--- title: "Cumulative Frequency Curves" description: "Build cumulative totals from grouped data and specify their plotting points at class boundaries. Use supplied coordinates or a cumulative frequency graph to estimate quartiles, spread and counts, whil" canonical: https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST source: https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST.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`. # Cumulative Frequency Curves Build cumulative totals from grouped data and specify their plotting points at class boundaries. Use supplied coordinates or a cumulative frequency graph to estimate quartiles, spread and counts, while keeping class boundaries separate from actual extremes. Subject: Mathematics · Area: Data & Statistics · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST ## Ready when they can - Build cumulative totals, specify plotting points at class boundaries and interpret count or relative-frequency heights - Estimate the median, quartiles and interquartile range using each group's total frequency - Estimate counts above thresholds or within stated intervals using cumulative totals ## Lesson: Cumulative totals and grouped-data estimates You compared numerical summaries using medians and quartiles. Now use grouped race times. Wren Lane's 80 finishers have frequencies 6, 14, 26, 22 and 12 in the five classes below. Cumulative frequency is a running total. For these right-closed classes, the cumulative height at a boundary counts times at or below it. A point (time, count) puts time horizontally and count vertically. *(drawing: These are separate class frequencies, not cumulative totals. Times are in minutes; the frequencies add to 80.)* Keep adding the frequencies: 6, 20, 46, 68, 80. The final height is the total number of observations. Recover a class frequency by subtracting neighbouring totals: 46 - 20 = 26 in the class ending at 60. Frequencies are zero or positive, so cumulative totals cannot decrease. An empty class leaves the total unchanged. *(drawing: These cells list coordinates, not a drawn curve. Start with zero at the first lower boundary, then use upper class boundaries.)* The final height N gives the group size. For the median estimate, find N/2 on the frequency axis, read across to the graph, then down to the value axis. For 80 runners, use height 40; the time estimate is about 58 minutes. Do not halve the largest time. Different group sizes need different heights: totals 40 and 200 use heights 20 and 100. Their median estimates can still match. **Example.** For quartiles, use heights N/4 and 3N/4. With 80 runners, these are 20 and 60. Read across to the graph, then down to the time axis: the quartile estimates are 50 and about 66 minutes. IQR = upper quartile - lower quartile, so the estimated IQR is 66 - 50 = 16 minutes. Subtract horizontal values, not frequency heights. Compare different groups using their own quartile heights and then their IQRs. An at-or-below height counts values up to and including x. Subtract it from N to estimate the count above x. At 65 minutes, height 57 gives 80 - 57 = 23 later finishers. Reverse this using height = N - count above. For a < x ≤ b, subtract height at a from height at b. Add disjoint lower and upper groups to count both ends. On paper, label both axes and use equal spacing for equal amounts. Plot each cumulative total at its class's upper boundary, not its midpoint. Include zero at the first lower boundary. Join points without falling, using straight segments or a nondecreasing smooth curve. A table does not supply midpoint counts. For classes a ≤ x < b, the total at b counts values below b; check which endpoints are included. Straight segments model an even spread within each class. To interpolate, find the fraction of the frequency rise needed, then move that fraction of the class width. The race median height 40 lies between (50, 20) and (60, 46). The fraction is (40 - 20)/(46 - 20) = 20/26. The time estimate is 50 + (20/26) × 10 ≈ 57.7 minutes. A smooth curve may give a slightly different estimate. Boundary totals are known from the table; positions within classes are not. Grouped median and quartile readings estimate raw statistics. Different raw lists can fit the same table. The actual minimum lies inside the first nonempty class, and the maximum inside the last. Boundaries need not be actual observations. A box plot with actual extremes needs those values separately; using outer boundaries gives an approximation. A steep rise means many observations in a short value interval. A steady straight rise models equal numbers in equal-width intervals. An exactly horizontal stretch adds no observations; a nearly horizontal stretch adds few. For an at-or-below graph, the count in 20 < x ≤ 30 is height at 30 minus height at 20. Compare curve shapes on matching scales and account for group sizes. Cumulative relative frequency uses proportions instead of counts. Divide a cumulative count by N to get a proportion; multiply a proportion by N to recover a count. For example, 0.6 × 80 = 48. Read quartiles at 0.25 and 0.75 and the median at 0.5, then across to the graph and down to the value axis. A percentage scale uses 25%, 75% and 50%. **Recap.** Plot running totals at upper class boundaries. Use frequency shares to estimate quartiles. Subtract cumulative heights to count intervals. Grouped boundaries need not be actual extremes. *The lesson, practice questions and flashcards were written for this learning dataset.* ## Practice 30 questions on this page, each with its working shown. ## Needs first - [Box-plot features from five-number summaries](https://lightmysky.com/learn/mathematics/box-plot-features-from-five-number-summaries-mt_G_SnfEllzQ) ## Opens up - [Histograms and frequency density](https://lightmysky.com/learn/mathematics/histograms-and-frequency-density-mt_0QAOgzD-Qw) - [Finding Probabilities and Values from a Normal Model](https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos)