🎲 True Mean (Population Mean)

True Mean (Population Mean)#

The actual mean of the whole population that a sample mean estimates.

Important

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What it is#

The true mean, or population mean \(\mu\), is the actual arithmetic average of an entire population. It is a fixed value but usually unknown — we can rarely measure everyone — so we estimate it with the sample mean \(\bar{x}\).

Parameter vs estimate#

\[\mu = \frac{1}{N}\sum_{i=1}^{N} x_i, \qquad \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.\]

Here \(\mu\) is a parameter (fixed, unknown) and \(\bar{x}\) a statistic (random, changing from sample to sample). By the Law of Large Numbers, as \(n\) grows, \(\bar{x} \to \mu\).

Example#

For the population \(\{2, 4, 6, 8, 10\}\), the true mean is \(\mu = (2+4+6+8+10)/5 = 6\). A sample \(\{4, 10\}\) gives \(\bar{x} = 7\) — an estimate of the true 6, off by sampling luck.

Inference about μ#

Everything in classical inference targets \(\mu\): a hypothesis test checks a claim like \(H_0 : \mu = 100\), and a confidence interval says “we’re 95% confident the true mean \(\mu\) lies between \(X\) and \(Y\).” The sample mean is the best unbiased estimator of \(\mu\), and tests and intervals quantify how far it might be from the truth.

The proportion analogue#

For yes/no outcomes the same parameter-vs-estimate story holds with the true conversion rate \(p\) estimated by \(\hat{p}\) — \(\mu \leftrightarrow p\), \(\bar{x} \leftrightarrow \hat{p}\) — the mean and proportion versions of one idea.

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

True Conversion Rate · Standard Error (SE) · Parameter(s) of Interest · Margin of Error (MoE) · Frequentist · Bootstrap Confidence Intervals (CIs)

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion