Sampling Distributions

Sampling is a technique where we consider a portion (known as a sample) of the distribution and see the distribution of some property (such as mean, variance, st. deviation) of those samples. We compute the statistic of the new distribution from some property of the samples such as mean or variance.

The new probability distribution of this new statistic known as a sampling distribution.

Variability of a Sampling Distribution

This is measured by the variance of it. Do not confuse this with the variance of the samples. This is the variance of the distribution computed using the statistic made with the property of samples. its std. deviation is also used for this.

This variability will depend on;

• $N$ : The number of observations in the population.
• $n$ : The number of observations in the sample.
• The method used to select the samples at random. eg. If the population is geographically based, if we take samples from one province only, the distribution variability changes a lot from another distribution made with taking samples from multiple provinces.

Central Limit Theorem

The Central Limit Theorem (CLT) states that the probability distribution of any statistic (or the sampling distribution for any statistic) will be normal or nearly normal, if the sample size is "large enough". Thus the CLT permits approximate calculations for a variety of distributions.

• sample size of 30 is “large enough” as a rule of thumb.

Sampling Distribution of Mean

Suppose we get samples from population whose mean is $\mu$ and each sample has mean of $\bar{x}_i$ The distribution of $\bar{x}_i$ values is known as sampling distribution of mean.

Mean

Average of All sample means, (i.e. Average of all averages of each sample) noted as $\bar{x}$

$$\bar{x} = \mu$$

Standard Error

Also known as "Standard Deviation",

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

• $\sigma$ is the true population standard deviation
• Standard Error decrease as $n$ increase

Approx. to Normal Distribution

If the below conditions are met:

• The population is normally distributed OR the sample size is large enough.
• The population standard deviation $\sigma$ is known.

Then:

$$\bar{x}\sim\mathcal{N}\left(\mu,\frac{\sigma^2}{n}\right)$$

Usually population’s standard deviation σσ is unknown. In such cases, Student's T Distribution is used.

Sampling Distribution of Proportion

proportion is expressed as:

$$p = \frac{x}{n}$$

• $x$ is No. of successes
• $n$ is total cases

If a population of size $N$ has proportion of success $p$, If we draw all possible samples of size $n$ from that population and get their proportion of success $p_i$, we get the sampling distribution of proportions.

Mean

Average of All sample proportions, noted as $\mu_p$ ,

$$\mu_p = p$$

Standard Error

$$\sigma_p = \sqrt{\frac{p\,(1-p)}{n}}$$

Approx. to Normal Distribution

when,

• the sample size is sufficiently large
• the population probability of success ($p$) is known,

$$p \sim \mathcal{N}\left(p, \frac{p\,(1-p)}{n} \right)$$