--- title: "Estimating the Mean from Grouped Data" description: "Estimate a mean from grouped data using class midpoints, say why the answer is only an estimate, and identify the modal class and the class holding the median." canonical: https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw source: https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw.md retrieved: 2026-09-02 --- > **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`. # Estimating the Mean from Grouped Data Estimate a mean from grouped data using class midpoints, say why the answer is only an estimate, and identify the modal class and the class holding the median. Subject: Mathematics · Area: Data & Statistics · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw ## Ready when they can - Use the midpoint of each class as the value standing for that class - Explain why the result is an estimate rather than the true mean - Identify the modal class and the class that contains the median ## Lesson: A Mean When You Only Have the Bands Last stop the visits table gave exact answers, because every row held one value. The second question is harder. It asked how long members stay on site, and nobody wrote 34 minutes. They ticked a band. In 20 ≤ t < 40 the ≤ end takes 20 itself and the < end stops just short of 40, so a visit of exactly 40 minutes goes in the next band. The five bands came back with 6, 21, 15, 12 and 6 members, sixty in all, and not one exact time among them. *(drawing: Sixty members sorted into five bands. Each bar says how many, and none of them says how long any one visit was.)* You cannot add up times you do not have, so pick one value to stand for each band. The fairest single choice is the middle of the band, called the midpoint: lower boundary plus upper boundary, divided by 2. For 20 ≤ t < 40 that is (20 + 40) / 2 = 30. Do the same down the column and the five midpoints are 10, 30, 50, 70 and 90. Each one is now the value the whole band gets treated as having, exactly the way a value stood for its row last stop. **Example.** From here the machine is the one you already have, with the midpoint m in place of the value x. Multiply each midpoint by its frequency: 6 × 10 = 60, 21 × 30 = 630, 15 × 50 = 750, 12 × 70 = 840 and 6 × 90 = 540. Add them: 60 + 630 + 750 + 840 + 540 = 2820 minutes. Divide by how many members: 2820 / 60 = 47. So members stay about 47 minutes on average. *(drawing: The same three moves as last stop. Only the first one is new, because the value has to be invented before it can be used.)* That 47 is an estimate, and no amount of care with the arithmetic makes it exact. The 21 members in the 20 ≤ t < 40 band are all being treated as 30-minute visitors. If most of them actually left after 22 minutes, the real mean is lower than 47. If most stayed until 38, it is higher. The estimate is only as good as the assumption that values sit evenly inside each band. Narrower bands would tighten the estimate, and only the actual times would make it exact. Two more summaries, and both name a band rather than a number. The modal class is the band with the largest frequency, which is 20 ≤ t < 40 with 21 members. It is not 21, and there is no single mode, because no time was ever written down. For the median, run a total down the frequencies: 6, 27, 42, 54, 60. Position (60 + 1) / 2 = 30.5 sits past 27 and inside the block ending at 42. So the median lies in 40 ≤ t < 60. You get the band, not the minute. **Recap.** When data arrives in bands you invent a value for each one: the midpoint, halfway between the boundaries. Multiply each midpoint by its frequency, add those, and divide by the total frequency for an estimated mean. It is an estimate because every member of a band is treated as sitting at its middle. The modal class is the band with the largest frequency, and a running total says which band holds the median. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Averages from Frequency Tables](https://lightmysky.com/learn/mathematics/averages-from-frequency-tables-mt_5WF_Rhdb1X) ## Opens up - [Quartiles and the Interquartile Range](https://lightmysky.com/learn/mathematics/quartiles-and-the-interquartile-range-mt_vN10d1I9dA)