Represent data with dot plots, histograms, and box plots, and read the shape of a distribution from them (LA A1: S-ID.A.1).

What this topic is asking

Standard A1: S-ID.A.1 asks you to represent data with dot plots, histograms, and box plots, and to read the shape of a distribution. On LEAP 2025 these are Type I items in the Additional and Supporting Content category. The embedded calculator can build displays on the calculator sessions, but you must read and interpret them.

The three displays

The five-number summary behind a box plot

A box plot is built from five values. To find them, order the data, then locate:

The box spans Q1 to Q3 (the middle 50 percent), and its width is the interquartile range (IQR).

Reading shape

• Symmetric: values balanced around the center (a roughly even dot plot or histogram).
• Skewed right: a longer tail toward high values (mean pulled above the median).
• Skewed left: a longer tail toward low values.

A box plot shows skew by the position of the median in the box and the relative whisker lengths.

How LEAP examines this topic

• Equation response. Compute the five-number summary or the IQR.
• Multiple choice. Match a display to its purpose, or read shape from a graph.
• Drag and drop. Build a box plot from a five-number summary, or match displays to data sets.

A clarifying idea: choose the display by the question. Exact small data favor a dot plot; overall shape of many values favors a histogram; comparing spread and center across groups favors box plots.

Why different displays reveal different things

Each display is a deliberate trade-off between detail and summary, which is why S-ID.A.1 asks you to choose and read all three rather than picking a single favorite. A dot plot keeps every individual value, so you can see exact repeats and gaps, but it becomes unreadable for hundreds of points. A histogram sacrifices the individual values to reveal the shape of the whole distribution, where the data pile up, whether it is symmetric or skewed, but the bin choice can hide or exaggerate features, so the same data can look different under different bin widths. A box plot throws away even more detail, keeping only five summary numbers, but in return it makes comparison easy: several box plots side by side instantly show which group has the higher median or the wider spread. None of these is "the right" display; the right one depends on whether you need exact values, overall shape, or a quick comparison. Understanding what each display preserves and what it discards is what lets you both choose wisely and avoid being misled, for instance, recognizing that a symmetric-looking histogram and a box plot with a centered median are telling the same story about shape, while a skewed data set will show its tail in the histogram and its off-center median in the box plot.

Try this

Q1. For $2, 4, 4, 6, 9$, find the median and the IQR. [2 points]

• Cue. Median $4$; Q1 $= 3$ (avg of $2, 4$), Q3 $= 7.5$ (avg of $6, 9$), IQR $= 4.5$.

Q2. Which display best shows each exact value in a set of 8 numbers? [1 point]

• Cue. A dot plot.