Measures of Central Tendency: Mean, Median, and Mode

$\mu = \frac{\sum_{i=1}^{N} x_i}{N}$

Where $N$ is the population size and $x_i$ represents each individual value.

Sample Mean

Because we often rely on a subset of a population (a sample) to make inferences, we calculate a sample mean, denoted as $\bar{x}$.

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Where $n$ is the observed sample size.

Note: While the mean is often considered our "best guess" or the "expected value" of a data series, it can sometimes be misleading, especially if the data contains extreme values.

The Median

The median is the midpoint value of a data series. When the data is arranged in order (either smallest to largest or largest to smallest), the median is the value that separates the observations into two equal halves.

Finding the Median Position

To find the median, you must first determine its position in the ordered list using the following formula:

$\text{Median position} = \left(\frac{n + 1}{2}\right)$

The method for identifying the actual value depends on whether your sample size ($n$) is odd or even:

• Odd Sample Size: The median is the exact middle value. For example, in the series $[1, 3, 4, 8, 13]$, the median is $4$.
• Even Sample Size: The median is the average of the two middle values. For example, in the series $[1, 3, 4, 8, 13, 18]$, the median is $\frac{4+8}{2} = 6$.

The Mode

The mode is the most frequently observed value in a data series. Unlike the mean and median, which are used to represent an "expected" value, the mode is simply a measure of what is most common within the data. A data set may have one mode, multiple modes, or no mode at all if no value repeats.

Summary Comparison

The relationship between these three measures can change depending on the "shape" of your data. For instance, in a right-skewed distribution, the mean is often pulled toward the higher end by extreme values, while the mode remains at the peak of the distribution.

Exam Tip: Always remember to order your data series before attempting to find the median or the mode! If the data is not sorted, your position calculations will be incorrect.