Descriptive Statistics

Before you can draw conclusions, make predictions, or test hypotheses, you need to understand your data. Descriptive statistics are the tools that let you take a raw pile of numbers and distill it into a handful of meaningful summaries — a center, a spread, a shape. They are the first step in every statistical analysis, whether you are examining test scores, tracking daily temperatures, or evaluating the performance of a manufacturing process.

The two most important questions you can ask about a data set are: "Where is the middle?" and "How spread out are the values?" The mean, median, and mode each answer the first question in a slightly different way. The mean uses every data point and is sensitive to extremes; the median ignores magnitude and focuses on position; the mode identifies the most common value. Choosing the right measure of center depends on the shape of your data — symmetric data favor the mean, while skewed data or data with outliers favor the median.

Equally important is understanding variability. Two data sets can share the same mean yet look completely different if one is tightly clustered and the other is widely dispersed. The range gives the quickest snapshot — just the gap between the largest and smallest values — but it is easily distorted by a single extreme observation. The variance and standard deviation give a much richer picture by averaging the squared distances from the mean, and the interquartile range (IQR) captures the spread of just the middle 50% of the data, making it resistant to outliers.

In this lesson you will learn to calculate each of these measures by hand, interpret what they tell you about a data set, and understand when one measure is more appropriate than another. These skills are foundational — every topic in the rest of this statistics course, from probability to regression, builds on the ability to summarize and describe data clearly.