Descriptive Statistics

Mean

The population mean $\mu$$\large \mu$ or sample mean $\overline{x}$$\large \bar{x}$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Note: $\sum$$\large \sum$ represents $\sum _{i=1}^N$$\large \sum\limits^N_{i = 1}$.

Median

The median of a dataset is the middle value of the sorted dataset.

• If $N$$\large N$ is odd, the median is the middle value
• If $N$$\large N$ is even, the median is the mean of the middle two values

Mode

The mode of a dataset is the value that occurs most frequently.

• If there are multiple values that occur most frequently, they are all modes
• If all values occur at equal frequency, there is no mode

Variance & Standard Deviation

The variance and standard deviation of a dataset are measures of the degree of variation of the data from the mean value.

Population Variance

The population variance ${\sigma }^2$$\large \sigma^2$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Sample Variance

The sample variance $s^2$$\large s^2$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Population Standard Deviation

The population standard deviation $\sigma$$\large \sigma$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Sample Standard Deviation

The sample standard deviation $s$$\large s$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Skewness

The skewness of a dataset is a measure of its asymmetry.

Fisher-Pearson Population Skewness

The Fisher-Pearson population skewness ${\tilde{\mu }}_3$$\large \tilde{\mu}_3$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Fisher-Pearson Sample Skewness

The Fisher-Pearson sample skewness $g_1$$\large g_1$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Adjusted Fisher-Pearson Sample Skewness

The adjusted Fisher-Pearson sample skewness $G_1$$\large G_1$ of a dataset $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ is given by:

Covariance & Correlation

The covariance and correlation coefficient of two datasets are measures of linear relationship between the datasets.

Population Covariance

The population covariance ${\sigma }_{xy}$$\large \sigma_{xy}$ of datasets $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ and $y_1,y_2,...,y_N$$\large y_1, y_2, ..., y_N$ is given by:

Sample Covariance

The sample covariance $s_{xy}$$\large s_{xy}$ of datasets $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ and $y_1,y_2,...,y_N$$\large y_1, y_2, ..., y_N$ is given by:

Pearson Correlation Coefficient

The Pearson correlation coefficient $r_{xy}$$\large r_{xy}$ of datasets $x_1,x_2,...,x_N$$\large x_1, x_2, ..., x_N$ and $y_1,y_2,...,y_N$$\large y_1, y_2, ..., y_N$ is given by: