Fourier Fit

The red Powerball is a single integer between 1 and 26. Plot every draw and you get a zigzag; smooth it with a rolling average and the line settles near 13.5 with small wiggles. Wiggles look like waves — and waves have formulas. We sweep five smoothing windows, fit a Fourier series at each, and ask the harder question: is this signal, or is it filtered noise pretending to be signal?

Verdict

Zero peaks above the 95% noise floor at any of the 5 smoothing windows — the wave is filtered noise at every level.

Smoothing window(count) = peaks above noise floor

Power spectrum

Power per frequency bin from the FFT. The dashed line is the 95% noise floor — the highest power Gaussian noise would produce after the same filtering, corrected for testing all 662 bins simultaneously.

noisetop-K peak (in fit)above noise floor

Fitted formula

The Fourier series we just fit, written out. Each cos term is one of the top peaks from the power spectrum, written in amplitude–period–phase form. As you change K, terms appear and disappear here too.

f̂(t) = μ + Σk=1..5 ak·cos(2π·t/pk + φk)
f̂(t) = 13.58
+ 0.70 · cos(2π·t/147.1 − 1.25)period ≈ 49.0w
+ 0.63 · cos(2π·t/77.9 − 1.98)period ≈ 26.0w
+ 0.59 · cos(2π·t/94.6 − 2.82)period ≈ 31.5w
+ 0.49 · cos(2π·t/189.1 − 3.00)period ≈ 63.0w
+ 0.46 · cos(2π·t/264.8 − 0.11)period ≈ 88.3w

t = draw index, oldest → newest. Periods in draws (3 draws ≈ 1 week). Phase φ in radians.

Powerball + rolling average + Fourier fit

The faint zigzag is the actual Powerball drawn each night (1–26). The mid-weight line through the middle is the rolling average for the active window. The bold line is the sum of the top 5 highest-power sine waves fit to that rolling average. Drag K to add or remove harmonics; pick how many recent draws to show.

Show lastshowing 200 of 1,324 draws

K (harmonics)5

actual Powerball draws20-draw rolling averageFourier fit (K=5)

Residuals

What's left after subtracting the fit. Should look like noise centered on zero. The shaded band is ±1σ of filtered Gaussian noise (raw σ / √window) — what we'd expect from fair draws.

residual σ = 1.33expected σ = 1.72ratio = 0.77×

Draws analyzed
1,343

Series length (N)
1,324
after dropping warm-up

Window
20 draws
rolling average

Mean (μ)
13.58
of rolling avg

Filtered σ
1.62
rolling-avg stdev

Expected σ
1.72
raw σ / √20

Residual σ (K=5)
1.33
after fit

Significant peaks
0
of 662 bins, α=0.05 corrected

How it works

The series. For every Powerball draw since the format change to a 1–26 red ball pool (Oct 2015), we take the red Powerball value. We then sweep five rolling-average windows — 5, 10, 20, 50, 100 draws — to ask the same question at multiple smoothing levels. Each window drops its first W−1 warm-up points (their averages cover fewer than W draws) and subtracts the mean.

The transform. A discrete Fourier transform decomposes each de-meaned signal into a sum of sines and cosines at every frequency from one full cycle across the whole history up to the Nyquist limit (period of two draws). The power at each frequency tells us how much of the signal's wiggle lives there.

The null. A rolling average is a low-pass filter — even pure white noise, filtered this way, will produce a non-flat power spectrum with apparent peaks. So per window we run a Monte Carlo: 200simulations of Gaussian noise with the same standard deviation as real draws, run through the same pipeline. The 95th-percentile-of-max power across all bins is the global threshold for that window. Bars above it are 95%-significant after correcting for the fact that we're looking at hundreds of frequency bins at once.

The fit. We pick the top 10highest-power frequencies and reconstruct them as a smooth curve. Drag K to add or remove harmonics; switch windows to ask the question at a different smoothing level. If the draws are fair, every window's spectrum should be flat against its own threshold — zero peaks, at any smoothing.