Book Isle: Separate bin width, frequency and density in unequal-bin histograms

With unequal histogram bin widths, height need not directly count observations. Read the vertical-axis definition before area rather than treating the widest or tallest bar as the most populated. Locate caption and definition in Book Isle and calculate a small example externally, without automatic chart recognition, data statistics or bin selection.

Fix caption, units and both axis definitions

Record book, edition, chapter, figure and location. Read caption and surrounding explanation to establish horizontal quantity, vertical quantity, sample and units. Adjacent rectangles alone do not identify this density histogram; bar charts and other rectangle diagrams may use different definitions. Create five external fields: interval, width, frequency, height and area. If axis or unit is unclear, mark the definition pending and consult the prose rather than assuming height is frequency or treating reader zoom as an axis scale.

Use unequal widths with equal frequencies

Invent eight values: 0.2,0.4,0.6,0.8,1.2,1.6,2.2,2.8 in exercise units. Use intervals zero inclusive to one exclusive, and one inclusive to three exclusive. Each contains four observations, with widths one and two. Preserve values and boundary convention. Avoiding endpoint values focuses this example on width; it does not make boundaries unimportant. Apply the actual textbook’s endpoint rules to its data instead of imposing these. Total frequency is eight, with no real-population or probability-estimation claim.

Divide frequency by width for frequency density

For an axis explicitly defined as frequency density, heights are four divided by one equals four, and four divided by two equals two. The first bar is twice as high, but counts are equal; it does not contain twice as many observations. Height is observations per exercise unit, while frequency is observations. Enter four and two as heights and retain four/four counts separately, rather than inferring from pixels. Relative-frequency density has a normalization definition that must be recorded separately, not assumed numerically identical from a similar name.

Multiply height by width to recover frequency

Areas are one times four equals four and two times two equals four. Under this definition, area rather than height represents frequency. Illustrative drawing or heavy strokes do not alter supplied values; label a non-scale sketch illustrative instead of claiming measured area. Equal-looking areas in another diagram do not replace its axis definition, since area may have no statistical meaning there. Recover source frequencies from width times height before comparing counts. Here the bins tie; do not misuse height to force a single winner.

Check total area after normalization

For relative-frequency density, each relative frequency is four eighths, one half. Dividing by widths gives heights one half and one quarter. Each area is one half, totaling one; the height sum three quarters need not equal one. Name this version separately from frequency density. Total area eight versus one reflects different units, not contradiction. This represents relative frequencies of a finite supplied sample, without claiming a true continuous-population density or adding unsupported inference.

Reassign observations when a bin boundary changes

In an external copy, move the middle boundary from one to two, retaining endpoint rules. Zero-to-two contains six observations and two-to-three two. Widths two/one give frequency-density heights three/two and areas six/two, still eight. Do not widen an old bar while retaining four observations. Changed boundaries reassign data rather than add two observations. Preserve both versions and source values. Without original values, mark regrouping not computable instead of inventing it.

Check axis definition, boundaries and display when results disagree

For height/count disagreement, check density definition, widths and endpoint membership before disputing data. Recover the full page for a cropped axis and retain unread status if zoom fails; do not assume an app calculation defect. Logarithmic or otherwise transformed axes require their own definitions rather than this direct multiplication. Use paper or an existing external tool, explicitly not Book Isle automation. Preserve location, definition and arithmetic, leaving unresolved questions pending rather than replacing the source with a tidier drawing.

Reconstruct the explanation from five fields after returning

Leave the reader and return to the figure and axis definition. Independently explain why width-one/height-four and width-two/height-two both represent four observations, then check half-area relative-frequency bars. Recheck six/two after regrouping without mixing versions. Preserve units and sources for all five fields, unknowns separately and the book independently. Completion traces bars to intervals and values and identifies how to compare counts, rather than memorizing tallest or widest always wins or claiming statistical-significance analysis.

Related guides and sources

Read error bars in Book Isle: identify SD, SE and confidence intervals first; Read logarithmic plots in Book Isle: interpret equal distances as ratios, not differences.

Checked on November 9, 2026: Apple public version information lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Feature scope follows the official App Store description and the installed version. This article provides reading and checking methods you can perform; it does not claim step-by-step physical-device or private-server testing. Manual records, external tools and actually visible controls are identified separately, and personal analysis or code observations are not presented as verified automatic capabilities.

For the concept or external-tool scope, see Stata: Histograms with varying bin widths. Samples, checking sequences and observation records here are original reading exercises. External references establish no new automatic recognition, editing or inference in the app and do not replace checking the actual file or passage.