playing time in both seasons, computes this year's wRC+ minus last year's, weights by the harmonic mean of the two seasons' plate appearances, and fits a straight line in age; the age where that line crosses zero is where improvement turns to decline.

Survivor bias runs through all of it. The snapshot only sees players good enough to be playing; the delta method only sees players who kept a job for two seasons, which leaves out those who declined their way out of the league and makes the late-career drop look gentler than it is. Biographies exist only for players on a current roster. Imported players arrive older and pre-selected for performance. One pair of seasons is a small base for an aging curve; a career-length study would need many more years on disk.