Median and mode

Unlike the mean, the median is robust to outliers. Consider income in the US; the top 1% is much higher than the rest of the population, so this will skew the mean to be higher and distort the perception of the average person's income. However, the median will be more representative of the average income because it is the 50 percentile of our data; this means that 50% of the values are greater than the median and 50% are less than the median.

The median is calculated by taking the middle value from an ordered list of values; in cases where we have an even number of values, we take the mean of the middle two values. If we take the numbers 0, 1, 1, 2, and 9 again, our median is 1. Notice that the mean and median for this dataset are different; however, depending on the distribution of the data, they may be the same.

The mode is the most common value in the data (if we, once again, have the numbers 0, 1, 1, 2, and 9, then 1 is the mode). In practice, we will often hear things such as the distribution is bimodal or multimodal (as opposed to unimodal) in cases where the distribution has two or more most popular values. This doesn't necessarily mean that each of them occurred the same amount of times, but rather, they are more common than the other values by a significant amount. As shown in the following plots, a unimodal distribution has only one mode (at 0), a bimodal distribution has two (at -2 and 3), and a multimodal distribution has many (at -2, 0.4, and 3):

Understanding the concept of the mode comes in handy when describing continuous distributions; however, most of the time when we're describing our continuous data, we will use either the mean or the median as our measure of central tendency. When working with categorical data, on the other hand, we will typically use the mode.

Knowing where the center of the distribution is only gets us partially to being able to summarize the distribution of our data—we need to know how values fall around the center and how far apart they are. Measures of spread tell us how the data is dispersed; this will indicate how thin (low dispersion) or wide (very spread out) our distribution is. As with measures of central tendency, we have several ways to describe the spread of a distribution, and which one we choose will depend on the situation and the data.

We will discuss about range and variance in the  next post