Averages
Mean, median, and weighted averages.
In your head, jot if needed · no calculator why?
Opens at level 12. You're level 1. You can read and practice here now.
the idea
An average is one number that stands in for a whole list. The mean shares the total out equally: add the values, then divide by how many there are. The median is the middle value of the sorted list, so one huge value can't drag it far.
A weighted average lets some parts count more: multiply each score by its weight, its share of the whole, before you add. And since mean × count = total, you can work backward from a target mean to the score you need.
techniques
Total, then share it out
- Mean: add every value, then divide by how many there are.
- Target mean: multiply it by the number of scores, the new one included, for the total you need.
- Subtract the total you already have.
worked example
Your first 3 test scores are 72, 85 and 90. What score on test 4 makes the mean of all 4 scores exactly 84?
- Total needed: 4 × 84 = 336.
- You have 72 + 85 + 90 = 247.
- 336 − 247 = 89.
Answer: 89
Sort, then find the middle
- Sort the values from smallest to largest.
- Odd count: the median is the middle value.
- Even count: average the two middle values. The answer may end in .5.
worked example
What is the median of 9, 2, 14, 5, 11, 6?
- Sort: 2, 5, 6, 9, 11, 14.
- Six values, so average the middle two: (6 + 9) ÷ 2 = 7.5.
Answer: 7.5
Weight each part, then add
- Write each weight as a decimal: 20% is 0.2.
- Multiply each score by its weight, then add. The weights make 100%, so there is nothing to divide.
worked example
Homework counts 40% of a grade and the final exam 60%. You scored 90 on homework and 70 on the exam. What is your overall grade?
- 0.4 × 90 = 36 and 0.6 × 70 = 42.
- 36 + 42 = 78.
Answer: 78
watch out for
- Miscounting the values, then dividing by the wrong number.
- Taking the middle of the list before sorting it.
- Ignoring the weights. A plain average of the grade above gives 80, not 78.
- Giving the target mean as the missing score. The new score must bring the total up to the target mean × the number of scores.
practice
Mean
worked example
What is the mean of 58, 94, 40, 84, 49?
Answer: 65
- Add them: 58 + 94 + 40 + 84 + 49 = 325.
- Divide by how many there are, 5: 325 ÷ 5 = 65.
Median
worked example
What is the median of 49, 44, 30, 25, 42, 24?
Answer: 36
- Sort: 24, 25, 30, 42, 44, 49.
- With an even count, average the middle two: (30 + 42) ÷ 2 = 36.
Score needed for a mean
worked example
Your first 2 test scores are 61 and 94. What score on test 3 makes the mean of all 3 scores exactly 80?
Answer: 85
- You need a total of 3 × 80 = 240.
- You have 61 + 94 = 155, so you need 240 − 155 = 85.
Weighted average
worked example
Your course grade is weighted: quizzes 60%, midterm 40%. You scored 63 on quizzes and 55 on the midterm. What is your overall grade?
Answer: 59.8
- Multiply each score by its weight and add: 0.6 × 63 + 0.4 × 55
- = 37.8 + 22 = 59.8.