2.5 Descriptive Measures for Population and Sample

Definition: The collection of all individuals under consideration in a study.

Population size [latex]N[/latex]= the total number of individuals in the population.

Population mean [latex]\mu[/latex]: Suppose the measurement of each individual is [latex]x_1, x_2, \cdots, x_N[/latex], the population mean is defined as

[latex]\mu=\frac{x_1+x_2+\cdots+x_N}{N}=\frac{\sum_{i=1}^N x_i}{N}=\frac{\sum x_i}{N}.[/latex]

Population standard deviation, [latex]\sigma[/latex], is the square root of the population variance [latex]\sigma^2[/latex]. It is defined as

[latex]\sigma=\sqrt{\frac{\sum_{i=1}^N (x_i-\mu)^2}{N}}=\sqrt{\frac{\sum (x_i-\mu)^2}{N}}.[/latex]

The following formula is helpful in calculating the population standard deviation,

[latex]\sigma=\sqrt{\frac{\sum_{i=1}^N x^2_i}{N}-\mu^2}=\sqrt{\frac{\sum x^2_i}{N}-\mu^2}[/latex]

A descriptive measure for a population, such as [latex]\mu[/latex] and [latex]\sigma[/latex], is called a parameter.

Sample

Definition: Part of or a subset of the population from which information is obtained.

Sample size [latex]n[/latex]= the total number of individuals in the sample.

Sample mean [latex]\bar x[/latex]: Suppose the measurements of the sample are [latex]x_1, x_2, \cdots, x_n[/latex], the sample mean is defined as

[latex]\bar x=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum_{i=1}^n x_i}{n}=\frac{\sum x_i}{n}.[/latex]

Sample standard deviation, [latex]s[/latex], is the square root of the sample variance [latex]s^2[/latex]. It is defined as

[latex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar x)^2}{n-1}}=\sqrt{\frac{\sum (x_i-\bar x)^2}{n-1}}.[/latex]

The following formula is helpful in calculating the sample standard deviation,

[latex]s=\sqrt{\frac{\sum_{i=1}^n x^2_i-\frac{(\sum x_i)^2}{n}}{n-1}}=\sqrt{\frac{\sum x^2_i-\frac{(\sum x_i)^2}{n}}{n-1}}[/latex]

A descriptive measure for a sample, such as [latex]\bar x[/latex] and [latex]s[/latex], is called a statistic.