Averages
Mean, median, and weighted averages.
In your head, jot if needed · no calculator why?
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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.
more of the idea
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.
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.
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.
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
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Mean
worked example
What is the mean of 58, 94, 40, 84, 49?
- Add them: 58 + 94 + 40 + 84 + 49 = 325.
- Divide by how many there are, 5: 325 ÷ 5 = 65.
Answer: 65
Median
worked example
What is the median of 49, 44, 30, 25, 42, 24?
- Sort: 24, 25, 30, 42, 44, 49.
- With an even count, average the middle two: (30 + 42) ÷ 2 = 36.
Answer: 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?
- You need a total of 3 × 80 = 240.
- You have 61 + 94 = 155, so you need 240 − 155 = 85.
Answer: 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?
- Multiply each score by its weight and add: 0.6 × 63 + 0.4 × 55
- = 37.8 + 22 = 59.8.
Answer: 59.8