AcademyProbability
Academy
Sample Mean
Level 1 - Math II (Physics) topic page in Probability.
Principle
The sample mean is a random variable before data are observed and a number after data are recorded. Averaging independent measurements preserves the population mean and reduces variance by a factor of \(n\).
Notation
\(X_1,X_2,\ldots,X_n\)
independent identically distributed measurements
\(\overline{X}\)
sample mean random variable before observation
\(\overline{x}\)
observed sample mean after data are recorded
\(n\)
sample size
\(\mu\)
population mean E[X_j]
\(\sigma^2\)
population variance Var(X_j)
Method
Step 1: Define the sample mean
Add the measurements and divide by the number of measurements.
Sample mean random variable
\[\overline{X}=\frac{1}{n}\sum_{j=1}^{n}X_j\]
Observed sample mean
\[\overline{x}=\frac{1}{n}\sum_{j=1}^{n}x_j\]
Step 2: Derive the mean
Use linearity of expectation. Independence is not needed for this mean calculation.
Start
\[E[\overline{X}]=E\left[\frac{1}{n}\sum_{j=1}^{n}X_j\right]\]
Pull out constant
\[E[\overline{X}]=\frac{1}{n}E\left[\sum_{j=1}^{n}X_j\right]\]
Linearity
\[E[\overline{X}]=\frac{1}{n}\sum_{j=1}^{n}E[X_j]\]
Identical means
\[E[\overline{X}]=\frac{1}{n}\sum_{j=1}^{n}\mu\]
Simplify
\[E[\overline{X}]=\frac{n\mu}{n}=\mu\]
Step 3: Derive the variance
Use independence so the variance of the sum is the sum of variances.
Start
\[\operatorname{Var}(\overline{X})=\operatorname{Var}\left(\frac{1}{n}\sum_{j=1}^{n}X_j\right)\]
Scale variance
\[\operatorname{Var}(\overline{X})=\frac{1}{n^2}\operatorname{Var}\left(\sum_{j=1}^{n}X_j\right)\]
Independence
\[\operatorname{Var}(\overline{X})=\frac{1}{n^2}\sum_{j=1}^{n}\operatorname{Var}(X_j)\]
Identical variances
\[\operatorname{Var}(\overline{X})=\frac{1}{n^2}\sum_{j=1}^{n}\sigma^2\]
Simplify
\[\operatorname{Var}(\overline{X})=\frac{n\sigma^2}{n^2}=\frac{\sigma^2}{n}\]
Rules
Sample mean
\[\overline{X}=\frac{1}{n}\sum_{j=1}^{n}X_j\]
Observed mean
\[\overline{x}=\frac{1}{n}\sum_{j=1}^{n}x_j\]
Expected sample mean
\[E[\overline{X}]=\mu\]
Variance of sample mean
\[\operatorname{Var}(\overline{X})=\frac{\sigma^2}{n}\]
Standard deviation of sample mean
\[\operatorname{SD}(\overline{X})=\frac{\sigma}{\sqrt n}\]
Examples
Question
Eight independent dice-pair sums are recorded. If one dice-pair sum has mean
\[7\]
what is the expected sample mean?Answer
The measurements are identically distributed with
\[\mu=7\]
Therefore \[E[\overline{X}]=\mu=7\]
Checks
- \(\overline{X}\) is random before data are observed; \(\overline{x}\) is the observed number.
- The sample mean has expected value \(\mu\).
- Independence is needed to add variances in the variance derivation.
- The variance of the sample mean is \(\sigma^2/n\).
- The standard deviation of the sample mean is \(\sigma/\sqrt n\), not \(\sigma/n\).