A histogram groups numbers but cannot preserve every value

Start with the list

NIST’s histogram guide describes a common construction: divide a data range into equal-sized intervals, then count the observations in each interval. Its horizontal axis represents the measured variable; its vertical axis can show the counts.

The following exercise supplies its own fictional numbers and grouping choices.

Imagine twenty paper tokens, each bearing one invented number: 2, 4, 5, 7, 8, 9, 10, 11, 12, 14, 14, 15, 16, 18, 19, 20, 22, 24, 27 and 29. These are not grades, waiting times, student records or observations of any locality.

We give them no physical unit. They are simply a fixed list for testing what a grouping retains and what it conceals.

Keep both copies of 14: two tokens have that value, so deleting one would change the exercise.

Give a boundary one home

Make three boxes labeled 0 up to but not including 10, 10 up to but not including 20, and 20 up to but not including 30. In compact notation these are [0, 10), [10, 20) and [20, 30).

The square bracket includes its endpoint; the round bracket excludes it. We are defining that rule for this worksheet, not assuming every published chart uses the same convention.

Write the words beside the symbols if the notation is unfamiliar.

The first box receives 2, 4, 5, 7, 8 and 9: six tokens. The second receives 10, 11, 12, 14, 14, 15, 16, 18 and 19: nine tokens.

The third receives 20, 22, 24, 27 and 29: five tokens. Six plus nine plus five is twenty, matching the original list.

The token marked 10 belongs only in the second box; 20 belongs only in the third. Including a boundary in both neighboring boxes would count a single observation twice.

Draw and then narrow the boxes

Draw adjacent bars over those three equal-width intervals, with heights 6, 9 and 5. Label the vertical scale as token count.

A height of nine means nine tokens fell within that interval; it does not mean a token had value nine. The highest bar covers numbers from 10 through values below 20.

Its left edge is not an average, and its center is not a recorded token. The list remains the authority for the exact twenty values.

Now split every interval in half. The six counts for [0, 5), [5, 10), [10, 15), [15, 20), [20, 25) and [25, 30) become 2, 4, 5, 4, 3 and 2.

Check the pairs: 2 plus 4 restores six; 5 plus 4 restores nine; 3 plus 2 restores five. This drawing shows more of our list’s arrangement because we supplied narrower boxes.

It does not add a twenty-first observation or make the original tokens more accurate. Two drawings can differ while describing exactly the same invented list.

Move the edges without moving a token

Try a third grouping, this time with width eight: [0, 8), [8, 16), [16, 24) and [24, 32). The counts are 4, 8, 5 and 3.

Place 8 in the second box, 16 in the third and 24 in the fourth. Again the total is twenty.

The tallest bar now has height eight rather than nine. Nobody removed a token; a different set of boundaries collected different neighbors together.

Label this drawing separately instead of silently swapping it for the first.

To compare these worksheets fairly, put their interval labels and count scales beside them. Our ten-wide chart has three bars, while the eight-wide chart has four.

Merely counting the bars or comparing their tallest heights cannot establish that one list contains more observations. Summing each chart’s counts answers that question here.

We have deliberately kept the widths equal within each individual drawing. A chart with unequal widths raises another design question and is outside this exercise; do not reuse these height comparisons as a rule for that different case.

Ask a question the broad chart cannot settle

Return to the first three-box chart and ask how many tokens have values below 15. The original list answers eleven: the six below 10 and five more bearing 10, 11, 12, 14 and 14.

But the three bar heights alone do not reveal that split inside the middle interval. All nine middle tokens are somewhere from 10 up to 20.

Their exact positions were not written into that chart. The narrower five-wide drawing can answer this particular question because 15 is one of its stated boundaries.

Prove the loss of detail by making two alternative fictional lists. Keep the first six and final five tokens unchanged.

In one alternative, replace all nine middle tokens with 10. In the other, replace all nine with 19.

Both alternatives still produce broad counts 6, 9 and 5. Yet their counts below 15 are fifteen and six respectively.

Our original eleven is between those answers. Matching bars therefore do not prove matching raw values: we have constructed three different lists with the same broad drawing.

Keep the drawing and its evidence together

Finish with a small reproducible packet: the twenty-number list, the three sets of interval boundaries, their inclusion rule and the resulting counts. Ask another reader to rebuild one drawing without seeing yours.

If a token lands differently, compare the written boundary rule before changing a number. If the totals disagree, look for a missing token, a duplicated boundary token or an erased repeated value.

Those checks test this worksheet; they do not certify a real survey’s sampling or measurement method.

The lesson is not to reject grouped charts. Our first drawing made the broad concentration easy to see, and the second answered a more specific threshold question.

The useful choice depends on the question we wrote down. Keep an exact list when later readers may need exact values, and describe a grouped drawing as grouped.

A new shape after changing the boxes is evidence of a changed presentation in this exercise, not evidence that the fictional observations changed or that a real community’s circumstances improved.