--- title: "Histograms and frequency density" description: "Calculate frequency density and class counts from supplied numerical data for unequal-width histogram classes. Estimate partial-class counts using an even-spread assumption." canonical: https://lightmysky.com/learn/mathematics/histograms-and-frequency-density-mt_0QAOgzD-Qw source: https://lightmysky.com/learn/mathematics/histograms-and-frequency-density-mt_0QAOgzD-Qw.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Histograms and frequency density Calculate frequency density and class counts from supplied numerical data for unequal-width histogram classes. Estimate partial-class counts using an even-spread assumption. Subject: Mathematics · Area: Data & Statistics · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/histograms-and-frequency-density-mt_0QAOgzD-Qw ## Ready when they can - Calculate class widths, frequency densities and class frequencies, including missing counts and merged-class densities - Estimate the frequency in part of a class by using area and assuming an even spread - Interpret inclusive and exclusive class boundaries, explain area distortion from frequency heights with unequal widths, and explain why an empty class alone does not prove impossibility ## Lesson: When the bars are different widths You used class boundaries on a cumulative frequency curve. Wren Lane's next 10k has 80 finishers: 8 in 30 ≤ t < 40, 12 in 40 ≤ t < 45, 15 in 45 ≤ t < 50, 25 in 50 ≤ t < 60 and 20 in 60 ≤ t < 80. Here t is time in minutes. Each class includes its lower boundary and excludes its upper boundary. A time of exactly 45 belongs to 45 ≤ t < 50. *(drawing: This chart compares class counts correctly. Its equal bar widths do not show the different time intervals.)* The 60 to 80 class has 20 runners over 20 minutes: an average of 1 per minute. The 40 to 45 class has 12 over 5 minutes: 2.4 per minute. It has fewer runners but a higher average rate. A histogram draws numerical widths, so these bars must be 20 and 5 minutes wide. If height were frequency, area would be frequency x width. Wider classes would receive more area per runner. Density heights make areas equal counts. Class width = upper boundary - lower boundary. For a density histogram, height = frequency / width, the average frequency per unit. Area = density x width = frequency. If density is positive, width = frequency / density, and a wider class at that density has more values. Equal-width frequency heights preserve the shape. Density equals frequency numerically when width is 1; an empty class has both zero at any positive width. To draw the histogram, label the horizontal axis time in minutes and the vertical axis frequency density. Draw each rectangle from its lower to its upper boundary. The five widths are 10, 5, 5, 10 and 20. Their heights are 0.8, 2.4, 3, 2.5 and 1. For 45 to 50, width 5 x height 3 gives frequency 15. The boundary line below shows the numerical spacing; it is not the histogram. **Example.** For part of a class, estimate with density x the requested width. From 50 to 55: 2.5 x 5 = 12.5, or about 13 runners. From 42 to 48, split at 45: 2.4 x 3 + 3 x 3 = 16.2, or about 16. Split at every boundary crossed and add the contributions. A complete class contributes its known frequency. Partial classes require an even-spread assumption. Keep decimals during calculation and round at the end if requested. A categorical count chart uses heights for counts; its widths do not measure category sizes. A density histogram uses numerical widths and areas for counts. Adjacent nonempty classes meet at their shared boundary. An empty class has density zero and may leave a blank interval. Emptiness alone proves neither possibility nor impossibility. Frequency = density x width. Classes count each value once, so areas sum to the total. Find a missing count by subtracting known frequencies from that total. To merge adjacent classes, add their frequencies and widths separately. Divide the combined frequency by the combined width. Widths 4 and 6 with densities 2 and 3 give frequencies 8 and 18. The merged density is (8 + 18) / (4 + 6) = 2.6. Adding densities or taking their unweighted mean does not generally work. The merged bar preserves total frequency but loses the density change at the old boundary. Splitting a class into equal-width pieces under the even-spread assumption keeps the estimated density unchanged. Each piece receives an equal share of the class frequency. The grouped count alone does not tell you the actual count in each piece, or where individual observations lie. **Recap.** For a density histogram, frequency = density x width = bar area. Density = frequency / width. Estimate partial classes using an even spread, splitting at boundaries and adding the contributions. *Lesson and practice text written for this dataset. Histogram conventions, average density, class areas and grouped-data estimates are explained with original examples.* ## Practice 33 questions on this page, each with its working shown. ## Needs first - [Cumulative Frequency Curves](https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST)