PRB-28

Normal MLE σ

MLE with n (not n-1).

Reading speed

GoverningNormal MLE σ

Governing equation

\\hat\\sigma=\\sqrt{\\sum (x_i-\\bar x)^2/n}

$n \in \left[2, 40\right]$

where

ss: ss (—)
n: n (—)
sig: Normal MLE σ (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\\hat\\sigma=\\sqrt{\\sum (x_i-\\bar x)^2/n}

$n \in \left[2, 40\right]$

$n = 2$ $sig = 2.449$

Along the curve

②

Domain of the variable — blue

n_{\min} = 2,\quad n_{\max} = 40

$n \in \left[2, 40\right]$

$n = 2$ $n = 2$

Along the curve

③

Response curve — green

\\hat\\sigma=\\sqrt{\\sum (x_i-\\bar x)^2/n}

$n \in \left[2, 40\right]$

$n = 2$ $sig = 2.449$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}sig}{\mathrm{d}n}

$n \in \left[2, 40\right]$

$n = 2.239$ $sig^{\prime} = -0.521$

Along the curve

Lecture brief

Historical brief

Bernoulli trials, Poisson, Gauss, Bayes and Shannon entropy turned chance into a calculus of belief and noise. The lab computes likelihood, interval and information in SI-free counts. This sheet (PRB-28 — Normal MLE σ) is the form associated with Normal MLE σ. Working symbols: $ss$ , $n$ $\rightarrow$ $sig$ . MLE with n (not n-1). Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

sig

from

ss

n

in Probability & statistics via

\\hat\\sigma=\\sqrt{\\sum (x_i-\\bar x)^2/n}

MLE with n (not n-1). Use it when a real probability & statistics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

ss = 12.000\,\mathrm{—}

n = 8.000\,\mathrm{—}

, the governing relation

\\hat\\sigma=\\sqrt{\\sum (x_i-\\bar x)^2/n}

yields

sig = 1.225\,\mathrm{—}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

ss ss12 —n n8 —

Outputs

Normal MLE σ sig1.225 —

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

MLE with n (not n-1). Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube