Averages, Rates and Unit Conversions
Averages and rates as fractions
The mean is a total divided by a count; the median is the middle value of an ordered list; the mode is the most frequent; the range is largest minus smallest. When groups of different sizes are combined, use a weighted mean: multiply each group mean by its size, add, then divide by the total size. Rates are fractions with different units on top and bottom. Speed = distance ÷ time; unit price = cost ÷ quantity; productivity = output ÷ hours. Convert units before dividing, and keep track of what the denominator means.
Traps
The classic trap is averaging speeds arithmetically. A journey at 60 km/h out and 40 km/h back has an average speed of 48 km/h, because more time is spent at the slower speed. Convert minutes to hours as decimals (45 minutes = 0.75 h), not as 0.45. Do not average percentages or averages without weighting them by their bases. Check that the units of the answer match what the question asks for.
Worked example
A courier drives 120 km at 60 km/h, then returns 120 km at 40 km/h. Outbound time = 2 h; return time = 3 h; total distance = 240 km; total time = 5 h. Average speed = 240 ÷ 5 = 48 km/h, not (60 + 40) ÷ 2 = 50 km/h.
Rule of thumb: for average speed, total the distances and total the times first, then divide.