Variance

4.2 Distribution of Sample Proportion

Sampling Distribution of a Small Proportion

Recall that

$\text { Sample proportion, }\ \hat{p}=\frac{\text { number in population with attribute }}{\text { population size }}$

and can be treated as a random variable.

In a small enough population, we are able to list out all the possible samples, the probability of getting each sample, and therefore the sample proportion.

Therefore, we are able to construct the probability distribution of $\hat{p}$ .

The distribution of a statistic which is calculated from a sample (such as the sample proportion) is called a sampling distribution.

Expected Value (Mean)

The expected value for a continuous random variable X is given by

$E(X)=\int_{-\infty}^{+\infty} x \cdot f(x) d x=\int_{c}^{d} x \cdot f(x) d x$

where $[c,\ d]$ is the range of $x$ where, outside of this range, we have $f(x)=0$ .

As usual $E(X)$ can also be denoted as the Greek letter $\mu$ (mu).
Expanding the formula, we have

Variance

The actual outcome will often differ from the mean (or $E(X)$ ). Sometimes the difference is large, sometimes it is not (or even sometimes no difference).

The variance of a random variable captures the spread of the probability distribution about its mean value. It is defined as

$\operatorname{Var}(X)=E\left[(X-\mu)^{2}\right]$

and $\operatorname{Var}(X)$ represents the variance of $X$ .

$\operatorname{Var}(X)$ can also be denoted as the Greek letter $\sigma^{2}$ (called sigma square).

It is also considered as the long-run average value of the square of the distance from $X$ . Also, notice that there is always $\operatorname{Var}(X) \geq 0$ .

Continuous Random Variables

This tutorial covers material encountered in chapter 15 of the VCE Mathematical Methods Textbook, namely: Continuous random variables Probability density functions Mean, variance and standard… Read More »Continuous Random Variables

The Binomial Distribution

This tutorial covers material encountered in chapter 14 of the VCE Mathematical Methods Textbook, namely: Bernoulli distribution Binomial distribution Mean and variance of a binomial… Read More »The Binomial Distribution

Introduction to Probability

This tutorial covers material encountered in chapter 13 of the VCE Mathematical Methods Textbook, namely: Basic set and probability theory Addition and multiplication (Bayes’ Theorem)… Read More »Introduction to Probability

Variance

4.2 Distribution of Sample Proportion

Sampling Distribution of a Small Proportion

3.2 Mean, Variance, Standard Deviation, Interquartile Range

Expected Value (Mean)

2.3 Variance and Standard Deviation

Variance

Continuous Random Variables

The Binomial Distribution

Introduction to Probability