Averages, Weighted Means and Mixtures

Average is total divided by count

Mean = sum of observations / number of observations. The total is often the most useful intermediate quantity. If 8 people have an average score of 45, their total is 360.

Adding or replacing a value

If a ninth score of 63 is added, the new mean is (360 + 63)/9 = 47. If a score of 38 among the original eight is replaced by 54, the total rises by 16 and the mean rises by 16/8 = 2, becoming 47.

Weighted means

Group A has 20 learners averaging 60; group B has 30 learners averaging 80. Combined mean = (20×60 + 30×80)/50 = 72. Taking (60+80)/2 = 70 would incorrectly give equal weight to unequal groups. A combined mean with positive weights lies between the group means. A result outside that interval signals a calculation or interpretation error.

Mixture example

For all mixture calculations in this lesson, concentrations are percentages by volume; assume volumes are additive and the solute amount is conserved. Mix 2 litres of a 20% solution with 3 litres of a 40% solution. Solute amount = 0.4 + 1.2 = 1.6 litres; total = 5 litres; concentration = 32%. The simple average 30% is wrong because the volumes differ.

To obtain 30% from 20% and 50% solutions, let quantities be x and y. Then 0.20x + 0.50y = 0.30(x+y), so x = 2y. Required ratio is 2:1. The result must use more of the lower concentration because 30 is closer to 20 than to 50.

Average speed warning

For equal distances travelled at different positive speeds, average speed is not the arithmetic mean of those speeds. Covering 60 km at 30 km/h and 60 km at 60 km/h takes 2 + 1 = 3 hours. Average speed = 120/3 = 40 km/h. For equal time intervals, the arithmetic mean works.

Practice

1. Mean of 5 numbers is 18. Removing 10 leaves what mean? (90−10)/4 = 20.
2. Ten scores average 25; fifteen average 35. Combined mean? 775/25 = 31.
3. A group of 12 has mean 40. One value rises by 24. New mean? 42.
4. Equal volumes of 10% and 30% solutions yield what concentration? 20%, under the additive-volume assumption.

Common traps

Averaging averages without weights; forgetting the count changes after adding a value; confusing average speed with the average of two speeds; mixing percentage units with absolute amounts. Write totals and units before simplifying.