Find the mean, median, mode and range from a frequency table by working with the counts instead of listing every value.
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.
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.
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.
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.
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.
Practice questions and the readings "Two columns, two jobs" and "Finding the middle person" were written for this dataset. The rewritten readings were written for this dataset. Adapted summaries: Khan Academy, "Mean and Median". The two videos are published by Khan Academy. Source licenses are not specified.
The 60 you shared out between the three groups have sent their forms back, counted into six rows. Visits last week: 0 from 9 members, 1 from 12, 2 from 17, 3 from 12, 4 from 6, 5 from 4. Leo wants the mean and he has it already: add the visit numbers 0, 1, 2, 3, 4 and 5, which makes 15, then divide by the six rows. Mean 2.5, he says.
Leo's calculation does not average the members' visits. What has he averaged?
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