Empirical data for sequences of measurements

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Two height samples both average $175 cm$ . One list wanders from $168$ to $182$ ; the other sticks to $174$ and $176$ . If the average alone cannot tell them apart, what single number records how spread out the trials are — and how do you draw the shape of a longer list?

A sequence of measurements

A quantity that changes from trial to trial is a random variable $X$ . Measuring $X$ once per trial produces a finite list

$x_{1}, x_{2}, \dots, x_{n} .$

Height of a person is one such $X$ . Eight men, for example, may give either

$168, 170, 172, 175, 176, 177, 180, 182$

or the much tighter list

$174, 174, 174, 174, 176, 176, 176, 176$

(both in centimetres). Both lists will turn out to share the same empirical mean $175$ . The next section names that mean; the section after it names the spread.

Empirical mean

For any sequence $x_{1}, \dots, x_{n}$ the empirical mean is the ordinary average

$x ˉ := \frac{1}{n} (x_{1} + x_{2} + \dots + x_{n}) .$

On the wide list,

$x ˉ = \frac{1}{8} (168 + 170 + 172 + 175 + 176 + 177 + 180 + 182) = 175.$

On the tight list the same recipe again returns $175$ . So $x ˉ$ alone does not see the difference between “scattered about $175$ ” and “almost glued to $175$ ”.

Empirical standard deviation

The second characteristic quantity is built from squared deviations from that mean. Its square — the empirical variance — is

$(Δ x)^{2} := \frac{1}{n - 1} [(x_{1} - x ˉ)^{2} + (x_{2} - x ˉ)^{2} + \dots + (x_{n} - x ˉ)^{2}] .$

The denominator is $n - 1$ , not $n$ : with that choice $Δ x$ is an unbiased estimator of the theoretical standard deviation of the underlying random variable $X$ . For large $n$ the two choices hardly differ. The positive square root $Δ x$ is the empirical standard deviation.

Drag a. At a = 0 the eight green marks match the tight height sample (Δx ≈ 1.07); at a = 1 they match the wide sample (Δx ≈ 4.8). The blue mark is the shared empirical mean 175. The orange segment is ±Δx about that mean.

Drag $a$ . At $a = 0$ the green marks sit on the tight sample and $Δ x \approx 1.07$ (because $(Δ x)^{2} = 8/7$ ). At $a = 1$ they sit on the wide sample and $Δ x \approx 4.8$ (because $(Δ x)^{2} = 162/7 \approx 23$ ). The blue mark never moves: both endpoints keep $x ˉ = 175$ . The orange segment is the interval $x ˉ \pm Δ x$ ; it widens exactly when the marks spread.

Rule of thumb

The smaller $Δ x$ is, the smaller the variation of the measured values about $x ˉ$ . In the extreme case $Δ x = 0$ , every measurement equals $x ˉ$ .

Drag a. At a = 1 the green marks are the wide height sample; drag toward 0 and every mark slides onto the blue mean at 175. The orange segment is ±Δx; it shrinks to a point when Δx = 0.

Drag $a$ down from $1$ . The wide sample collapses onto the blue mean; the orange $\pm Δ x$ segment shrinks to a point. That is the rule of thumb on the board: shrink the deviations and $Δ x$ follows.

Distribution — the histogram

Mean and standard deviation are two numbers. To see the shape of a longer sequence, group the values into neighbouring classes $K_{1}, \dots, K_{s}$ . Let $m_{r}$ be the number of measurements that fall in class $K_{r}$ . The relative frequency of that class is $m_{r} / n$ . A histogram draws, over each class interval, a rectangle whose height is that relative frequency.

One hundred heights, for example, may fill six classes as follows (centimetres):

Class	Interval	$m_{r}$	$m_{r} /100$
$K_{1}$	$150 \leq x < 165$	$2$	$0.02$
$K_{2}$	$165 \leq x < 170$	$18$	$0.18$
$K_{3}$	$170 \leq x < 175$	$30$	$0.30$
$K_{4}$	$175 \leq x < 180$	$32$	$0.32$
$K_{5}$	$180 \leq x < 185$	$16$	$0.16$
$K_{6}$	$185 \leq x < 200$	$2$	$0.02$

The end classes are wider than the middle ones; the board still uses height $m_{r} / n$ , not a density scaled by width.

Drag t. At t = 0 every class height is 1/6 (flat); at t = 1 the six green bars match the 100-person height table (relative frequencies 0.02, 0.18, 0.30, 0.32, 0.16, 0.02). Bar height is mᵣ/n, not density.

Drag $t$ . At $t = 0$ every bar has height $1/6$ — a flat relative-frequency picture with no preferred class. At $t = 1$ the bars match the table above: most mass sits in $K_{3}$ and $K_{4}$ around $175$ , and the tails $K_{1}$ and $K_{6}$ stay rare. The living shape is exactly what $x ˉ$ and $Δ x$ alone cannot draw.

What this page put in place

A measurement sequence $x_{1}, \dots, x_{n}$ of a random variable is summarized first by the empirical mean $x ˉ$ and the empirical standard deviation $Δ x$ (with denominator $n - 1$ ). The same mean can hide very different spreads; $Δ x = 0$ forces every trial onto $x ˉ$ . A histogram then reads the shape through class counts and relative frequencies $m_{r} / n$ . The next natural question is how a theoretical distribution for $X$ relates to these empirical summaries — that step is left for the following section.