# Sampling in Statistical Inference

The use of randomization in sampling allows for the analysis of results using the methods of statistical inference. Statistical inference is based on the laws of probability, and allows analysts to infer conclusions about a given population based on results observed through random sampling. Two of the key terms in statistical inference are parameter and statistic:

A parameter is a number describing a population, such as a percentage or proportion.

A statistic is a number which may be computed from the data observed in a random sample without requiring the use of any unknown parameters, such as a sample mean.

Example

Suppose an analyst wishes to determine the percentage of defective items which are produced by a factory over the course of a week. Since the factory produces thousands of items per week, the analyst takes a sample 300 items and observes that 15 of these are defective. Based on these results, the analyst computes the statistic , 15/300 = 0.05, as an estimate of the parameter p , or true proportion of defective items in the entire population.

Suppose the analyst takes 200 samples, of size 300 each, from the same group of items, and achieves the following results:

```Number of Samples	Percentage of Defective Items
20			3
30			4
50			5
45			6
35			7
30			8
```
The histogram corresponding to these results is shown below: These results approximate a sampling distribution for the statistic , or the distribution of values taken by the statistic in all possible samples of the size 300 from the population of factory items. The distribution appears to be approximately normal, with mean between 0.05 and 0.06. With repeated sampling, the sampling distribution would more closely approximate a normal distribution, although it would remain discontinuous because of the granularity caused by rounding to percentage points.

### Bias and Variability

When a statistic is systematically skewed away from the true parameter p, it is considered to be a biased estimator of the parameter. In the factory example above, if the true percentage of defective items was known to be 8%, then our sampling distribution would be biased in the direction of estimating too few defective items. An unbiased estimator will have a sampling distribution whose mean is equal to the true value of the parameter.

The variability of a statistic is determined by the spread of its sampling distribution. In general, larger samples will have smaller variability. This is because as the sample size increases, the chance of observing extreme values decreases and the observed values for the statistic will group more closely around the mean of the sampling distribution. Furthermore, if the population size is significantly larger than the sample size, then the size of the population will not affect the variability of the sampling distribution (i.e., a sample of size 100 from a population of size 100,000 will have the same variability as a sample of size 100 from a population of size 1,000,000).