doing math

courses › Probability & Statistics

Averages

level 12 course

Mean, median, and weighted averages.

In your head, jot if needed · no calculator why?

Learn first (about 3 minutes)

Opens at level 12.

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Builds on: Division (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

A mean, or the score needed to reach a target mean.

  1. Mean: add every value, then divide by how many there are.
  2. Target mean: multiply it by the number of scores, the new one included, for the total you need.
  3. Subtract the total you already have.
worked example

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?

  1. Total needed: 4 × 84 = 336.
  2. You have 72 + 85 + 90 = 247.
  3. 336 − 247 = 89.

Answer: 89

Sort, then find the middle

Any median.

  1. Sort the values from smallest to largest.
  2. Odd count: the median is the middle value.
  3. Even count: average the two middle values. The answer may end in .5.
worked example

Example: What is the median of 9, 2, 14, 5, 11, 6?

  1. Sort: 2, 5, 6, 9, 11, 14.
  2. Six values, so average the middle two: (6 + 9) ÷ 2 = 7.5.

Answer: 7.5

Weight each part, then add

Parts that count for different percentages, like a course grade.

  1. Write each weight as a decimal: 20% is 0.2.
  2. Multiply each score by its weight, then add. The weights make 100%, so there is nothing to divide.
worked example

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?

  1. 0.4 × 90 = 36 and 0.6 × 70 = 42.
  2. 36 + 42 = 78.

Answer: 78

Tips by skill

  • TipMean: Add every value, then divide by how many there are. Count the values twice.
  • TipMedian: Sort the list first. With an even count, average the two middle values.
  • TipScore needed for a mean: Target mean × number of scores, new one included, is the total you need. Subtract the scores you have.
  • TipWeighted average: Turn each weight into a decimal, multiply it by its score, and add. The weights already make 100%.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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