Choose a display that preserves the question

Try

Seven recorded durations are 0, 4, 5, 5, 9, 10 and 15 minutes. Before drawing a histogram, decide where a reading exactly on a bin boundary belongs. If both neighboring bins include it, you double count; if neither includes it, you lose an observation. A display begins with a definition, not a decorative shape.

Predict

Predict the counts for bins [0,5), [5,10) and [10,15]. Here a square bracket includes an endpoint and a parenthesis excludes it. The last bin includes 15 deliberately. Also predict what information would disappear if you retained only the minimum, quartiles, median and maximum.

Experiment

The three bin counts are 2, 3 and 2. Draw adjoining rectangles with width 5 and those heights, labeling the vertical axis frequency. Now compare an unequal-bin example: 20 observations in [0,2) and 30 in [2,8). To make rectangle areas represent counts, use frequency-density heights 20/2=10 and 30/6=5. Raw heights 20 and 30 would make area misleading.

For a separate original dataset 1,2,4,7,8,9,10,15, compute the median as (7+8)/2=7.5. Using medians of the lower and upper halves gives Q1=3 and Q3=9.5, so IQR=6.5. State that convention beside your drawing.

Understand

Histograms describe a quantitative distribution through intervals. Their horizontal order and bin width have numerical meaning. Bar charts for categories need not join their bars and do not treat category width as a measurement interval. Dotplots retain individual values and are particularly useful for small datasets; stem-and-leaf displays preserve digits while organizing magnitude. Each display trades detail for compactness.

Describe a distribution’s shape, center, spread and unusual features in context. Shape can include skew, symmetry, multiple modes or gaps. The tail direction names skew: a right-skewed distribution has a long right tail. Changing bin boundaries or widths can hide or create apparent bumps, so inspect more than one sensible binning before declaring a real subgroup.

A boxplot compresses the distribution into quantiles. Quartile algorithms differ, particularly for small samples. In this unit use median-of-halves quartiles, excluding the central observation from both halves when the sample size is odd. A software package may interpolate differently; a disagreement is not automatically an arithmetic error.

For a modified boxplot, potential-outlier fences are Q1−1.5IQR and Q3+1.5IQR. Whiskers extend to the most extreme observed values inside the fences, not necessarily to the fences themselves. In the eight-value example the fences are −6.75 and 19.25, so all observations remain inside and the whiskers reach 1 and 15. A flagged value deserves investigation, not automatic deletion.

Identical five-number summaries can hide different interior arrangements and modes. A boxplot supports compact comparisons but cannot reconstruct the complete data. Keep the unit, sample size and measurement rule visible. If a graph truncates or changes scales, explain how that affects visual comparison.

Apply

Draw a dotplot and modified boxplot of the eight-value dataset. Mark the calculated fences separately from actual whisker endpoints. Then replace 15 by 40 and recompute the effect on the display. Explain why a potential outlier could be a valid rare observation, a data-entry error or a meaningful subgroup; the graph alone cannot decide.

Continue

Next connect these visual summaries to numerical measures of center and spread. Keep a graph nearby so a compact number does not erase the distribution it summarizes.