View from a tee down a rolling fairway toward bunkers and a pond, with the sun rising over the trees.

Golf Lab

How the Lab calculates.

Every Golf Lab model: the equations in words, the published data each was checked against, and how far off it is.

Ball Flight Explorer

Ball flight model, version 1

What it calculates

Given ball speed, launch angle, spin and spin axis, the model flies the ball until it lands and reports carry, sideways distance, maximum height, landing angle and hang time. It works in small time steps of 4 milliseconds (fourth-order Runge–Kutta).

The forces

Gravity.
Drag, against the direction of travel. Its coefficient has two parts: one that falls from 0.5 to 0.176 as the ball speeds up through the "drag crisis" that dimples create, and one that rises with spin.
Lift from backspin (the Magnus force), at right angles to the ball's path and its spin axis. Its coefficient rises with the ratio of spin to speed.
Spin decay. Spin falls as the ball flies, faster when it is moving faster.

Tilting the spin axis tilts the lift sideways, which curves the ball. Air density is fixed at 1.194 kg/m³, sea level at about 21 °C. There is no wind.

How it was fitted

The forms of the forces come from the published physics below. Six constants in them were fitted to TrackMan's PGA Tour averages for 12 clubs, matching carry, maximum height and landing angle together. The LPGA Tour averages were then used as a check: they played no part in the fit.

How far off it is

Average error (root mean square) against the published averages
Data	Carry	Max height	Land angle	Worst carry
PGA Tour, 12 clubs (fitted)	2.9 yd	6 ft	2°	5.5 yd
LPGA Tour, 11 clubs (check)	3.9 yd	6 ft	2°	7 yd

PGA Tour averages (used to fit the model): model minus published
Club	Carry (yd)	Max height (ft)	Land angle (°)
Driver	−2.5	−1	+4.0
3-wood	+5.5	+3	+1.7
5-wood	+3.6	+2	−0.2
Hybrid	−5.0	+2	−1.6
3-iron	+1.7	−3	−2.7
4-iron	0.0	−7	−3.2
5-iron	−0.8	−11	−2.6
6-iron	−1.6	+2	+0.4
7-iron	−3.6	−1	+0.4
8-iron	−0.7	+6	+0.5
9-iron	+1.6	+10	−0.7
PW	−1.9	+10	−0.7

LPGA Tour averages (not used in the fit): model minus published
Club	Carry (yd)	Max height (ft)	Land angle (°)
Driver	+2.7	+1	+2.4
3-wood	+5.3	−13	−4.4
5-wood	+3.3	0	+0.1
Hybrid	+4.5	+3	−0.2
4-iron	−7.0	−8	−0.8
5-iron	−6.5	−8	−2.1
6-iron	−2.3	+1	+1.0
7-iron	−0.4	0	+1.2
8-iron	+2.0	+2	+2.6
9-iron	−1.0	−1	+1.7
PW	−0.5	−7	+0.5

A plus sign means the model flies the ball farther, higher or lands it steeper than the published average.

Where it stops

The Explorer accepts 85–175 mph ball speed, 9–26° launch and 2,400–9,500 rpm spin: the span of the published shots it was checked against, with a small margin. It refuses numbers outside that.
The sideways curve has not been checked against published data, so it is labelled model only.
It models carry only: no bounce or roll. It assumes a centered strike, one generic ball, still air at sea level.
Golf balls differ. Lyu and colleagues measured an 18-meter spread in carry across 13 balls in one simulated drive.

Sources

TrackMan, “New PGA & LPGA Tour Averages”, 2023 season data. published May 2, 2024; read October 11, 2026.
R. D. Penner, “The physics of golf”, Reports on Progress in Physics 66 (2003) 131–171. forces on a ball in flight.
R. D. Penner, “The optimum loft of a driver”, American Journal of Physics 69 (2001) 563. spin decay law, as given by Smits & Smith (1994).
J. Lyu, J. Kensrud, L. Smith and T. Tosaya, “Aerodynamics of golf balls in still air”, Proceedings 2 (2018) 238. measured drag and lift of current balls, and the drag crisis.
A. M. Nathan, golf trajectory calculator, University of Illinois. the drag and lift forms used here.

The full constants are in the model's source: drag 0.5 → 0.17585 plus 0.51385·S − 0.43466·S²; lift 0.51129·S^0.47825; spin decay 1.419e-5, where S is spin speed at the ball's surface divided by ball speed.

Shot Shape Lab

Shot shape model, version 1

Open the Shot Shape Lab

What it calculates

Given a face angle and a club path (both measured to the target), a club and its attack angle, it works out where a centered strike starts and how its spin axis tilts. The Golf Lab flight model then flies the ball and reports the curve, where it lands and the carry. It names the result with the classic shot shapes.

The geometry, in words

Face to path is the face angle minus the path angle. Positive means the face points right of the path for a right-handed golfer.
Loft to path is the dynamic loft minus the attack angle: the vertical angle between face and path.
The two combine into one obliqueness of the strike, Φ: cos Φ = cos(face to path) × cos(loft to path). TrackMan calls the same 3-D angle spin loft.
The ball leaves between the path and the face, much nearer the face. The share it keeps is Tutelman's launch relation, 0.96 − 0.0071 × Φ. So start direction = path + share × face to path.
Spin-axis tilt: tan(tilt) = tan(face to path) ÷ tan(loft to path). The same face to path tilts the axis less as loft rises.
For a left-handed golfer the same numbers are mirrored: an open face points left.

The club numbers

Ball speed, launch angle, spin and attack angle are TrackMan's PGA Tour averages for the club, unchanged by face and path. TrackMan's table gives no dynamic loft, so the model solves Tutelman's launch relation for the loft that gives the published launch at the published attack angle. That loft is labeled Model. Against the two dynamic lofts TrackMan does publish, it runs low:

Model loft against TrackMan's published PGA Tour figures
Club	Dynamic loft, model	TrackMan	Spin loft, model	TrackMan
Driver	12.1°	12.8°	13.0°	14.7°
6-iron	18.3°	20.2°	22.0°	24.3°

A lower loft to path tilts the spin axis more, so the model's curve is, if anything, on the large side.

Checks against published numbers

Tutelman's worked example. A 23° loft to path, face 2° right of target and path 5° left gives Φ = 24.0°, a share of 0.790, a start 0.5° right and a spin-axis tilt of 16.1°. Tutelman gives 24°, 0.790, 0.5° and a spin ratio of 0.289 (a tilt of 16.1°).

Share of the start direction set by the face
Club	This model	Penner impact model	TrackMan rule of thumb
Driver	87%	81%	85% (driver)
7-iron	78%	80%	75% (mid-irons)

Penner's figures use his reference 200 g head at 45 m/s. The TrackMan figures are a rule of thumb quoted by Titleist, not a measured table.

The sideways curve comes from the flight model, whose carry, height and landing angle were checked against TrackMan's averages. No published averages exist for curve, so it has not been checked and is labeled Model.

How the shape is named

By the start direction (to 0.1°) and the curve (to the yard), as shown in the readouts. For a right-handed golfer:

Pull hook: Starts left of the target and curves farther left.
Pull: Starts left of the target and flies straight.
Fade: Starts left of the target and curves right.
Hook: Starts on the target line and curves left.
Straight: Starts on the target line and flies straight.
Slice: Starts on the target line and curves right.
Draw: Starts right of the target and curves left.
Push: Starts right of the target and flies straight.
Push slice: Starts right of the target and curves farther right.

Where it stops

Face and path from -10° to +10°. Attack angle from -5° to +5°, which spans the published averages from −4.7° (PGA Tour pitching wedge) to +2.8° (LPGA Tour driver) and Winfield and Tan's +5° upward driver path. Numbers outside are refused.
Centered strikes only. Off-center gear effect needs each head's mass, inertia and center of gravity, which makers rarely publish.
Ball speed, launch and spin stay at the club's average. In a real swing, more loft or a different attack angle also changes them. Dynamic loft is not an input for the same reason.
Tutelman's launch relation is a curve fit: within 0.1° up to 40° of loft and about 1° up to 60°, backed by measured data to 45°. No strike this tool allows exceeds 48°.
No wind, sea level, one generic ball. Carry only.

Sources

D. Tutelman, “3-Dimensional Launch Conditions from Impact Conditions”. the 3-D geometry: obliqueness, start direction and spin-axis tilt; worked example. Published 6 March 2007, last modified 23 April 2023.
D. Tutelman, “The Golf Swing: impact” (launch angle from loft). the launch relation 0.96 − 0.0071 × loft and its range.
TrackMan, “New PGA & LPGA Tour Averages”, 2023 season data. ball speed, launch, spin and attack angle per club. Published 2 May 2024.
TrackMan, “Dynamic Loft” and “Spin Loft”. PGA Tour driver and 6-iron dynamic loft and spin loft, used as a check.
Titleist, “The Physics of Ball Striking in Golf”. face 85% of start direction with a driver, 75% with mid-irons, from TrackMan data.
R. D. Penner, “The optimum loft of a driver”, American Journal of Physics 69 (2001) 563. rigid-body impact model, eqs 5, 7, 8 and 10, used as a cross-check.
R. D. Penner, “The physics of golf”, Reports on Progress in Physics 66 (2003) 131–171. Winfield and Tan’s +5° upward driver path (p. 162).

Shot Pattern Lab

Carry, spread and overlap, version 1

Open the Shot Pattern Lab

What it calculates

Nothing about the ball is modeled. Every shot is the golfer’s own, read from a launch monitor file or a pasted table: its carry, how far it finished left or right of the target line (sideways), and, when the file has them, total distance, ball speed, launch angle and spin. For each club the tool works out the number of shots kept, the average carry and sideways, their standard deviations, and the ellipses that a normal spread with those numbers would draw. Across the session it lines the clubs up by average carry and flags neighbors whose spreads overlap.

The statistics, in words

Average. The sum of the kept shots’ values divided by how many there are.
Standard deviation (SD). The square root of the variance, where the variance is the sum of each value’s squared distance from the average divided by the number of shots minus one (NIST e-Handbook 1.3.5.6). It needs at least 2 shots.
How carry and sideways move together. The covariance is the sum of (carry − average carry) × (sideways − average sideways), divided by the number of shots minus one; the correlation is the covariance divided by the two SDs (NIST 6.5.4.2). It needs at least 3 shots and tilts the ellipse when long shots also tend to go one way.
The ellipses. Their axes point along the eigenvectors of the 2 × 2 covariance matrix, and each half-axis is k times the square root of its eigenvalue. For a two-dimensional normal spread, the share of shots inside the ellipse k SDs out is 1 − e^(−k²/2). So the one-SD ellipse holds only about 39% of shots; the tool draws k = 1.51 for 68% and k = 2.45 for 95%.
The carry ladder. Each club is a band from its average carry minus one SD to plus one SD: about 68% of its carries, if they are normal. Two neighboring clubs overlap when the shorter club’s band reaches past the bottom of the longer club’s band; the overlap is shown in yards.
Units. Meters become yards by dividing by 0.9144 and km/h become mph by dividing by 1.609344; m/s become mph by dividing by 0.44704. All three are exact definitions (NIST SP 811).

The normal-spread assumption

The averages and SDs are plain arithmetic and hold for any pattern. The ellipses, the 68% and 95% labels, and the “about 68%” on each band assume the shots spread like a normal distribution: most near the middle, fewer farther out, the same on both sides. A session with a few big mishits, or two different swings, does not look like that, and its ellipses will be larger than where most shots actually land. That is why leaving a shot out is offered, shot by shot, and never done for the golfer.

Checked against published numbers

Share of a normal spread inside the ellipse k SDs out: this tool’s formula against the published values
k	This tool	Published	Source
1	39.35%	39.35%	Wang, Shi & Miao (2015)
2	86.47%	86.47%	Wang, Shi & Miao (2015)
3	98.89%	98.89%	Wang, Shi & Miao (2015)
2.448	95.00%	95% at k² = 5.991	NIST chi-square table, 2 degrees of freedom

The unit tests also draw 20,000 simulated shots from a known tilted normal spread and confirm that the 68% and 95% ellipses fitted to them each hold their share to within 1.5 percentage points.

How rough a small session is

An SD from a few shots can be far from the golfer’s real spread. The 95% interval for the true SD runs from the session’s SD times √((N − 1)/χ²₀.₉₇₅) to times √((N − 1)/χ²₀.₀₂₅) (NIST 1.3.5.8, with chi-square values from NIST’s table). Any club with fewer than 10 shots is flagged.

The true spread could be this many times the session’s SD (95% interval, normal spread)
Shots	From	To
5	0.60 ×	2.87 ×
10	0.69 ×	1.83 ×
20	0.76 ×	1.46 ×
30	0.80 ×	1.34 ×

Reading the file

In the browser. The file or the pasted text is read on the golfer’s device. Nothing is uploaded, and nothing is stored until Save to My Lab.
Generic, then confirmed. The header is the first row that names at least two known columns; a row of units under it (“[yds]”, “mph”) is read too. Column names are matched against common names, including the names TrackMan gives its numbers (Carry, Carry Side, Total, Ball Speed, Launch Angle, Spin Rate). The golfer then confirms or corrects every column, the units and the clubs’ short names. Two columns can’t be set to one number, and carry, sideways and units must be chosen.
Formats. TrackMan’s TPS exports a “Trackman CSV” from Shot Analysis; TrackMan’s article does not list its columns or units, so no layout is assumed. Foresight’s FSX exports session reports as PDF. Garmin’s R10 manual shows session statistics in the Garmin Golf app and describes no file export; on Garmin’s forum, a verified Garmin reply says the export request is still under review. Any table with a header row works.
Left and right. TrackMan’s definitions give no sign for left and right, so the golfer says what a positive number means. Values written with L or R are read as written.
Strict. A row is a shot only with a club, a carry from 10 to 400 yards and a sideways number within 100 yards. Other rows, and summary rows such as “Average”, are skipped and listed with their row number and reason. An optional value outside its range (ball speed 40–200 mph, launch 0–45°, spin 500–12,000 rpm) is left out of that shot, with a note. Nothing is clamped.

What Save and share hold

Save to My Lab writes each club’s average carry, rounded to the yard and marked as a launch monitor number; the average ball speed, launch and spin when all three are in the file; and the spread: the carry and sideways SDs and the number of shots (at least 2). A club already in My Lab keeps its loft and model. Typing a new carry for that club elsewhere in the Lab drops the saved spread, which belonged to the session.
A share link holds each club’s averages, SDs, correlation and shot count in the address after the #, which is never sent to Par 3 Gear. It never holds a single shot.

Limits

Carry and sideways are as the launch monitor reported them. The tool doesn’t correct for wind, temperature, altitude, the ball or the device.
Sideways is where the shot carried, left or right of the target line; how much of that was start line and how much was curve isn’t separated.
It describes a session. It doesn’t say what a club should carry, or which clubs to carry.

Sources

TrackMan, “Shot Analysis: How To Export A CSV File From TPS”. Shot Analysis, Table View, Export, “Trackman CSV”. The article does not list the file’s columns or units. Read October 12, 2026.
TrackMan, “Practice | Trackman Data Parameter Definitions”. The names TrackMan gives its numbers: Carry, Carry Side, Total, Ball Speed, Launch Angle, Spin Rate. It gives no sign for left and right. Read October 12, 2026.
Foresight Sports, “Exporting Session Reports (FSX Mobile / FSX Pro iPad)”. Session reports export as PDF, to email, Files or print. No CSV is described. Read October 12, 2026.
Garmin, Approach R10 Owner’s Manual. Session statistics are viewed in the Garmin Golf app, which syncs to Garmin Connect. The manual describes no file export. Read October 12, 2026.
Garmin Forums, Approach R10, “Golf Sim Sessions’ Data Export”. Garmin’s verified reply: the export feature request “is still being considered and reviewed by our engineering team”. Read October 12, 2026.
NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.6 Measures of Scale. Variance and standard deviation with N − 1. Read October 12, 2026.
NIST/SEMATECH e-Handbook, 6.5.4.2 The Multivariate Normal Distribution. The bivariate normal: two means, a covariance matrix, and the correlation σ₁₂ / (σ₁σ₂). Read October 12, 2026.
Wang, Shi & Miao, “Confidence Analysis of Standard Deviational Ellipse…”, PLOS ONE 10(3): e0118537 (2015). For a planar normal spread, the 1, 2 and 3 standard-deviation ellipses hold about 39.35%, 86.47% and 98.89%. Read October 12, 2026.
NIST/SEMATECH e-Handbook, 1.3.6.7.4 Critical Values of the Chi-Square Distribution. Chi-square critical values, used for the 95% ellipse check and the small-sample table. Read October 12, 2026.
NIST/SEMATECH e-Handbook, 1.3.5.8 Chi-Square Test for the Variance. The confidence interval for a standard deviation. Read October 12, 2026.
NIST Guide to the SI, Appendix B.9. 1 yd = 0.9144 m; 1 mph = 0.44704 m/s = 1.609344 km/h (exact). Read October 12, 2026.

Gapping & Bag Builder

Gaps, bands and lofts, version 1

Open the Gapping & Bag Builder

What it calculates

Nothing is modeled. The tool takes the carry you enter for each club, sorts the clubs from longest to shortest, and subtracts each carry from the one above it. Carries are rounded to the nearest yard first, so the gaps add up to the spread of the bag exactly.

Band. A gap narrower than the band’s lower end or wider than its upper end is flagged, with a text label and an outline. The band starts at 7 to 15 yards so the chart has something to show; it is a setting, not a target, and the golfer sets it anywhere from 1 to 40 yards.
Overlap. Two clubs overlap when they carry the same whole-yard distance, or when both lofts are known and the club with less loft carries shorter than the club with more.
Club count. Rule 4.1b of the Rules of Golf (“Limit of 14 Clubs; Sharing, Adding or Replacing Clubs During Round”) says a player must not start a round with, or have during it, more than 14 clubs. The counter compares the bag with that number.

Lofts and “Fill this gap”

A loft is the golfer’s own, or the maker’s when the golfer picks the model from Par 3 Gear’s product records and uses one of the lofts the maker publishes. Change it and it is labeled as yours. For iron sets, the loft of each club comes from the maker’s spec table where we have recorded it; otherwise only the 7-iron loft is known and the other clubs show “not recorded”. Today 32 of 34 iron sets have every club’s loft and 0 have the 7-iron only.

For a gap whose two lofts are known (or typed in), the tool lists every fairway wood, hybrid and wedge in our data with a published loft strictly between them, 71 products in all. An adjustable head is listed when its published range reaches into the gap. The list is ordered by type, then brand, then name. It is not a ranking, and it does not say what carry a loft will give: that depends on the golfer’s swing, the shaft and the strike.

Checked against the published example

The example bags are TrackMan’s tour-average carries. The ladder reproduces each published carry exactly; the gaps below are those carries subtracted.

PGA Tour averages, flagged against the starting band of 7 to 15 yd
Gap	Carries	Gap	Flag
Dr to 3W	282 − 249	33 yd	over the band
3W to 5W	249 − 236	13 yd	inside
5W to Hy	236 − 231	5 yd	under the band
Hy to 3i	231 − 218	13 yd	inside
3i to 4i	218 − 209	9 yd	inside
4i to 5i	209 − 199	10 yd	inside
5i to 6i	199 − 188	11 yd	inside
6i to 7i	188 − 176	12 yd	inside
7i to 8i	176 − 164	12 yd	inside
8i to 9i	164 − 152	12 yd	inside
9i to PW	152 − 142	10 yd	inside

LPGA Tour averages, flagged against the starting band of 7 to 15 yd
Gap	Carries	Gap	Flag
Dr to 3W	223 − 200	23 yd	over the band
3W to 5W	200 − 189	11 yd	inside
5W to Hy	189 − 178	11 yd	inside
Hy to 4i	178 − 175	3 yd	under the band
4i to 5i	175 − 166	9 yd	inside
5i to 6i	166 − 155	11 yd	inside
6i to 7i	155 − 143	12 yd	inside
7i to 8i	143 − 133	10 yd	inside
8i to 9i	133 − 123	10 yd	inside
9i to PW	123 − 111	12 yd	inside

Limits

Carry only. It ignores roll, and the spread of shots around each carry.
Each carry is only as good as where it came from, so each one carries its source: launch monitor, range estimate or on the course.
The tool never suggests a carry distance and never calls a gap right or wrong.

Sources

R&A, Rules of Golf, Rule 4.1b, “Limit of 14 Clubs; Sharing, Adding or Replacing Clubs During Round”. Read October 11, 2026.
TrackMan Tour averages (2023 data), published May 2, 2024; read October 11, 2026. Carries only; TrackMan publishes no lofts with them.
Par 3 Gear product records: each maker’s published lofts, with the page and the date it was read, shown next to every product the tool lists.

Wedge Matrix

Carries, gaps and the card, version 1

Open the Wedge Matrix

What it calculates

Nothing is modeled. The golfer enters a carry, in whole yards, for each wedge at each swing length they name (up to 8 wedges and 6 swing lengths; the names start as Full, Three-quarter, Half and are theirs to change). The tool puts every carry on one ladder, longest first, and subtracts each from the one above it.

Gap. One carry minus the next one down the ladder, whichever wedge or swing each comes from. The distances in between, one yard past the shorter carry to one yard short of the longer, are named as missing.
Band. A gap wider than the golfer’s band is outlined and labeled in words. The band starts at 10 yards so the chart has something to show; it is a setting, not a target, and runs from 3 to 30 yards.
Overlap. Two carries 2 yards apart or closer are flagged as an overlap, with a dashed outline and a label.
Ties. Equal carries are listed stronger loft first, then the longer swing.

Filling a gap

When both carries come from one wedge, the open distance sits between two of its swing lengths, so the tool names that option. When they come from two wedges whose lofts are entered, it lists every wedge in Par 3 Gear’s product records with a published loft strictly between the two, 21 wedges in all, by brand and then name. It is not a ranking, and it never says what carry a loft will give: that depends on the golfer’s swing, the shaft and the strike. The Wedge Finder hand-off names the finder’s own choice whose loft range meets the gap: Gap wedge (48° to 52°); Sand wedge (54° to 58°); Lob wedge (58° to 64°).

The card and the Rules

The card prints at 85.6 × 54 mm a side, the size of a bank card, front and back side by side to cut out and fold, or as a full page. It holds the golfer’s carries and the gaps between them, and nothing about the greens.

The Rules of Golf allow it. Rule 4.3a(3), “Information Gathered Before or During Round”, lists as allowed: “Using information that was gathered before the round (such as playing information from previous rounds, swing tips or club recommendations).” Separate limits apply only to green-reading materials (Clarification 4.3a/1), which the card does not contain.

Checks

There is no published dataset to compare with, because every number is the golfer’s own. The arithmetic is checked by unit tests: the ladder order, each gap, the band and overlap edges, the missing range, the loft lists, and the strict reading of saved cards and share links (unknown fields, out-of-range numbers and malformed links are refused).

Limits

Carry only, in yards. It ignores roll, and the spread of shots around each carry.
Each carry is only as good as how it was measured; the tool never suggests one.
Carries run from 10 to 200 yards and lofts from 30° to 64°; anything outside is refused with the reason.

Sources

Rules of Golf, Rule 4.3a(3), “Information Gathered Before or During Round”, R&A and USGA, 2023 edition (in effect from 1 January 2023). The wording quoted above was read in the full 2023 book (PDF) on October 12, 2026.
USGA, Clarification 4.3a/1 (Green-Reading Materials), questions and answers. Read October 12, 2026.
Par 3 Gear product records: each maker’s published wedge lofts, with the page and the date it was read, shown next to every wedge the tool lists.
The Wedge Finder’s rule file (data/rules/wedge-finder.v1.yaml), for the finder’s wedge choices.

Conditions Lab

Conditions model, version 1

Open the Conditions Lab

What it calculates

The Conditions Lab flies one shot twice with the ball flight model above. The baseline is the air the model was fitted in, with no wind, landing on flat ground. Today's flight uses the air, wind and green height the golfer sets. It reports today's carry, the change, the sideways drift, and what a target at the baseline carry "carries like": the baseline carry minus the change.

The equations, in words

Pressure from altitude. The International Standard Atmosphere's troposphere formula: sea-level pressure times (1 − lapse rate × height ÷ 288.15 K) raised to g·M ÷ (R × lapse rate), about 5.256. The tool uses this standard pressure for the altitude, not the day's barometer.
Density of humid air. Dry air and water vapor treated as two ideal gases: density = dry-air pressure ÷ (287.058 × T) + vapor pressure ÷ (461.495 × T), with T in kelvin. The vapor pressure is the humidity times the saturation pressure from Tetens' equation, 0.61078 × exp(17.27 (T − 273.15) ÷ (T − 35.85)) kPa. The stated error is under 0.2% from −10 °C to 50 °C, which is why the temperature input stops there.
Density, not viscosity. The air changes the flight through its density: drag and lift both scale with it. The drag curve keeps the viscosity it was fitted with. Its fitted rise below a Reynolds number of 100,000 cannot be told apart from a change with speed, and wind-tunnel data put a dimpled ball's drag crisis near 50,000 with drag nearly constant above it. Fed the thinner air's viscosity, that rise made carry fall above 8,000 ft and in hot air, which no published study shows.
Wind. One steady wind, the same at every height and everywhere on the hole. Drag, lift, spin parameter, Reynolds number and spin decay all use the ball's velocity relative to the moving air. Real wind is slower near the ground and gusts; no golf-specific height profile was found in published sources, so none is used.
Green height. The flight ends when the ball comes down through the green's height. If the shot never climbs that high, the tool says so and shows no carry.

The baseline

70 °F, 50% humidity and sea-level pressure (101,325 Pa) give 1.1939 kg/m³ by these formulas: the 1.194 kg/m³ the ball flight model was fitted with. As checks, dry air at 15 °C gives 1.2250 kg/m³ against the standard atmosphere's 1.2250, and the pressure at 5,280 ft is 83.43 kPa against 83.51 kPa from NASA's separate formula.

Against published figures

We found one published set of on-course weather figures: a TrackMan study by teacher Andrew Rice, reported by GOLF.com. Its launch conditions differ from ours: a 155 mph drive at "optimal launch", and a Tour drive carrying 300 yards for the wind figures. So the comparison below uses percentages and the model's PGA Tour driver (280 yd). The two articles give different altitude figures; both are shown.

Carry change for a driver: published study and this model
Change	Rice / TrackMan	Model
20 mph headwind	−13.7% (300 → 259 yd)	−14.6%
20 mph tailwind	+11.0% (300 → 333 yd)	+8.7%
Each 10 °F warmer	+0.8% to +1.0% (2 to 2.5 yd of 250)	+1.2%
Each 1,000 ft higher	+1.0% or +1.8% (2.5 or 4.5 yd of 250)	+2.0%
7,000 ft and 90 °F	+8.0% (250 → 270 yd)	+11.9%

Wind agrees within a few percent. The model's temperature and altitude effects run a little above the study's figures. The model changes only the air: it holds ball speed, launch and spin fixed, so anything in the study that changed them is not in the model. Read its temperature and altitude figures as the effect of the air alone.

Where it stops

Inputs: wind 0–25 mph, temperature 14–122 °F (−10 to 50 °C), altitude −300 to 12,000 ft, humidity 0–100%, green 40 yd below to 40 yd above the tee. The shot itself stays inside the ball flight model's checked range.
Carry only: no bounce or roll, which slope, firmness and wind also change.
Steady, uniform wind. No gusts, no wind shear, no shelter from trees.
The air only: ball speed, launch and spin are the same in every condition.
Thinner air lifts the ball less as well as dragging it less. A low-spin, low-launch shot leans on lift, so it gains less: the LPGA Tour 3-wood's carry levels off above about 6,000 ft and its apex drops.
Adjusted distances are for practice and planning. Rule 4.3a(1) does not allow measuring elevation changes during a round, and Model Local Rule G-5 lets a committee ban distance-measuring devices.

Sources

Wikipedia, “Density of air”: humid air, and variation with altitude (International Standard Atmosphere constants). the humid-air density formula, Tetens’ equation, and the troposphere pressure formula with p0 = 101,325 Pa, T0 = 288.15 K, L = 0.0065 K/m, g = 9.80665 m/s², M = 0.0289652 kg/mol, R = 8.31446 J/(mol·K); read 11 October 2026.
NASA Glenn Research Center, “Earth Atmosphere Model”. an independent troposphere formula used to check the pressure.
J. Crabill, F. Witherden and A. Jameson, “High-Order Computational Fluid Dynamics Simulations of a Spinning Golf Ball”, arXiv:1806.00378 (2018). its summary of Bearman and Harvey’s 1976 wind-tunnel data: dimples lower the drag crisis to a Reynolds number near 50,000, and the drag coefficient stays nearly constant above it.
National Weather Service, El Paso, “Vapor pressure”. a second form of Tetens’ equation, used to check the first.
A. Rice, GOLF.com, 24 June 2019. wind, temperature and altitude figures from the author’s year-long study with TrackMan.
L. Kerr-Dineen, GOLF.com, 14 December 2018. earlier temperature and altitude figures from the same study.
The R&A, Rules of Golf, Rule 4.3a(1). measuring elevation changes is not allowed during a round.
USGA Player Services, “Are distance measuring devices permitted?”. Model Local Rule G-5 prohibits distance-measuring devices where a committee adopts it.

Launch Window Explorer

Launch window map, version 1

Open the Launch Window Explorer

What it calculates

For one ball speed, the modeled carry at 35 launch angles (9° to 26°, every 0.5°) and 30 spin rates (2,400 to 9,500 rpm, about every 245 rpm): 1,050 flights of the ball flight model above, in its sea-level, still air. The map shows them as bands of carry. Between grid points the band edges are drawn by linear interpolation.

The longest carry is the grid point with the longest carry inside the club's spin band (below), flown again at the model's full time step. When it sits inside the map it is a peak. When it sits on an edge, carry is still rising there, so the tool calls it the longest on the map, not a peak.
The spin band. Drivers, woods and hybrids are searched from 2,400 rpm up to 4,587 rpm, the highest spin in TrackMan's published wood and hybrid averages; irons and wedges from 4,404 rpm, the lowest published iron spin, up to 9,500 rpm. The rest of the map is dimmed.
The green line is the contour 5 yards below the longest carry: everything inside it carries within 5 yards of it.
The what-ifs fly the golfer's point again with one number changed (±2 mph, ±1° launch, ±500 rpm) and report the difference in yards. A what-if that would leave the checked range is not shown.

Speed

The map uses a 40-millisecond time step instead of the model's 4 ms. Across all 23 published tour shots, that moves carry by at most 0.01 yd. A new map is computed a few rows at a time, so the page keeps responding; the default map is computed when the site is built.

What the model gives for the tour averages

Modeled carry at the published numbers, and the longest modeled carry at the same ball speed
Shot	Carry at published numbers	Longest	At
PGA Tour driver	280 yd	283 yd	14.0°, 2,400 rpm (edge: not a peak)
LPGA Tour driver	226 yd	230 yd	17.5°, 2,400 rpm (edge: not a peak)
PGA Tour 7-iron	172 yd	179 yd	15.5°, 4,603 rpm (edge: not a peak)
LPGA Tour 7-iron	143 yd	146 yd	16.0°, 9,500 rpm (edge: not a peak)

For these shots the longest carry sits on the low-spin edge of the band: in the model, carry keeps rising as spin falls, and no published shot in the checking data spins less than2,506 rpm, so the map does not go lower. At lower ball speeds the model's carry also rises again toward very high spin: on the whole map, below about 94 mph the high-spin corner is the longest. The club's band keeps a driver's longest carry at driver spins; for irons and wedges it can sit on either edge of the band. No published map of carry across launch and spin was found to check the map against as a whole; each point carries the ball flight model's own error, shown above.

Where it stops

Carry only. The longest carry is not always the best shot: the map ignores roll, and how widely shots spread.
Ball speed, launch and spin are treated as independent. On a real swing they are linked, so changing one usually changes the others.
A what-if is a change in the model, not a claim about any club, shaft or ball.
Ranges: 85–175 mph, 9–26°, 2,400–9,500 rpm, the span of the published shots the model was checked against (TrackMan Tour averages (2023 data)).

Green Reading Lab

Putting model, version 1

Open the Green Reading Lab

What it calculates

For a putt of a given length, from a given position around the hole, on a flat, tilted green of a given slope and speed, the model finds the starting direction and speed that take the ball to the hole at the chosen pace. It reports where that start line passes the hole (the aim), how far the path bends from the straight line (the break), the starting speed, and the level-green putt that needs the same speed. In the around-the-hole view it does this for putts from every 15° and finds where their start lines cross.

The equations, in words

The model follows Grober’s Appendix A, which builds on Penner’s rolling-ball model.

The green is a plane tilted along one direction, the fall line. Slope is rise over run.
The ball rolls without slipping. For a solid sphere this leaves 5/7 of gravity’s pull down the slope acting on it: (5/7)·g·sin θ.
The grass slows the ball with a constant drag, always against its direction of travel, the same on every slope.
Grober applies the 5/7 factor to gravity only. The drag is defined from how quickly the ball slows on a level green, so it needs no rolling correction.

Green speed to drag

Green speed is how far, in feet, a ball rolls on a level green after leaving a USGA green-speed ramp, a grooved incline. If the ramp releases the ball at speed v and it rolls d feet, the constant drag is v² ÷ 2d. The release speed has never been published as a measurement, and the published values disagree:

6.0 ft/s (1.83 m/s), Holmes’s calculation, used by Grober and given by Wikipedia. Grober expects the true value is slightly lower.
6.4 ft/s, the USGA-funded Science of Golf putting toolkit.

Version 1 uses 6.0 ft/s, as Grober does, so its results can be checked against his. The higher value means more drag (2.05 ft/s² instead of 1.8 on a 10 ft green), so putts break less: the starting putt’s aim would be 7.3 in instead of 8.2 in.

Pace

Finishes 18 in past is Grober’s convention. The ball crosses the center of the hole at the speed that would carry it 18 inches further on its line: v² = 2 × 1.5 ft × (drag + (5/7)·g·sin θ·sin φ), where φ is its direction at the hole.
Dies at the hole is ours: the ball’s speed runs out exactly at the center of the hole. Its last inches curl downhill, as a dying putt does.

How it solves a putt

18 in past. As Grober does, the model starts at the hole with the terminal speed and runs time backwards until the ball is the putt’s length away. It searches the direction of arrival (bracketing, then regula falsi) until that point is the chosen position, to within a millionth of a degree.
Dies at the hole. It rolls the ball forwards and adjusts the starting direction and speed together (Newton’s method) until the ball stops within 0.01 in of the center.
Both use fourth-order Runge–Kutta steps of 4 milliseconds or less. Rolling the answer forward again lands it within 0.001 in of the hole’s center in every test.

The numbers it shows

Aim: the distance from the hole’s center to the start line where it passes the hole, Grober’s measure. The hole is 4.25 in across, so the edge is 2.125 in from the center.
Break: the largest distance between the path and the straight line from ball to hole.
Level-green length: the length of a putt on a level green, at the same pace, that needs the same starting speed.
Capture: Holmes found that a putt heading straight at the center of the hole drops at up to 1.63 m/s (5.35 ft/s) on a level green. Penner corrects this for slope (his eqs 28–31); the model applies his correction using the slope along the ball’s direction at the hole. The table under the tool compares it with the ball’s speed at the hole’s edge.
Target point and diamond: for each direction, the point nearest the start lines of five putts spanning ±30° (Grober’s Appendix B, a least-squares fit, written here with unit normals so a line along the fall line is handled). The diamond’s size is the mean of its height and width, Grober’s measure.

Checked against the published figures

Grober reports his results as figures. We read the values from his figure images on October 11, 2026, calibrating each figure’s axes (about ±0.2 in). The model has no fitted constants: nothing was tuned to match.

Middle of the target diamond, inches above the hole (Grober Fig. 24; 18 in past): model minus published
Putt	5 ft-%	10 ft-%	15 ft-%	20 ft-%	25 ft-%	30 ft-%
5 ft	−0.06	−0.05	−0.07	0.00	+0.01	+0.03
7.5 ft	−0.05	−0.05	−0.04	−0.03	−0.03	+0.03
10 ft	−0.03	0.00	−0.06	−0.02	0.00	−0.08
12.5 ft	−0.03	−0.06	−0.04	−0.02	−0.09	−0.15
15 ft	−0.04	−0.03	−0.01	−0.06	−0.14	−0.24

Over all 30 points: average error (root mean square) 0.07 in, worst 0.24 in. For the 10 ft putt on a 20 ft-% green, Grober’s figure gives 11.00 in and the model 10.98 in.

Other published checks
Check	Published	Model
Diamond size, 10 and 15 ft putts, 5–30 ft-% (Grober Fig. 25), worst of 12	within 0.09 in (RMS 0.05 in)
Aim point, 10 ft from 8 o’clock, 20 ft-% (Grober Fig. 1b), feet from the hole	0.38 left, 0.56 up	0.38 left, 0.57 up
Starting speed, straight 10 ft putt that stops at the hole, level green (Penner Fig. 3)	2.37 m/s	2.37 m/s
Starting speed, straight 10 ft putt that stops at the hole, 5° uphill (Penner Fig. 3)	3.05 m/s	3.05 m/s
Starting speed, straight 10 ft putt that stops at the hole, 5° downhill (Penner Fig. 3)	1.36 m/s	1.36 m/s
Capture limit, centered, 5° uphill (Penner §3)	1.71 m/s	1.71 m/s
Capture limit, centered, 5° downhill (Penner §3)	1.56 m/s	1.56 m/s

Grober’s text describes the Fig. 1 putt as aiming “5 inches left of the hole”. His figure places the aim point 4.6 in left of the hole and 6.7 in above it, 8.1 in from the center; the model gives 8.2 in. Penner’s speeds are run through this model with his drag for an average green (ρg = 0.131).

Where it stops

Putts of 2.5–15 ft, the lengths Grober calculated. He calls a plane “often a reasonable approximation within 10-15 feet of the hole”. Green speeds of 4–15 ft (Penner’s “very slow” 4 ft to Oakmont’s 15 ft) and slopes of 0.5–6%. Green speed × slope may not pass 30 ft-%, the top of Grober’s calculations. The tool refuses anything outside these.
Real greens are not planes. Grain, undulation, footprints and wind are not modeled.
The ball is taken to roll from the moment it leaves the putter. In reality it skids and bounces for about the first 20% of the putt (Penner, citing Cochran and Stobbs, and Daish).
The drag is constant. Hubbard and Alaways measured about 10% more drag at low speed over a 14 ft putt (as reported by Penner).
The answer is only as good as the slope and speed you enter. The tool does not know any real green.

Sources

R. D. Grober, “The Geometry of Putting on a Planar Surface” (2011). arXiv:1106.1698. The equations of motion (Appendix A), the 18-inch pace, the target point and diamond (Appendix B), the 6.0 ft/s ramp speed and the figures checked below.
A. R. Penner, “The physics of putting”, Canadian Journal of Physics 80 (2002) 83–96. the rolling-ball model the paper builds on, its validation speeds, and Holmes’s capture limit with its slope correction (eqs 23, 28–31).
B. W. Holmes, “Putting: how a golf ball and hole interact”, American Journal of Physics 59 (1991) 129. the 1.63 m/s capture limit, as quoted by Penner.
Wikipedia, article on the USGA green-speed device. the 6.00 ft/s release speed (citing Holmes 1986) and the USGA’s 8, 10 and 12 ft guide speeds for regular play and 15 ft readings at Oakmont.
USGA, The Science of Golf: Test Lab Toolkit, The Swing: Putting (grades 9–12). its classroom exercise assumes the ball leaves the ramp at 6.4 ft/s.
The R&A, Clarification 4.3a/1, Limitations on Using Green-Reading Materials. with Rule 4.3; scale and size limits in effect since 1 January 2019.

All sources read October 11, 2026. Version history: v1 (October 11, 2026), first release.

My Bag Report

Findings from your own numbers, version 1

Open My Bag Report

What it calculates

Nothing is modeled and nothing is stored. The report reads what My Lab already holds (the bag with its lofts, carries, how each carry was measured, launch numbers and spreads; the gap band; the wedge card and its band; the usual miss) and writes findings. Each one is arithmetic on those numbers, labeled Model, or a difference from a named, dated source, labeled Published. The golfer’s own numbers are labeled Yours.

Club count. The number of clubs saved, against the limit of 14 in R&A, Rules of Golf, Rule 4.1b.
Carry span. The longest carry minus the shortest, each rounded to a yard. Clubs without a carry are named.
How carries were measured. A count of each carry source the golfer chose (launch monitor, range estimate, on the course), when there is more than one.
Carry gaps. The Gapping & Bag Builder’s own ladder: clubs by carry, longest first, and each gap is one carry minus the next. A gap outside the golfer’s band is flagged with the yards it is over or under; without a saved band the starting setting, 7 to 15 yards, is used and named as such. Equal carries, or a shorter-carrying club with less loft, are flagged as overlaps.
Overlapping spreads. For two neighbors on the ladder that both have a spread saved from Shot Pattern Lab, each club’s range is its carry plus or minus one sample standard deviation. When the shorter club’s range reaches past the bottom of the longer club’s, the yards they share are reported. Clubs that are not neighbors are never paired.
Loft steps. Known lofts in order, each minus the one before; the smallest and largest steps are reported, and two clubs at one loft are named.
The wedge card. The Wedge Matrix’s ladder with the card’s own band (10 yards when none is saved): the largest gap, the distances it leaves uncovered, how many gaps are over the band, and carries 2 yards apart or closer.
Launch numbers. For each club with saved ball speed, launch angle and spin, the difference from TrackMan’s PGA Tour or LPGA Tour average for the same club (the golfer chooses which). Only the difference is shown, with no judgement. TrackMan publishes one hybrid average and none for wedges, 7-woods or 2-irons, so those are not compared.
The miss. The miss the golfer chose in My Lab, never inferred, with links to where it is explained.

Hand-offs

Each finding links to one to three places that deal with it: a Lab tool, a finder, a guide or a category. Gap hand-offs follow the Gapping & Bag Builder’s rules for the clubs on each side. No product is named or ranked.

The example and the share link

With nothing saved, the report can show TrackMan’s published PGA Tour or LPGA Tour average carries as an example, labeled Published. They include no lofts, spreads or launch numbers, so the example has only club-count, span and gap findings. A share link holds what the Gapping & Bag Builder’s does: club names, types and carries, nothing else.

Checks

Unit tests check every finding against hand-worked numbers: the gap and band edges, overlaps by carry and by loft, spread pairs (only neighbors), loft steps, the wedge card’s ladder, the launch differences against TrackMan’s rows, the club mapping, every hand-off’s route, and that no finding uses a judgement word.

Limits

A finding is only as good as the number behind it, and the report never suggests one.
Carry only: roll, the sideways spread and conditions on the day are not part of the gaps.
A tour average is a reference point from one published season, not a target.

Sources

TrackMan Tour averages (2023 data), published May 2, 2024. Read October 11, 2026.
R&A, Rules of Golf, Rule 4.1b, “Limit of 14 Clubs; Sharing, Adding or Replacing Clubs During Round”. Read October 11, 2026.

Practice Plans

Sessions from your own numbers, version 1

Open Practice Plans

What it builds

A dated plan of practice sessions. It reads My Lab and the findings My Bag Report writes from it, and turns each finding it can measure into a drill: a stated number of balls with named clubs, what to write down, and the Lab tool where the result is recorded. A drill is a measurement procedure. Its targets are the golfer’s own numbers (Yours), our arithmetic on them (Model) or, in a sample plan, TrackMan’s published tour averages (Published). No drill is attributed to a coach or a study, and none predicts a result.

The drills

Gap check, for every gap the report flags: 5 balls with each of the two clubs, alternating; the targets are both saved carries and the gap between them, against the golfer’s band.
Overlap check, for two neighbors whose carry ranges (carry ±1 sample standard deviation, from Shot Pattern Lab) overlap: 10 balls with each; count the carries that land in the shared range.
Wedge card: 3 balls at each swing length saved for a wedge, against the card. When the card’s largest gap is over its band, a search: 5balls at a swing the golfer chooses, counting those that land in the yards the card leaves uncovered.
Carry check: 10 balls with a club that has no carry saved, a carry not measured on a launch monitor, or any other carry, against the saved number. Shot Pattern Lab marks a club with fewer than 10 shots, so checks use 10.
Dispersion check, for up to two clubs with a saved side spread: 10 balls at a target line; count those inside a corridor as wide as twice the saved side standard deviation (±1 SD), rounded to a yard. It is tied to the miss the golfer set in My Lab, or to the club’s spread finding.
Launch check, for each club the report compares with TrackMan’s tour averages: 10 balls on a launch monitor, against the golfer’s saved ball speed, launch and spin.
One loft: two clubs saved at the same loft get a gap check of their own.

The schedule

The golfer chooses 1 to 5 sessions a week, 20 to 120 minutes a session, 1 to 8 weeks, the first day and the time. Sessions fall on evenly spread days of each seven-day week: two a week on thefirst, fifth days, three on the first, third, sixth. The plan budgets 1 minute a ball, including setting up and writing the number down: a planning assumption, not a measurement.

Drills are placed in priority order (gaps, overlaps, the wedge card, missing carries, dispersion, launch, one loft, then other carries) until a session’s minutes are used. A drill that does not fit waits for the next session. When every drill has been placed once, the list starts again, so later weeks measure the same things a second time (round 2).

Calendar file, print and saving

The calendar file is written in the browser (iCalendar, RFC 5545): one event per session at the chosen local time, with every drill, its targets and its links in the description. Nothing is sent anywhere. Save to My Lab keeps only the five settings; the sessions are rebuilt from the saved numbers each time, so a new carry changes the targets.

Checks

Unit tests check each drill against hand-worked numbers (gaps, shared ranges, corridor widths, wedge targets), the order and fill of the schedule, the session dates, the calendar file’s escaping and line folding, and the strict reading of saved settings.

Limits

A plan is as good as the numbers saved in My Lab, and a carry written down on the range is an estimate.
One ball a minute will be slow for some golfers and fast for others.
A sample plan uses tour averages as stand-ins for a golfer’s own carries, so it has gap and carry checks only.

Sources

TrackMan Tour averages (2023 data), published May 2, 2024. Read October 11, 2026. Sample plans only.
RFC 5545, Internet Calendaring and Scheduling Core Object Specification (iCalendar). The calendar file’s format.

Aim & Dispersion Planner

Shot pattern simulation

Open the Aim & Dispersion Planner

What it calculates

For a shot aimed at a point, it gives the chance of finishing on the green or fairway, in each hazard, and elsewhere, and which side the misses fall on. It then finds the aim that best serves a goal the golfer picks, and says exactly what that goal is. It does not turn these chances into strokes or scores.

The model, in words

Coordinates are yards from the middle of the target: sideways (plus is right) and along the line of play (plus is long).
The shots form a bivariate normal pattern: a bell curve sideways with one standard deviation, another along the line with its own, and no link between the two. Its center is the aim plus the golfer’s average miss.
A green is an oval of the width and depth set. A fairway is a corridor of the width set, running the length of the hole.
Each hazard is a rectangle on one side. On a green, side hazards run alongside its depth and front and back hazards along its width, each measured from the edge at the widest point. On a tee shot, side hazards run the length of the fairway and short and long hazards cross the whole hole, measured from the target distance. Out of bounds runs on without end.
Where hazards overlap (only at a tee shot’s corners), the harsher one counts: out of bounds, then water, then sand, as their penalties under the Rules rank them.
A ball finishes where it lands. There is no roll, bounce or slope.

The simulation

50,000 shots: 12,500 pairs of standard normal numbers from a seeded generator (mulberry32, seed 20080214, Box–Muller), each used four times with its sideways and distance parts mirrored. The sample is exactly symmetric, so a symmetric hole gives exactly equal chances left and right.
The same shots are reused at every aim, so the numbers change smoothly and the same setup always gives the same answer. Because the four mirrored shots of a draw can agree, the sampling error is counted on the 12,500 draws: at most ±0.9 points at 95%, and much less for chances near 0 or 100%.
A chance shown as “<0.1%” means none of the 50,000 shots finished there.

The best aim

Most on the green (or fairway): the aim with the highest chance of finishing on the target. It may sit off the target only by as much as the average miss it undoes.
Least in one hazard, or fewest in any hazard: among aims on the target, the lowest chance of finishing there. Chances within 0.05 of a point of the lowest count as equal; of those, the aim with the most on the target wins. The second goal counts a bunker the same as water or out of bounds.
Every aim is a whole yard. A tee shot is searched sideways at the distance the golfer aims; an approach, across the whole green. The search compares exact chances, not simulated ones, so sampling noise cannot move the answer: rectangles under an uncorrelated normal are products of normal-distribution differences, overlaps are removed by inclusion–exclusion, and the oval green is integrated numerically. The readouts at the best aim come from the simulation, like every other number on screen.

Published patterns

Broadie measures direction error as the angle between the line from start to finish and the line to the target, and reports its standard deviation for long tee shots by group (Table 1). An angle becomes yards at the shot’s distance: sideways SD = distance × tan(angle). His own example: 4° is 14 yd on a 200 yd shot and 21 yd on a 300 yd one.

Broadie 2008, Table 1, long tee shots
Group (scores)	Direction SD	75% tee distance	Sideways SD at 220 yd
PGA Tour players (64–79)	4.0°	297 yd	15.4 yd
Low-handicap amateurs (70–83)	5.4°	248 yd	20.8 yd
Middle-handicap amateurs (84–97)	6.4°	237 yd	24.7 yd
High-handicap amateurs (97–120)	8.1°	216 yd	31.3 yd

The paper gives no distance spread for these shots, so the planner does not invent one: with a published pattern, short and long hazards need the golfer’s own distance spread. The planner offers these patterns for tee shots only, where they were measured.

Checked against exact answers

Sideways chances on a corridor, and any rectangle under an uncorrelated pattern, have exact answers from the normal distribution. On the starting tee shot (a 30 yd fairway, water from the left edge to 25 yd beyond it, aimed at the middle 220 yd out):

Simulated chances against exact answers
Pattern	Fairway, exact	Simulated	Water, exact	Simulated
PGA Tour players, 4.0°	67.0%	67.4% (+0.4)	16.0%	15.8% (−0.2)
Low-handicap amateurs, 5.4°	52.9%	52.9% (0.0)	20.8%	20.9% (+0.1)
Middle-handicap amateurs, 6.4°	45.7%	45.2% (−0.5)	21.9%	22.2% (+0.3)
High-handicap amateurs, 8.1°	36.8%	36.3% (−0.5)	21.5%	21.9% (+0.4)

A round green 24 yd across under a round pattern with a 10 yd standard deviation: exact 51.3% (1 − e^−R²/2σ²), simulated 51.8%. The largest difference above is 0.48 points.

Limits

Real patterns are not perfectly normal. Broadie counts the “awful” shots that cost amateurs the most (Table 3); a bell curve has thinner tails than a round with a few of those in it, so the planner can understate the chance of a big miss.
Sideways and distance errors are treated as unrelated. A pattern that tilts (long misses going left, say) is not modeled.
Broadie’s groups are by score range, from rounds on six courses in 2005–2007, and his direction SD comes from where tee shots finished. A group is not a handicap, and one golfer can differ a lot from their group.
A Shot Pattern Lab spread is carry: where the ball lands. The planner treats it as where the ball finishes.
Wind, slope, lie and roll are not in the model. Add a known shift as an average miss.
The chances say where the ball is likely to finish, not what each result costs. Which miss is worse in strokes needs scoring baselines such as Broadie’s, which are copyrighted; the planner does not use them.
Inputs are limited: spreads 1 to 60 yd, tee shots 150 to 340 yd, approaches 40 to 250 yd. Anything outside is refused.

Sources

M. Broadie, “Assessing Golfer Performance Using Golfmetrics”, Science and Golf V: Proceedings of the 2008 World Scientific Congress of Golf. Table 1 (direction SDs and 75% tee distances by group), the definition of direction error and its 4° example, Table 3 (awful shots). Read October 14, 2026.

Putt Odds

Make odds, version 1

Open Putt Odds

What it calculates

The chance of holing a putt of a given length on a flat green, split into its two parts: the start line and the pace. The model is the geometric one in Andrew Gelman’s Stan case study (first published with Deborah Nolan in 2002), with the pace term Mark Broadie added. We rebuilt it in TypeScript from the equations as printed. No code was copied.

The equations, in words

The window. The ball’s center must pass within the hole’s radius less the ball’s: R − r = (4.25 − 1.68) ÷ 2 in = 1.285 in. From x feet, the start angles that do this lie within ± asin((R − r) ÷ x) of the center.
Line. The start angle is normal around the aim with standard deviation σ_angle. The chance it is inside the window is 2Φ(window ÷ σ_angle) − 1.
Pace. The golfer aims to roll the ball 1 ft past. With no hole it would stop at u, normal with mean x + 1 and standard deviation (x + 1) × σ_distance. It drops only if u is between x and x + 3: chance Φ(2 ÷ ((x + 1)σ_distance)) − Φ(−1 ÷ ((x + 1)σ_distance)).
Make. The two are independent, so the chance of holing is line × pace.

Broadie fixed the 1 ft and the 3 ft; they were not fitted. The tool also shows the two ways the pace fails: short of the hole (u < x) and more than 3 ft past (u > x + 3).

The published fit

Gelman fits the model to summary data from Mark Broadie on putts by pro golfers: the share holed at 31 distances from under a foot to 75 ft, with an extra error term σ_y on each point. Posterior means: σ_angle = 1.020° ± 0.006, σ_distance = 0.080 ± 0.001, σ_y = 0.003 ± 0.001. The tool starts from 1.02° and 8%. These are the pros’ numbers, so the starting odds are pros’ odds.

Checked against the published numbers

We ran the published fit against the data file published with the case study, on October 14, 2026. Over all 31 distances the model is within 0.35 points on average (root mean square) and 0.79 at worst (5.94 ft). The case study puts its fit at about 0.3 points (σ_y = 0.003). The data themselves are not reproduced here.
Refitting the model to that data from scratch (posterior mode, same half-normal priors) gave σ_angle = 1.020° and σ_distance = 0.0801, inside the published uncertainty. The angle-only model refitted to the case study’s older dataset (Berry, 1995) gave 1.53°; the case study reports 1.53° ± 0.02.
The case study says a 0.5° spread would make “over 80%” of 10 ft putts. Its formula gives 78.0%, read here as a rounded reading of its chart. Its other statements hold: 58.7% at 15 ft (“over 50%”), and 12.2% at 2 ft with a 20° spread (“missing more than half”).

Make %, published fit
3 ft	5 ft	10 ft	20 ft	40 ft
95.4%	75.7%	38.9%	14.4%	4.2%

Where it stops

Putts of 1–75 ft; the fit covers putts to 75 ft. Start-line spreads of 0.5–5° and pace spreads of 2–25%: the case study’s illustration starts at 0.5°, and both upper limits are well past the pros’ fit. The tool refuses anything outside these.
The green is flat. Slope, grain and green speed are not modeled; the Green Reading Lab covers slope and speed.
As the case study says, the errors are assumed independent, the pace error proportional to the length, and the start-line spread the same at every length. Broadie’s 1 ft and 3 ft are fixed guesses, not fitted.
The data are aggregated over pro golfers, courses and conditions. Changing the spreads models a different stroke; no dataset of amateur putts is used.

Sources

A. Gelman, “Model building and expansion for golf putting”, Stan case study (24 Sep 2019). the angle model, the angle-and-distance model with Broadie’s 1 ft and 3 ft, the fitted values (posterior means and sds), the ball and hole sizes, the statements checked below, and the putting data file published with it, which we used only to check the fit.

All sources read October 14, 2026. Version history: v1 (October 14, 2026), first release.