Display the same numerical data in three different ways

Learning goal: Construct and interpret dot plots, histograms and box plots for numerical data, explaining what each display preserves or summarizes. Use the same complete data set so differences between the pictures reflect representation choices, not changed observations. Choosing and drawing a display is step 3, analyze data, of the four-step GAISE investigation model from the companion lesson.

The twelve invented durations in seconds are 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 11. The attribute is duration, measured in seconds in this worked example. A display title and axis label should communicate that context, and the observation count should remain twelve in every representation. The values are invented. They model attempts timed from the participant’s first touch of the cards to “finished” and rounded to whole seconds, the procedure planned in the investigation lesson.

In a dot plot, each dot represents one observation at its numerical value. Place one dot at two, two at three, three at four, two at five, two at six, one at seven and one at eleven. Stack dots vertically when values repeat. Count 1 + 2 + 3 + 2 + 2 + 1 + 1 = 12 to verify that no value has been lost or duplicated.

The dot plot preserves individual recorded values and shows the peak at four. It also makes the gap at eight through ten and the isolated eleven clear. The vertical stack height counts observations, while horizontal position gives duration. A dot higher in a stack is not a longer duration than another dot at the same x-coordinate.

A histogram groups values into numerical intervals. Use equal-width bins 0 ≤ t < 3, 3 ≤ t < 6, 6 ≤ t < 9 and 9 ≤ t < 12. The left endpoint is included and the right endpoint excluded in each bin. Thus three belongs in the second bin, not the first, and six belongs in the third. Explicit boundary rules ensure each observation is counted once.

The bin counts are one, seven, three and one. The first contains only two. The second contains 3, 3, 4, 4, 4, 5, 5. The third contains 6, 6, 7, and the fourth contains eleven. Draw adjacent bars with equal bin widths and heights matching those frequencies. Their counts sum to twelve.

A histogram shows grouped shape while hiding exact locations within each bin. The seven observations in the second bin do not all equal four or five; the raw data or dot plot is needed to recover them. Different reasonable bin choices can emphasize different features, so state the intervals and do not infer more precision than the display supplies.

A box plot summarizes five values: minimum two, Q1 = 3.5, median 4.5, Q3 = 6 and maximum eleven. Here quartiles use the median-of-halves method; with an odd count, the overall median is left out of both halves. Draw a box from 3.5 to six, a median line at 4.5 and whiskers reaching two and eleven. This lesson uses minimum-to-maximum whiskers and does not separately mark outliers with another whisker rule.

The length of the box along the number line represents IQR = 6 − 3.5 = 2.5 seconds, the spread of the middle half. The entire whisker span represents range = 11 − 2 = 9 seconds. The median line is not required to lie in the center of the drawn box. Its position follows the data, not a design preference for symmetry.

A box plot does not reveal the exact frequency at each value or the number of observations unless that count is stated separately. It also does not show every gap. The long upper whisker suggests the higher side stretches farther, but the dot plot gives more detail about the isolated eleven and absent intermediate values.

Choose a display for the question. A dot plot works well for this small set when individual values matter. A histogram can summarize a larger set in intervals. A box plot helps compare centers and middle spreads across groups using a common scale. No display is automatically best for every purpose.

Displays also guide the choice of numerical summaries. The dot plot shows a right tail with an isolated eleven, so the median, 4.5 seconds, and the IQR, 2.5 seconds, describe a typical attempt with less influence from that value than the mean has. The mean, 5 seconds, is the fair share: sixty total seconds shared equally among twelve observations. None of the three displays marks the mean directly, so report it beside the graph when it is needed.

Check consistency across representations. The dot count and histogram frequencies must each total twelve; all displays must include the observed span from two to eleven. The box's quartiles must match calculations from the sorted data. A mismatch signals a plotting, binning or calculation error that should be corrected before interpreting the distribution.

Worked boundary correction: A learner puts both threes in the first bin and both sixes in the second. With the stated intervals, which include the left endpoint and exclude the right one, three belongs in 3 ≤ t < 6 and six in 6 ≤ t < 9. Correct counts are one, seven, three and one. Boundary notation is part of the data representation.

Worked interpretation limit: “The histogram shows seven four-second attempts” is unsupported. It shows seven attempts from three seconds up to but not including six seconds. Only a display retaining individual values can reveal the exact counts at three, four and five.

Guided construction: Build the dot plot from the raw list, tally the histogram bins with explicit boundary rules and calculate the five-number summary for the box plot. Draw all three on consistently labeled numerical scales and verify observation totals.

Independent performance: Represent a new numerical data set using all three displays. State count, units, bin rules, quartile method and whisker convention. Describe center, spread, shape, clusters, gaps and unusual values only where the display supports them. Explain which graph best answers a particular question and identify one detail that another graph hides. Use actual investigation data when available and keep invented examples labeled.