Averages
Scenario
When you look at 'average package' stats for an engineering college, a few 40 LPA outliers drag up the 3.5 LPA mass.
If averages hide the real distribution, how do we manipulate the total sum to figure out the true numbers?
Mental model
Average is the 'equal distribution' point. If everyone threw their money in a pot and split it evenly, that's the average.
The sum of all observations divided by the number of observations. It acts as the center of gravity for the dataset.
Formula
Average = Sum / Count. Sum = Average x Count. Weighted Average = (w1*x1 + w2*x2 + ...) / (w1 + w2 + ...). New item joining: New Item = Old Average + (New Count x Change in Average). Removing an item: New Average = (Old Sum - Removed Value) / (Old Count - 1).
Most tricky average problems are solved by thinking in terms of Total Sum. When a new member joins and shifts the average by d, the new member's value = Old Average + (New Total Size x d). For weighted averages, the group size is the weight.
Pattern discovery
Observe what happens to the average when every single item in the dataset changes uniformly.
- Items: 10, 20, 30. Avg = 20
- Items: 12, 22, 32. Avg = 22 (+2 shift)
- Items: 20, 40, 60. Avg = 40 (x2 shift)
If you perform a basic arithmetic operation on every element, what happens to the average?
Rule: The exact same operation applies directly to the average. The entire 'center of gravity' shifts.
Worked example
Problem: The average score of 8 students in a test is 72. When the teacher's score is included, the average rises to 75. What is the teacher's score?
- Set up: Old sum = 8 x 72 = 576. New count = 9 (8 students + teacher).
- New sum = 9 x 75 = 675.
- Teacher's score = New Sum - Old Sum = 675 - 576 = 99.
- Sanity-check using the shortcut: Teacher must cover the old average (72) PLUS give everyone in the new group +3 points (9 x 3 = 27). 72 + 27 = 99. Matches.
Answer: The teacher's score is 99.
Problem: Average of 6 numbers is 14. One number 26 is replaced by another number, and the new average becomes 12. What is the replacement number?
- Old sum = 6 x 14 = 84.
- New sum = 6 x 12 = 72.
- The change in sum = 72 - 84 = -12.
- New number = 26 + (-12) = 14.
- Sanity-check: Replace 26 with 14 in the sum: 84 - 26 + 14 = 72. 72 / 6 = 12. Correct.
Answer: The replacement number is 14.
Shortcuts
- Assumed Mean (Deviation) Method: Guess an average. Find how much each number deviates from your guess. Average the deviations and add back to your guess. (Average of 102, 105, 98. Guess 100. Deviations: +2, +5, -2. Sum of dev = +5. Avg dev = +5/3. True avg = 101.67.)
- Inclusion / Exclusion: If a new item joins and increases the average by x, its value is: (Old Average) + (New Total Size x x). (10 items avg 20. New item makes avg 21. New item = 20 + (11 x 1) = 31.)
Common mistakes
- Average Speed Trap: Average speed is NOT the arithmetic mean of speeds (e.g., 40 km/hr and 60 km/hr does NOT give 50 km/hr average). It is Total Distance / Total Time.
- Ignoring Sample Size: You cannot average two averages directly unless they represent the exact same number of items. Use the weighted average formula.
Glossary
- observations
- The individual data points or numbers in a dataset.
- mean
- Another word for average, found by adding up values and dividing by the count.
- weighted
- When some items count more heavily towards the final average than others.
Recall questions
- What is the average of the first 5 prime numbers?
- If the average of 5 numbers is 10, what is their sum?
- Average of 4, 8, 12, 16 is 10. If we add 5 to each, what is the new average?
Questions & answers
The average age of a class of 30 students is 15 years. If the teacher's age is included, the average increases by 1. What is the teacher's age?
46 years
Approach: Teacher brings enough to match the old average (15) PLUS enough to give everyone including herself +1 year (31 people x 1 = 31). Age = 15 + 31 = 46.
The average of 50 numbers is 38. If two numbers, namely 45 and 55, are discarded, what is the average of the remaining numbers?
37.5
Approach: Total Sum = 50 x 38 = 1900. Discard sum = 100. New sum = 1800. New count = 48. New avg = 1800 / 48 = 37.5.
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