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Bates Distribution

The probability distribution of the arithmetic mean of n independent identically distributed uniform random variables, a scaled Irwin–Hall sum.

Version
v4 · 2026-09-19 · History
Domain-specific #
1353
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Probability Theory → Mathematics
Aliases
Bates(n) distribution, Uniform-mean distribution

Core Idea

Let \(U_1,\ldots,U_n\) be independent Uniform((0,1)) random variables. Their arithmetic mean

\[ X=\frac1n\sum_{i=1}^n U_i \]

has the Bates distribution with parameter (n). Since the sum has the Irwin–Hall distribution, Bates is its scale transformation by (1/n). It is supported on ([0,1]), has mean (½), and variance (1/(12n)).[1]

The recognition invariant is fixed number (n) + iid continuous uniforms + arithmetic mean + scaled convolution law.

How would you explain it like I'm…

The Average-Spinner Pattern

Spin a spinner that can land anywhere between 0 and 1, all spots equally likely. Do it a few times and find the average of your spins. The average usually lands near the middle, and the more spins you average, the more it bunches up there. The pattern of where that average lands is called the Bates distribution.

The Average of Random Picks

Imagine a random number maker that picks any number between 0 and 1, with every number equally likely. Pick n of these numbers, each on its own, and take their average. The Bates distribution describes how likely each possible average is. The average can be anywhere from 0 to 1, but it clusters around 1/2, and the more numbers you average, the tighter the cluster gets. It is closely related to the distribution of the sum of those numbers, just squeezed down by dividing by n.

Mean of Uniform Random Variables

Take n independent random numbers, each uniformly distributed on the interval from 0 to 1, and compute their arithmetic mean. The probability distribution of that mean is the Bates distribution with parameter n. It is closely related to the Irwin-Hall distribution, which describes the sum of the same numbers; since the mean is the sum divided by n, the Bates distribution is just the Irwin-Hall distribution scaled down by 1/n. The Bates distribution lives on [0, 1], has mean 1/2, and has variance 1/(12n), so it narrows as n grows. For n = 1 it is simply the uniform distribution. What identifies it is the fixed number of independent uniforms and the fact that you take their mean, not their sum.

 

Let U₁, …, U_n be independent Uniform(0, 1) random variables. Their arithmetic mean X = (1/n) Σ U_i has the Bates distribution with parameter n. Since the sum Σ U_i has the Irwin–Hall distribution, the Bates distribution is its scale transformation by 1/n, and its density is the n-fold convolution of the uniform density, rescaled. It is supported on [0, 1], with mean 1/2 and variance 1/(12n), so it concentrates around 1/2 as n increases; for n = 1 it reduces to the uniform distribution. The defining ingredients are a fixed number n, iid continuous uniforms on the unit interval, and the arithmetic mean; changing to a sum gives Irwin–Hall, and changing the component distribution gives a different law.

Structural Signature

  • Integer parameter \(n\ge1\).
  • Independent random variables.
  • Identical Uniform((0,1)) inputs, or an explicitly affinely transformed interval.
  • Arithmetic sum and division by (n).
  • Support ([0,1]).
  • Symmetry about (½).
  • Piecewise-polynomial density inherited from repeated convolution.
  • Mean (½) and variance (1/(12n)).
  • Irwin–Hall scaling relation.
  • Uniform case at (n=1).
  • Triangular density at (n=2).
  • Concentration and asymptotic normal approximation as (n) grows.
  • Parameter convention stated to prevent alternate rescalings.

What It Is Not

It is not the Irwin–Hall distribution, which describes the sum rather than the mean. It is not a mixture of uniforms; all draws contribute jointly through convolution. It is not exactly normal for finite (n), although a standardized version approaches normality.[2]

It is not the distribution of a mean from arbitrary bounded or nonidentical variables unless a generalized construction is explicitly defined.

Scope of Application

The distribution appears in simulation, business-statistics approximations, bounded random-number generation, sampling-distribution examples, and convolution exercises. Affine transformation gives the mean of iid Uniform((a,b)) variables on ([a,b]).[3]

For large (n), direct piecewise-polynomial evaluation can be numerically awkward; normal approximations or stable recurrence/convolution methods may be preferable.

Clarity

Scaling is load-bearing. If \(S=\sum U_i\), then \(S\in[0,n]\) is Irwin–Hall; \(X=S/n\in[0,1]\) is Bates. Densities transform with the Jacobian (f_X(x)=n f_S(nx)).

Independence and identical input laws are part of the named identity. Removing them changes variance and generally the entire distribution.

Manages Complexity

The family packages an (n)-fold convolution into one parameterized law. Moments, support, density shape, and limiting behavior can be reused without recomputing the distribution of each sample mean.

The compact name can obscure parameterization. Implementations should test support, scale, and moments to detect sum/mean confusion.

Abstract Reasoning

  1. Verify iid Uniform((0,1)) inputs and (n).
  2. Form the Irwin–Hall sum.
  3. Apply the (1/n) scale transformation and density Jacobian.
  4. Check support, symmetry, mean, and variance.
  5. Use exact piecewise polynomials for small/moderate (n).
  6. Quantify approximation error before using a normal limit.
  7. Apply affine transformation for Uniform((a,b)) inputs.

Knowledge Transfer

The portable structure is a sampling distribution derived by scaling a sum law. The proposed immediate parent is Probability Distribution.

Examples

n=1. The Bates distribution is Uniform((0,1)).

n=2. The mean has a symmetric triangular density on ([0,1]), peaked at (½).

Non-example. The sum of three uniforms on ([0,3]) is Irwin–Hall, not Bates until divided by three.

Structural Tensions

  • Sum versus mean parameterization.
  • Exact finite-(n) density versus normal approximation.
  • Simple moments versus piecewise-polynomial evaluation.
  • Named iid family versus generalized nonidentical inputs.
  • Bounded support versus Gaussian tail approximation.
  • Mathematical definition versus inconsistent software conventions.

Structural–Framed Character

Aggregation, scaling, convolution, support, and limiting behavior are structural. Uniform variables, densities, sample means, and distribution parameters are probability frame.

Structural Core vs. Domain Accent

The portable core is scaling an aggregate to form a reusable family. Independence, continuous uniform inputs, Irwin–Hall convolution, density pieces, and CLT approximation are constitutive domain accent.

This entry is a kind of Probability Distribution.

Probability Distribution is the broader abstraction this entry instantiates. Aggregation, Scaling, Convolution, Sampling, and Central Limit Theorem are related.

Relationships to Other Abstractions

Local relationship map for Bates DistributionParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Bates DistributionDOMAINDomain-specific abstraction: Probability Distribution — is a kind ofProbabilityDistributionDOMAIN

Current abstraction Bates Distribution Domain-specific

Parents (1) — outgoing typed relations

  • Bates Distribution is a kind of Probability Distribution Domain-specific

    Probability Distribution is the broader abstraction this entry instantiates.

Hierarchy paths (3) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Bates Distribution sits in a sparse region of the domain-specific corpus (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Composite Probability Distributions (5 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With

  • Irwin–Hall sum distribution.
  • Mixture distribution.
  • Exact Gaussian distribution.
  • Average of dependent or nonidentical variables.
  • Discrete uniform averaging.
  • A parameterization with unstated interval scaling.

References

[1] Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, 2nd ed., Wiley, 1995. registry ↩

[2] Luc Devroye, Non-Uniform Random Variate Generation, Springer, 1986. registry ↩

[3] Athanasios Papoulis and S. Unnikrishna Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed., McGraw-Hill, 2002. registry ↩

[4] William Feller, An Introduction to Probability Theory and Its Applications, vol. II, 2nd ed., Wiley, 1971. registry ↩