doing math

courses › Business Stats & Decisions

Averages & Spread

level 12 course

Mean, median, weighted averages, and how spread out results are.

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

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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 squeezes many results into one number, and the kind of average matters. The mean is the total divided by the count. The median is the middle value once they are sorted. One big job pulls the mean up but barely moves the median, so compare both before you call a number typical.

When groups differ in size, weight them: rebuild each group's total first. Standard deviation measures how spread out results are. Two crews with the same average can differ a lot in how reliable they are.

Techniques

Sort, then find the middle

The median of a list of values.

  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.
worked example

Example: A landscaping crew's six job values are $300, $120, $2,400, $180, $260 and $200. What is the median job value?

  1. Sorted: $120, $180, $200, $260, $300, $2,400.
  2. Even count, so average the middle two.
  3. ($200 + $260) ÷ 2 = $230.
  4. The mean, about $577, is pulled up by one job.

Answer: $230

Totals first

A mean, or an average across groups of different sizes.

  1. Mean: add every value, then divide by the count.
  2. For groups, rebuild each total: count × average.
  3. Add the totals, then divide by the total count.
worked example

Example: A bakery had 6 catering jobs averaging $200 and 2 averaging $600. What is the average across all 8 jobs?

  1. 6 × $200 = $1,200, and 2 × $600 = $1,200.
  2. $2,400 ÷ 8 = $300.
  3. Averaging $200 and $600 would wrongly give $400.

Answer: $300

Spread of three values

The sample standard deviation of three values.

  1. Find the mean.
  2. Square each value's gap from the mean, then add the squares.
  3. Divide by n − 1, which is 2 for three values.
  4. Take the square root.
worked example

Example: Three cleaning jobs took 3, 7 and 11 hours. What is the sample standard deviation, in hours, dividing by n − 1?

  1. Mean: 7 hours.
  2. Squared gaps: 16 + 0 + 16 = 32.
  3. 32 ÷ 2 = 16, and the square root of 16 is 4.

Answer: 4

Tips by skill

  • TipMedian: Sort the values first. Odd count: take the middle one. Even count: average the two middle ones.
  • TipMean: Add every value, then divide by how many there are. One big job pulls the mean up.
  • TipWeighted average: Turn each group's average into a total (count × average), add the totals, then divide by all the jobs.
  • TipStandard deviation of three values: Find the mean, square each gap from it, add, divide by n − 1 (here 2), then take the square root.

Watch out for

  • Calling the mean typical when one big job pulls it up. Check it against the median.
  • Picking one of the two middle values when the count is even. Average them.
  • Averaging two group averages when the groups differ in size. Weight each by its number of jobs.
  • Dividing by n instead of n − 1. That gives the population figure, not the sample one.

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

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