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Population Parameter

In statistics, a Population is a  collection of individuals, groups, items or objects, finite or infinite which can be analyzed using statistics and we can find their of mean, median and variance and other descriptive summaries.  The statistical inference can be made about the entire population or a sample of it.  If the sample is selected in such a way that it take the characteristic of the population then we can draw inference on the whole population. Parameters are used to analyze the about the population or the sample.

 

What is Population Parameter?

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Population parameters are statistical measures for the entire population and is usually derived using mathematical models as the populations are large and it is tedious to collect the entire data. Scientists use these models to interpret the population parameters.

The following symbols are used to represent population parameters.

Mean = $\mu$

Standard Deviation = $\sigma$

Variance = $\sigma^2$

As the whole population is difficult to study, samples are used to estimate, these samples provide information about the population. The representative sample can be used to know about the population and that is where sample statistics come into being.

Sample Statistic

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 A sample is a subset of the population. The measure taken from the sample is called the sample statistics. The sample statistic provides us with the characteristic of the sample like mean, variance and standard deviations. These can be used with proportion to estimate the population characteristics.

Population Parameters vs Sample Statisitcs

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• The population parameter is a fixed measure that describes population while statistic is a characteristic of a sample of the population.

• The statistic is a variable whose value can be calculated by formulas whereas the Population parameter values are already fixed but unknown numerical values.

• The population parameter mean is denoted by $\mu$, population proportion is denoted by $P$, the standard deviation is given by $\sigma$ and variance by $\sigma^2$.

• The sample mean is given by $\bar{x}$ (xbar) the sample proportion is $\bar{p}$ (pbar) and the sample standard deviation is given by the symbol $s$ and variance by $s^2$.

• When we estimate statistic from Population we are finding the probability. We estimate population from statistics we make an inference about the population.

An example to understand the differences in population and sample: Say we want to find out the how many people play at least two sport in colleges across the US.  The data we collect would be huge and the actual information would relate to the population parameters. To simplify this process we can select a small sample of colleges and analyze the data using sample statistics and then infer what the population parameters would be.

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