Why this chapter matters
Descriptive statistics turn raw values into an interpretable picture and guide which models and tests are reasonable next.
What you will learn
- Choose suitable visual summaries for categorical and quantitative variables.
- Compute and interpret mean, median, quartiles, range, and standard deviation.
- Describe distribution shape, skew, and potential outliers in context.
Understand the core ideas
Descriptive statistics convert a raw list of values into a structured picture. You summarize center to describe what is typical, spread to describe variation, and shape to describe how values are distributed around center. No single summary works best in all situations. The mean uses every value and is useful for roughly symmetric data, but it can move noticeably when a few extreme values appear. The median is resistant to outliers and often gives a better sense of a typical case in skewed distributions. Interquartile range complements the median by describing the middle fifty percent.
Visual displays are part of the analysis, not decoration. Histograms reveal skew and gaps, boxplots highlight potential outliers, and bar charts compare categories clearly. Good practice is to pair numbers with plots and check whether both tell the same story. If they conflict, review data quality and context before proceeding to inference. Strong descriptive work helps prevent model misuse later because you can see when assumptions like symmetry or stable spread are unlikely to hold.
Key terms
- Mean
- Arithmetic average found by summing values and dividing by the count.
- Median
- Middle ordered value that splits observations into two equal halves.
- Interquartile range
- Difference between third and first quartiles, measuring middle spread.
- Standard deviation
- A measure of typical distance of values from the mean.
Outlier effect on center
Study hours are 4, 6, 7, 8, and 25.
- Compute mean: (4 + 6 + 7 + 8 + 25) / 5 = 10.
- Order values and identify median as 7.
- Compare both centers and note that one high value pulls the mean upward.
- Choose median as more representative for typical hours in this skewed set.
A common misconception
Claim: If two groups share the same mean, their distributions are basically the same.
Correction: Groups can have the same mean but different spread, skew, and outlier patterns, which can change interpretation and decisions.
Lessons in this chapter
- Tables and graphs for distributionsBuild frequency tables, bar charts, histograms, and boxplots correctly. Read the full guide →
- Measures of centerCompare mean and median under symmetric and skewed data patterns.
- Measures of spreadUse IQR and standard deviation to quantify variability.
- Shape and outlier interpretationWrite clear distribution summaries that include center, spread, shape, and unusual points.
Study task
Chapter checkpoint
When might median be preferred over mean as the center measure?
When data are strongly skewed or contain outliers, because median is less affected by extreme values.