Sample vs. Population

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Population

A population is the entire group of individuals or objects that we are interested in studying. For example, the population could be all the people living in a certain city, all the students in a school, or all the cars on a certain road.

Sample

A sample is a subset of a population. It is a smaller group of individuals or objects selected from a larger population. The sample represents the population, and we use statistics to make inferences or predictions about the population based on the sample.

Statistic (NOT Statistics)

A statistic is a measure that describes a characteristic of a sample. For example, a sample's mean, median, mode, and standard deviation are all statistic. It is used to summarize and describe the sample data.

The sample mean is represented by x-bar, and it is calculated by summing all the values in the sample and dividing by the number of values in the sample. For example, if the values in the sample are 2, 3, 4, 5, and 6, then the sample mean would be:

$$\Large{\overline{x} = \frac{2 + 3 + 4 + 5 + 6}{5} = 4}$$

The sample standard deviation is represented by "s", and it is calculated by taking the square root of the sum of the squared differences between each value in the sample and the sample mean, divided by the number of values in the sample minus one. For example, if the sample mean is 4, and the values in the sample are 2, 3, 4, 5, and 6, then the sample standard deviation would be:

$$\Large{s = \sqrt{\frac{(2 - 4)^2 + (3 - 4)^2 + (4 - 4)^2 + (5 - 4)^2 + (6 - 4)^2}{(5 - 1)}} = 1.58}$$

Parameter

A parameter is a measure that describes a characteristic of a population. For example, the population mean, median, and standard deviation are all parameters. Parameters are used to describe the population.

The population mean is represented by mu, and it is calculated by summing all the values in the population and dividing by the number of values in the population. For example, if the values in the population are 2, 3, 4, 5, and 6, then the population mean would be:

$$\Large{\mu = \frac{(2 + 3 + 4 + 5 + 6)}{5} = 4}$$

The population standard deviation is represented by \(\sigma\), and it is calculated by taking the square root of the sum of the squared differences between each value in the population and the population mean divided by the number of values in the population. For example, if the population mean is 4, and the values in the population are 2, 3, 4, 5, and 6, then the population standard deviation would be:

$$\Large{\sigma = \sqrt{\frac{(2 - 4)^2 + (3 - 4)^2 + (4 - 4)^2 + (5 - 4)^2 + (6 - 4)^2}{5}} = 1.414}$$


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