Beyond the Average: Why Distribution Fitting Matters

Professor “Close Enough”

“Looks bell-shaped. Normal should be close enough.”

Similar shape. Similar probability?

Could “close enough” change our estimate of a rare event?

A closer look at the right tail

Beyond 3 SD: Normal ≈ 0.135% · Student t ≈ 0.586%

About 4× as likely under t.

A similar-looking curve can give a very different tail probability.

A distribution answers a question

A quantitative variable: a numerical quantity we measure or count.

A probability distribution: how probability is assigned to its possible values.

A distribution model describes shape, center, and spread, and helps us estimate probabilities and percentiles.

For repair labor hours…

What proportion of jobs require more than 8 hours?

What labor-hour threshold includes 90% of repair jobs?

Is 94 mph fast?

What are we measuring?

Four-seam fastball release speed, in miles per hour (mph).

Typically a pitcher’s fastest pitch.

Three MLB pitchers

Andrew Abbott · Cincinnati Reds
Tanner Bibee · Cleveland Guardians
Yoshinobu Yamamoto · Los Angeles Dodgers

Why understand the distribution?

Establish a pitcher’s usual velocity range.
Identify unusually fast or slow pitches.
Track changes over time.
Support coaching and training decisions.

A fastball is a pitch type, not a pitch exceeding a fixed speed.

All three pitchers play in MLB. The Reds and Dodgers are in the National League; the Guardians are in the American League.

Velocity changes can prompt further investigation, but do not by themselves establish fatigue or injury.

Establish a personal baseline: Knowing a pitcher’s typical velocity range is essential for determining whether today’s speeds are abnormal.
Track changes: If the overall velocity distribution shifts toward lower speeds, it warrants coach intervention to investigate the pitcher’s condition, mechanics, or training regimen. While this is a significant signal worth pursuing, it should not be used in isolation to diagnose fatigue or injury.
Evaluate pitch types alongside outcomes: Analyzing how the probability of a swing-and-miss or a hit changes based on velocity, location, and pitch type provides the necessary data to inform pitch-calling and training decisions.

2 minutes. Four-seam fastballs in regular-season Statcast records, March through September 2026. Ask for a show of hands about 94 mph. Speeds and counts are pitch-level observations, not independent player samples. Source: https://baseballsavant.mlb.com/statcast_search and https://baseballsavant.mlb.com/csv-docs

Does one distribution describe the whole season?

Pooling the season may hide changes over time.

Three candidate models for each pitcher

2026 regular-season four-seam fastballs. Normal, location-scale t, and two-parameter Weibull.

What does the comparison say?

Pitcher	Normal AIC	t AIC	Weibull AIC
Abbott	3200.29	3197.22	3244.30
Bibee	1877.28	1878.81	1943.72
Yamamoto	1970.04	1970.49	1995.87

Compare AIC within a pitcher. Lower means better relative fit after a parameter penalty.

Under our simplifying assumptions, AIC favors t for Abbott and Normal for Bibee and Yamamoto—but the Normal–t differences are small.

Lowest AIC among these candidates. Useful for our question?

A closer look at the quantiles

A personal baseline changes the answer

Pitcher	Observed share above 94 mph
Abbott	1.65%
Bibee	47.56%
Yamamoto	97.62%

An unusual pitch depends on the pitcher and the comparison period.

What did we assume?

We treat each pitcher’s velocity distribution as stable over the season.
We treat individual pitches as independent observations.

A good distributional fit does not establish independence or stability over time.

Cincinnati repair work

How much mechanic labor does a repair require?

What are these records?

City of Cincinnati fleet maintenance and repair work orders.
Our subset: 2,869 closed repair orders for police equipment, work-order year 2022.

What do we measure?

Recorded labor hours per repair order.

Why understand the distribution?

Plan staffing and labor budgets.
Estimate the proportion of labor-intensive jobs.
Allow for unusually demanding repairs.

Labor hours measure work effort—not elapsed repair time or vehicle downtime.

Source: Cincinnati Open Data · Fleet Preventative Maintenance & Repair Work Orders

Typical work and total work

Median: 1.6 labor hours

Mean: 3.10 labor hours

A few large jobs increase total labor demand.

32 orders record zero labor. Positive-labor models use the remaining 2,837 orders.

Three models for positive labor

Models use all positive values. Display shows 0–20 hours, with the longer tail outside the view.

Lognormal fits best among these candidates

Model	AIC
Gamma	12147.59
Lognormal	11526.07
Weibull	12166.14

Lower AIC means better relative fit. Lognormal tracks the long tail most closely.

Which model gets a large job right?

	Share above 8 hours	90th percentile
Observed	8.81%	7.6 hours
Gamma	7.19%	7.0 hours
Lognormal	6.98%	6.6 hours
Weibull	7.96%	7.3 hours

Positive-labor orders (n = 2,837).

Lognormal has the lowest AIC among these candidates; Weibull is closer to the observed values for these two questions.

Underestimating large jobs can reduce planned capacity. Acceptable error depends on the decision.

2 minutes. These answer the two questions from the opening slide: about 8.8% of jobs exceed 8 hours, and about 7.6 hours covers 90% of positive-labor jobs. All three models understate both, Lognormal the most. Ask whether lowest AIC makes every estimate most accurate. All estimates come from fitting and evaluating the same sample, so this is an illustration of task-specific fit rather than a forecasting contest. Eight labor hours is an illustrative staffing threshold, not an official service standard. Tail frequency alone is not total workload; job sizes and arrival volume also matter.

Talking point: the Gamma fit reproduces the sample mean exactly (3.14 hours among positive-labor orders; this is a property of the Gamma MLE), yet it still understates large jobs. Getting the average right does not mean getting the tail right. No claim that Cincinnati uses these models or was understaffed; these are teaching applications.

Flood claims after Hurricane Helene

Where do these data come from?

The Federal Emergency Management Agency (FEMA) manages the National Flood Insurance Program (NFIP).

Our sample

781 flood insurance claims associated with Hurricane Helene.
North Carolina · 2024 losses · Single-family homes.
Each policy has $250,000 in building coverage.

What do we measure?

Recorded building claim payments—not total flood damage.

Source: OpenFEMA · NFIP Redacted Claims, Version 3

Data snapshot: September 8, 2026 · Payments may change as claims develop.

Why model insurance claim payments?

An average payment does not tell us how often large claims occur.

How likely is a claim payment to exceed $100,000?
How often do payments reach the $250,000 coverage limit?

Underestimating large payments can lead to underestimating funding needs.

The policy helps shape the distribution

A mixed distribution for payments

A point mass at $0: 17.93%

A continuous interior between $0 and $250,000: 74.01%

A point mass at $250,000: 8.07%

One curve for the interior?

578 interior claims. Truncated Lognormal has lower AIC (ΔAIC ≈ 32.78).

Payment above $100,000: observed 27.4% · model 25.5%

Full model: point masses at $0 (17.93%) and $250,000 (8.07%), plus a truncated Lognormal interior (74.01%).

Professor “Close Enough” revisited

Close enough for a typical value?

Close enough for a tail probability?

Close enough for the decision you need to make?

You can make a presentation like this

This presentation uses Quarto.

A .qmd file combines Markdown, code, and presentation settings.

MTH 209 introduces the skills behind reproducible statistical documents.

All models are wrong, but some are useful

George Box

Useful for what?

Thank You!

Please do not hesitate to contact me (Tessa Chen) at ychen4@udayton.edu for questions or further discussions.