A.1 Basic statistical terms

Note that all the following statistical terms apply only to numerical variables, except the distribution which can exist for both numerical and categorical variables.

A.1.1 Mean

The mean is the most commonly reported measure of center. It is commonly called the average though this term can be a little ambiguous. The mean is the sum of all of the data elements divided by how many elements there are. If we have \(n\) data points, the mean is given by:

\[Mean = \frac{x_1 + x_2 + \cdots + x_n}{n}\]

A.1.2 Median

The median is calculated by first sorting a variable’s data from smallest to largest. After sorting the data, the middle element in the list is the median. If the middle falls between two values, then the median is the mean of those two middle values.

A.1.3 Standard deviation

We will next discuss the standard deviation (\(sd\)) of a variable. The formula can be a little intimidating at first but it is important to remember that it is essentially a measure of how far we expect a given data value will be from its mean:

\[sd = \sqrt{\frac{(x_1 - Mean)^2 + (x_2 - Mean)^2 + \cdots + (x_n - Mean)^2}{n - 1}}\]

A.1.4 Five-number summary

The five-number summary consists of five summary statistics: the minimum, the first quantile AKA 25th percentile, the second quantile AKA median or 50th percentile, the third quantile AKA 75th, and the maximum. The five-number summary of a variable is used when constructing boxplots, as seen in Section 2.7.

The quantiles are calculated as

  • first quantile (\(Q_1\)): the median of the first half of the sorted data
  • third quantile (\(Q_3\)): the median of the second half of the sorted data

The interquartile range (IQR) is defined as \(Q_3 - Q_1\) and is a measure of how spread out the middle 50% of values are. The IQR corresponds to the length of the box in a boxplot.

The median and the IQR are not influenced by the presence of outliers in the ways that the mean and standard deviation are. They are, thus, recommended for skewed datasets. We say in this case that the median and IQR are more robust to outliers.

A.1.5 Distribution

The distribution of a variable shows how frequently different values of a variable occur. Looking at the visualization of a distribution can show where the values are centered, show how the values vary, and give some information about where a typical value might fall. It can also alert you to the presence of outliers.

Recall from Chapter 2 that we can visualize the distribution of a numerical variable using binning in a histogram and that we can visualize the distribution of a categorical variable using a barplot.

A.1.6 Outliers

Outliers correspond to values in the dataset that fall far outside the range of “ordinary” values. In the context of a boxplot, by default they correspond to values below \(Q_1 - (1.5 \cdot IQR)\) or above \(Q_3 + (1.5 \cdot IQR)\).