--- title: "Averages from frequency tables" description: "Find the mean, median, mode and range from a frequency table by working with the counts instead of listing every value." canonical: https://lightmysky.com/learn/mathematics/averages-from-frequency-tables-mt_5WF_Rhdb1X source: https://lightmysky.com/learn/mathematics/averages-from-frequency-tables-mt_5WF_Rhdb1X.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`. # Averages from frequency tables Find the mean, median, mode and range from a frequency table by working with the counts instead of listing every value. Subject: Mathematics · Area: Data & Statistics · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/averages-from-frequency-tables-mt_5WF_Rhdb1X ## Ready when they can - Multiply value by frequency and divide by the total to get the mean - Find the median position from the running total of frequencies - Read the mode straight off the largest frequency ## Lesson: Averages without writing the list out Sixteen plot holders, twelve volunteers, thirty-two park members. The 60 are chosen and the forms are back. One question asked how many times each member visited last week, and the answers ran from 0 to 5. The office has already counted them, so nobody has to copy out 60 numbers. In a frequency table the value column holds the possible answers and the frequency column holds how many members gave each one. Here it reads 0 with 9 members, 1 with 12, 2 with 17, 3 with 12, 4 with 6 and 5 with 4. *(drawing: Sixty members, six answers. The bar heights are the frequencies, and they add up to 60.)* The mean is everything added up, divided by how many things there are. Write the value as x and the frequency as f. A row holding x = 3 with f = 12 is twelve members who each visited three times, so it contributes 3 × 12 = 36 visits. Do that for every row, then add: 0 × 9 + 1 × 12 + 2 × 17 + 3 × 12 + 4 × 6 + 5 × 4 = 126 visits. Divide by how many members: 126 / 60 = 2.1 visits each. Nobody visited 2.1 times. A mean does not have to be an answer anyone gave. Order the values and add frequencies to locate positions. With odd count n, take the value at (n + 1) / 2. With even n, (n + 1) / 2 is a half position between n / 2 and n / 2 + 1: locate both and average their values, not their positions. Here (60 + 1) / 2 = 30.5 lies between 30 and 31. Running totals 9, 21, 38 put both in the 2-visits row, so the median is 2. The middle row need not hold the middle observations. *(drawing: The running total says where each answer stops. Positions 30 and 31 are both inside the block that ends at 38.)* Find the largest frequency, 17, and read its value, 2 visits: that is the mode. The frequency counts members; the value names their answer. They can happen to be the same number, but play different roles. For the range, use only values with positive frequency: largest minus smallest. Here 5 - 0 = 5. This measures the gap between the smallest and largest observed answers. **Example.** You can reverse the mean calculation to find a total. A club records how many matches each member played, and reports a mean of 4 across 45 members. Multiply back: the total of all their matches is 4 × 45 = 180. A missing frequency can be worked out backwards too. If the frequencies read 6, 9, then a gap, then 5, and the total is 28, the gap must be 28 - 20 = 8, because the frequencies always add to how many people answered. **Recap.** For the median, locate the middle value or pair using running totals. Average the pair's values. Mean uses value × frequency; mode and range use observed values. *Original lesson, questions and flashcards written for this learning dataset.* ## Practice 34 questions on this page, each with its working shown. ## Needs first - [Calculating the mean](https://lightmysky.com/learn/mathematics/calculating-the-mean-mt_Cg8VPguS_V) - [Comparing data sets with averages and range](https://lightmysky.com/learn/mathematics/comparing-data-sets-with-averages-and-range-mt_XSXnTQoQ4l) ## Opens up - [Estimating the mean from grouped data](https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw)