Difficulty: Foundational | Prerequisites: None
Descriptive statistics is the starting point for any data analysis. Before fitting models or running hypothesis tests, you need to summarise a dataset's centre (where the data clusters) and spread (how far it stretches). Boxplots are one of the most common visual tools for doing this, and knowing how to read one is expected on every STAT 350 exam. This material underpins everything else in the course, so if any of this is unfamiliar, sort it out before moving forward.
Boxplots display the five-number summary (minimum, Q1, median, Q3, maximum) along with outliers. Choosing the right measure of centre and spread depends on whether the data is symmetric or skewed. Skewed data calls for median and IQR; symmetric data works well with the mean and standard deviation.
Five-number summary
The set of values: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. These five numbers capture the shape of a distribution at a glance.
In simple terms, it splits your data into four equal-sized chunks and tells you where each boundary falls.
Interquartile range (IQR)
IQR = Q3 - Q1. The range covered by the middle 50% of the data.
Think of it as the width of the box in a boxplot. It measures spread without being dragged around by extreme values.
Lower inner fence
Lower inner fence = Q1 - 1.5 * IQR. Any data point below this value is flagged as a potential outlier.
In simple terms, it is the cutoff below which data points are considered unusually small.
Upper inner fence
Upper inner fence = Q3 + 1.5 * IQR. Any data point above this value is flagged as a potential outlier.
Whiskers
The lines extending from the box in a boxplot. Each whisker reaches to the most extreme data point that is still within the inner fences (not beyond them).
In simple terms, whiskers stretch to the farthest non-outlier values.
Outlier (in the context of boxplots)
A data point that falls beyond the inner fences. Displayed as individual dots or circles outside the whiskers.
Sample mean
The arithmetic average of the data, denoted x-bar. Sensitive to outliers and skewness.
Sample median
The middle value when data is sorted. Resistant to outliers and preferred for skewed distributions.
Sample standard deviation
A measure of spread based on squared deviations from the mean. Like the mean, it is sensitive to extreme values.
Range
Maximum - Minimum. The simplest measure of spread, but heavily influenced by outliers.
The box spans from Q1 to Q3. The line inside the box marks the median.
Whiskers extend from the box to the most extreme observations still within the inner fences.
Points beyond the inner fences are plotted individually as outliers.
The lower whisker ends at the smallest data point that is at or above Q1 - 1.5 * IQR.
The upper whisker ends at the largest data point that is at or below Q3 + 1.5 * IQR.
The lower inner fence itself is a calculated boundary, not a data point. It equals Q1 - 1.5 * IQR.
If there are outliers below the lower whisker, the lower inner fence sits between the lower whisker and those outliers. The whisker ends at the smallest non-outlier; the outliers sit beyond the fence.
If no outliers exist on the lower side, the lower inner fence is at or below the minimum data point, and the whisker extends to that minimum.
Symmetric data (no outliers, no strong skew): Use the sample mean for centre and the sample standard deviation (or IQR) for spread.
Skewed data or data with outliers: Use the sample median for centre and the IQR for spread. The mean and standard deviation are pulled by extreme values and give a misleading picture.
When a boxplot shows outliers or obvious asymmetry (one whisker much longer than the other, or the median line off-centre in the box), that signals skewness, and the median + IQR are the better summary.
Compare their medians for centre and their box widths (IQR) for spread.
A boxplot with outliers on one side suggests skew; a boxplot that looks roughly symmetric with no outliers supports using the mean and standard deviation.
Students often think the whiskers always extend to the minimum and maximum of the dataset. They do not. Whiskers stop at the most extreme non-outlier values.
Students sometimes confuse the inner fence with the whisker endpoint. The fence is a calculated threshold; the whisker ends at an actual data point within that threshold.
Using the mean and standard deviation for skewed data is a frequent error. When you see outliers or asymmetry in a boxplot, switch to median and IQR.
The range is rarely the best measure of spread because it depends entirely on the two most extreme values.
⚠️ Exam questions regularly show a boxplot and ask you to pick the correct centre and spread measures. Look for outliers and asymmetry first.
⚠️ Understanding inner fences is critical for true/false questions. Know the formulas: Q1 - 1.5 IQR and Q3 + 1.5 IQR.
⚠️ Boxplot-reading questions appear in both true/false and multiple choice sections.
True or false: The whiskers of a boxplot always extend to the minimum and maximum values of the dataset.
Fill in the blank: The lower inner fence is calculated as ______.
True or false: For skewed data, the sample mean is the preferred measure of centre.
Fill in the blank: The IQR equals ______.
True or false: If a dataset has outliers on the lower end, the lower inner fence is located between the lower whisker and those outliers.
Q: A boxplot shows three outliers below the lower whisker. Where is the lower inner fence located relative to the whisker and the outliers?
A: The lower inner fence is between the lower whisker and the outliers. The whisker ends at the smallest data point still within the fence, and the outliers sit below the fence.
Q: A boxplot has a long right whisker, a median that sits close to Q1, and two outliers on the upper end. Which measures of centre and spread are most appropriate?
A: The sample median for centre and the IQR for spread, because the boxplot shows right skew and outliers.
Q: What is the difference between the inner fence and the whisker endpoint?
A: The inner fence is a calculated threshold (Q1 - 1.5 IQR or Q3 + 1.5 IQR). The whisker endpoint is an actual data point, the most extreme observation that falls within the fence.
Q: If a boxplot appears symmetric with no outliers, what measures of centre and spread could you use?
A: The sample mean and sample standard deviation are appropriate for symmetric data without outliers.
This connects to probability distributions because many named distributions (normal, exponential, binomial) have their own built-in measures of centre and spread (mean and variance), and knowing when data is symmetric or skewed helps you decide which distribution might fit.
It also connects to the normal distribution specifically: a symmetric, bell-shaped histogram with no outliers is a hallmark of normally distributed data, where mean and standard deviation are the natural summaries.
boxplot, box-and-whisker plot, five-number summary, quartiles, Q1, Q3, interquartile range, IQR, inner fence, lower fence, upper fence, whisker, outlier, sample mean, sample median, sample standard deviation, range, skewness, symmetric distribution, measures of centre, measures of spread, center and spread, STAT 350, Purdue, descriptive statistics