True Mean (Population Mean, μ)

The true mean, written \(\mu\), is the average of an entire population rather than of a sample. It is a parameter: fixed, and in almost every real setting unobserved. The number that appears in a report is the sample mean \(\bar{x}\), a statistic that varies from sample to sample and estimates \(\mu\).

A population of {2, 4, 6, 8, 10} has \(\mu = 30/5 = 6\). A sample of {4, 10} has \(\bar{x} = 14/2 = 7\). The sample mean missed, which is not a failure of the estimator; drawing a different pair would have missed in a different direction, and the average of all possible pairs is exactly 6.

What the parameter is attached to

There is no true mean until a population is specified, and specifying one is a modeling decision rather than a formality. Two different objects both get called \(\mu\).

Most business questions are the second kind, and that has a consequence people skip: \(\mu\) is defined relative to a set of conditions, so it is not a constant of nature. The true mean order value of a site is the mean of the process as it currently runs, for the customers it currently reaches. Change the traffic mix, the season or the catalog, and the estimate is no longer estimating the same number. Chasing a supposedly stable \(\mu\) across a year of product changes is chasing a moving target.

In the first kind the population is finite, and if a sample is a substantial fraction of it the standard error is smaller than the usual formula says — the finite population correction applies. That case is rare in web analytics and common in surveys of a known list.

What the sample mean does for it

The sample mean is unbiased for \(\mu\) — centered on it across repeated samples — provided the sample was drawn at random from the population in question. It is also consistent for independent draws under the law of large numbers, which needs the mean to exist. Dependence does not come free with that: a dependent sequence needs its own condition — ergodicity, or decaying correlation — and convergence is slower, or the normal approximation behind a \(t\) interval poorer, for heavy-tailed data. The exact \(t\) interval assumes independent normal draws; outside that it is a large-sample approximation whose quality depends on the shape of the distribution.

It is often called the best estimator of \(\mu\), and that claim needs its conditions. Among unbiased estimators, and under assumptions such as normally distributed observations, the sample mean has the smallest variance. Drop those assumptions and better estimators exist: on heavily skewed or contaminated data a trimmed mean is more stable, at the cost of estimating a slightly different quantity. What survives in general is the weaker and more useful statement — the sample mean is unbiased and consistent, which is why it is the default and not why it is always the best available.

Unbiasedness is a property of the sampling scheme as much as of the formula. If units enter the sample in a way related to their values — a survey answered by the enthusiastic, a log missing the sessions that crashed — then \(E[\bar{x}] \neq \mu\), and no sample size fixes it.

What an interval says about it

A 95 percent confidence interval is routinely read as a 95 percent chance that \(\mu\) lies inside it. Under the frequentist definition that reading is not available, because \(\mu\) is not random: it either is in this particular interval or it is not. What is random is the interval, and the 95 percent describes the procedure — intervals constructed this way from repeated samples contain \(\mu\) about 95 percent of the time.

The distinction is the same one the American Statistical Association makes about p-values, which “do not measure the probability that the studied hypothesis is true”. A statement of probability about the parameter itself requires a prior and gives a credible interval instead — see posterior probability of uplift for that machinery applied to a difference.

There is also a practical caveat worth carrying over from sample mean: on right-skewed data the usual interval does not deliver its stated coverage even at large samples, reaching only about 94 percent at a thousand observations in the calculation there. So the honest reading of a reported interval is weaker on two counts at once.

Why the distinction earns its keep

Every hypothesis test is a statement about a parameter, not about the numbers observed. A null hypothesis of \(\mu = 100\) is a claim about the population; the sample mean of 103 is evidence bearing on it, and the test asks whether a gap that size is surprising under the claim — see hypothesis testing, with z-test when the standard deviation is known and t-test when it is estimated.

The same distinction is what stops a dashboard from generating stories. A weekly average that moves from 98 to 103 to 99 may be a constant \(\mu\) being sampled three times, and the discipline of asking whether the movement exceeds the sampling variation is the difference between measurement and pattern-matching. The proportion version of the same argument is under true conversion rate.

References

• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.2 Are the data consistent with the assumed process mean?
• American Statistical Association, “ASA Statement on Statistical Significance and P-Values”, 2016