--- 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: 2026-09-02 --- > **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`. # 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. For the median, you need the middle answer once the 60 are lined up in order. With 60 values the middle sits at position (60 + 1) / 2 = 30.5, so it is halfway between the 30th and the 31st. Now run a total down the frequency column: 9, then 9 + 12 = 21, then 21 + 17 = 38. The first 9 members visited 0 times, members 10 to 21 visited once, and members 22 to 38 visited twice. Both the 30th and the 31st land in that block, so the median is 2. When the two middle answers differ, the median is halfway between them. *(drawing: The running total says where each answer stops. Positions 30 and 31 are both inside the block that ends at 38.)* The mode is the answer that came up most often, so it is the easiest of the three. Find the largest frequency, 17, then read across to the value beside it. The mode is 2 visits, not 17. The 17 is how many members, and the mode is always an answer somebody gave. The range comes from the value column too: largest value minus smallest, so 5 - 0 = 5. A bigger range means the answers sit further apart. Nothing in the frequency column belongs in a range. **Example.** The mean calculation runs backwards when you need it to. 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. That is worth having when a frequency is missing. 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.** A frequency table is the list already counted. Multiply each value by its frequency, add those up, and divide by the total frequency for the mean. Run a total down the frequency column to find which value holds position (n + 1) / 2 for the median. The mode is the value beside the largest frequency, and the range comes from the value column, never the frequency column. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Stratified Sampling](https://lightmysky.com/learn/mathematics/stratified-sampling-mt_CWzOOk0609) - [Calculating the Mean](https://lightmysky.com/learn/mathematics/calculating-the-mean-mt_Cg8VPguS_V) ## Opens up - [Estimating the Mean from Grouped Data](https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw)