Could the true average lie beyond everything you measured?

You time two deliveries. One takes four minutes, the other six. Their average is five minutes, comfortably between the readings.

Must the delivery service’s true average also lie between four and six?

Imagine that most deliveries arrive quickly, but a small fraction get stuck for an hour. Your two deliveries could easily have missed those delays. The average of the whole population can sit beyond everything in your notebook.

The surprise goes further. Even a perfectly symmetric population, with no strange tail or hidden second group, puts its mean outside a two-reading sample half the time.

This gives us a useful expedition: first watch a sample miss the middle of its world; then ask how a belief about that world can reach beyond the sample.

Two averages, two different jobs

The sample average is the arithmetic average of the values we collected. Write the readings as $x_1,\dots ,x_n$, where $n$ is their number. Then

$$\overline{x}=\frac{x_1+\cdots +x_n}{n}.$$

The bar in $\overline{x}$ means “average.” Because this is an ordinary average of recorded values, it always lies between the smallest and largest reading.

The population mean, written $\mu$, averages across the whole process that generates readings, including values we have not encountered. If $X$ denotes one random reading, we write $\mu =\mathbb{E}[X]$, where $\mathbb{E}$ means expectation, or probability-weighted average. We assume that this arithmetic mean exists.

Here is the distinction side by side:

Try the even-spread world first. Before revealing the true mean, predict whether your readings must surround it. Then take thirty fresh samples.

Take a sample, then reveal the world

Can the average escape your sample?

Restart sampling

Each reading is an independent draw from the chosen toy population. Predict whether its true average could lie beyond every reading, then try a batch of fresh samples.

An even spreadA long right tailA rare long delay

Every value between 4 and 6 is equally likely. Values are in minutes.

2 readings5 readings10 readings20 readings

Sample 1: the readings you collected

4.895.79

Observed range: 4.89 to 5.79. Sample average: 5.34.

Could the population average be outside this range?

Yes, it could It must be inside

Draw another sample Try 30 fresh samples

This experiment checks observed ranges, not uncertainty intervals. Samples use a repeatable random sequence; the exact percentage comes from the known toy population. Changing the world or sample size restarts the experiment. The batch limit is 120 samples.

Open the sampling experiment in the playground.

How a sample misses the middle

Suppose readings are independent draws from the same continuous, symmetric population. Its mean divides the population into two equally likely sides.

Two readings miss the mean when both land below it or both land above it:

$$Pr(\mu \text{ outside the sample range})={\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2=\frac{1}{2}.$$

The symbol $Pr$ denotes probability. With $n$ readings, the same reasoning gives $2^{1-n}$. Every extra independent reading halves the chance that the whole sample lands on one side.

Symmetry matters. Let $p$ be the probability that a reading falls below $\mu$. For a continuous population, the general formula is

$$\boxed{Pr(\mu \text{ outside the sample range})=p^n+(1-p{)}^n.}$$

The first term is the chance that all readings fall below the mean; the second is the chance that all fall above it. A long right tail can pull the mean above most ordinary observations, making $p$ much larger than one half.

In the demo’s rare-delay world, 98% of readings come from the interval 4 to 6 minutes, and 2% from 99 to 101 minutes. The population mean is

$$\mu =0.98\times 5+0.02\times 100=6.9\text{ minutes}.$$

Every ordinary reading is below that mean. Ten deliveries avoid the rare group with probability ${0.98}^{10}$, about 81.7%. A notebook full of five-minute trips can come from a service whose true average is nearly seven minutes.

These percentages concern the entire observed range. A narrower interval inside that range has even fewer chances to contain the truth.

What reweighting can reach

A natural way to express uncertainty is to let the observed values carry different shares of the population. Perhaps the four-minute reading represents a little more of the world; perhaps the six-minute reading does.

The Bayesian bootstrap, introduced by Donald Rubin, represents those shares with random positive weights. For $n$ independent unit readings, the weights have a Dirichlet distribution, written

$$(W_1,\dots ,W_n)\sim \mathrm{Dirichlet}(1,\dots ,1).$$

The symbol $\sim$ means “has this distribution.” Each $W_i$ is nonnegative and the weights sum to one. With these parameters, all allocations of shares are equally likely across the simplex, the geometric space of possible allocations.

One allocation gives a possible weighted mean,

$$M_W=\sum _i{W}_i{x}_i.$$

For readings four and six, this moves smoothly between four and six. It cannot reach seven. No arrangement of positive shares can put a weighted average beyond the values being averaged.

This is a property of the model’s support, the set of values it allows. The observed-support bootstrap describes uncertainty in the shares assigned to recorded values. Possibilities beyond those values require an additional choice about unseen parts of the population. The Hilbert-space reading of the same weights makes the same range restriction visible in another geometry.

In particular, a central 95% interval inside a two-reading range cannot have 95% repeated-sampling coverage for a continuous symmetric population: even the full range contains the mean only half the time. That is a statement about sampling performance. The weight law remains a well-defined mathematical object.

Lift a reading by the step it can move you

The Epistemic Physics continuous-mean exploration asks how to extend this geometry without choosing a fixed extra width in advance.

One answer is the jackknife lift. “Jackknife” means checking what happens when one reading is left out. Let ${\overline{x}}_{-i}$ be the average without reading $i$. The distance that deletion moves the average is

Vertical bars denote absolute value, or distance from zero. The nonnegative number $h_i$ is the reading’s lift height. A reading far from the average gets more height because removing it changes the answer more.

For four and six, the average is five. Remove either reading and the average moves one unit, so both heights are one. Add many ordinary readings and any one reading usually moves the average less.

Now keep the same Dirichlet weights. An allocation produces two quantities side by side:

The Greek letter $\gamma$ is gamma. Interpret this pair as the location and scale of a Cauchy distribution, a curve with slowly decaying tails. Average those curves over the possible weight allocations. The result is the lifted law about the population mean.

The recorded values stay fixed. The lift supplies a particular rule for extending the uncertainty law beyond them. Its scale ${\gamma }_W$ describes width; it is not a standard deviation, because a Cauchy distribution has no ordinary variance.

Select the two-reading preset below, then the cluster with one outlier. Tap individual readings to inspect their deletion steps. Predict which reading will have the greatest height before selecting it.

From readings to a mean

Give each reading a little room

Reset the readings

Reweight the recorded values in sky blue; lift them with a Cauchy spread in rose. Both use the same readings. Select a reading to see how far the average moves when you leave it out.

Two readingsA cluster and an outlierTen readingsAll readings equal

#14#26

Add a reading Add Remove last

Average of all 2 readings

5.00

Average without reading #1

6.00

That reading’s lift height

1.00

The comparison leaves one reading out temporarily. Both laws below still use all 2 readings, each with one unit of trial weight.

Reweight the readings Jackknife lift Sample average

Reweight the readings

Central 95% of the weight law

4.05 to 5.95

Beyond the readings: 0%

Jackknife lift

Central 95% of the lifted law

-7.73 to 17.73

Beyond the readings: ≈ 55.1%

Positive mass beyond the sample opens the door to unseen averages. The percentage assigned there belongs to this construction; repeated-sampling coverage is a separate test.

What the picture computes

The sky histogram approximates random positive weights that sum to one, with a Dirichlet(1, …, 1) distribution. For each weight allocation, the rose law is a Cauchy distribution centered at its weighted average, with scale equal to its weighted lift heights. We average its exact conditional density and distribution function across 2,048 fixed allocations; the two-reading case uses evenly spaced quadrature weights. Displayed intervals are approximate 2.5th and 97.5th percentiles. The tails continue beyond the picture and are not rescaled to fit it.

These synthetic weights add no observed trials. The lifted law is rebuilt from the readings, so it is not a posterior from one fixed prior-and-likelihood model. Its Cauchy tails have no ordinary mean or variance; the gold reference is the sample average. This demo assumes an unrestricted number line and supports 2 to 20 readings between −100 and 100.

Open the lift explorer in the playground.

For two distinct readings, the lift assigns about 55.1% of its probability outside their range. The symmetric sampling experiment had a 50% exterior frequency. Those are two different calculations whose comparison is informative; their proximity alone does not establish calibration.

The construction also travels with the ruler. Add ten to every reading and its entire law shifts by ten. Multiply every reading by a positive scale factor and every height and interval width scales by the same factor. No hard-coded “extra centimetre” sneaks into an experiment measured in metres.

The conditional curve, and the meaning of its center

For a candidate population mean $t$, the conditional Cauchy density is

$$f(t\mid W)=\frac{{\gamma }_W}{\pi [(t-m_W{)}^2+{\gamma }_W^2]}.$$

Here $\pi$ is the circle constant, and a density assigns probabilities through areas under its curve. The lifted density is the average of $f(t\mid W)$ across weight allocations. The demo integrates each conditional Cauchy curve exactly and approximates the average over weights.

The average of the conditional locations $m_W$ is $\overline{x}$. The lifted law itself has Cauchy tails and therefore has no ordinary mean or variance. Calling $\overline{x}$ its reference location avoids confusing that useful center with a nonexistent expectation of the full law. For asymmetric samples, its median need not equal $\overline{x}$.

The scale of a typical deletion step falls like $1/n$ under regular finite-variance sampling, while fluctuations of an ordinary sample mean fall like $1/\sqrt{n}$. This explains why the extra lift can matter greatly for tiny samples and become a smaller correction as readings accumulate. It does not promise a particular finite-sample coverage rate.

Three questions to ask about an interval

An interval can contain 95% of a constructed law without containing the real population mean in 95% of repeated experiments. To assess it, separate three questions:

The jackknife lift answers the first question directly. Its heights are rebuilt from the current sample, so the construction does not by itself supply a posterior from one fixed prior-and-likelihood model. Its central 95% band is a band of the constructed law. Coverage requires a separate investigation.

Other approaches change the construction in different ways. Studentization scales a resampled error by an estimated spread; this can help account for skewness and uncertain scale in regular populations. A model for unseen tails can express possibilities the sample barely touches. The appropriate choice depends on what is known about the generating process and which question matters. Current continuous-mean experiments do not establish a universally best rule.

The world can hide something larger

Return to the delivery service. Suppose we make delays rarer, but much longer. The observations become less likely to encounter them, even as they move the population mean farther away.

We can write the mechanism exactly. Start with a population of mean $\mu$. Give a rare value $R$ probability $\epsilon$, taking that probability from the original population. The new mean and the probability of missing the new component are

The Greek letter $\epsilon$, epsilon, is the rare-event probability. Make it tiny and a finite sample will probably miss the new component. Make $R$ sufficiently large and the mean can move as far as we choose. This is the obstruction behind the Bahadur-Savage nonexistence result: without restrictions on tails or other external information, finite data cannot give useful uniformly reliable bounds for an arbitrary population mean.

The lift leaves the door open, but its assigned tail probability cannot discover how often an unobserved breakdown occurs or how long it lasts. Known limits, a defensible tail model, previous measurements, and targeted sampling can supply information that geometry alone lacks. If all readings are identical, the demo exposes the same issue in miniature: there is no empirical scale to lift, yet the real population might still vary.

For a next experiment, compare two questions about those deliveries: “What is the average waiting time?” and “What fraction take longer than ten minutes?” The second turns each trial into a bounded yes-or-no observation. A hundred-minute delay and a ten-thousand-minute delay contribute the same answer to it, while their effects on the arithmetic mean are very different. Choosing the question changes which part of the unseen world you need to understand.

Further reading

• A “mean” and “variance” for the Cauchy that has neither explores location, scale, and the circle geometry behind Cauchy curves.
• When a probability table becomes a room follows positive weights into a Hilbert-space representation.
• Rubin, D. B. (1981), The Bayesian Bootstrap, develops the random-weight construction.
• Bahadur, R. R. and Savage, L. J. (1956), The Nonexistence of Certain Statistical Procedures in Nonparametric Problems, studies the limits of inference over very broad population classes.

The interactive worlds here are synthetic teaching examples. General context for the site’s exploratory connections is collected on the disclaimer page.