Averages & Spread
Mean, median, weighted averages, and how spread out results are.
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 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
- Sort the values from smallest to largest.
- Odd count: the median is the middle value.
- Even count: average the two middle values.
worked example
A landscaping crew's six job values are $300, $120, $2,400, $180, $260 and $200. What is the median job value?
- Sorted: $120, $180, $200, $260, $300, $2,400.
- Even count, so average the middle two.
- ($200 + $260) ÷ 2 = $230.
- The mean, about $577, is pulled up by one job.
Answer: $230
Totals first
- Mean: add every value, then divide by the count.
- For groups, rebuild each total: count × average.
- Add the totals, then divide by the total count.
worked example
A bakery had 6 catering jobs averaging $200 and 2 averaging $600. What is the average across all 8 jobs?
- 6 × $200 = $1,200, and 2 × $600 = $1,200.
- $2,400 ÷ 8 = $300.
- Averaging $200 and $600 would wrongly give $400.
Answer: $300
Spread of three values
- Find the mean.
- Square each value's gap from the mean, then add the squares.
- Divide by n − 1, which is 2 for three values.
- Take the square root.
worked example
Three cleaning jobs took 3, 7 and 11 hours. What is the sample standard deviation, in hours, dividing by n − 1?
- Mean: 7 hours.
- Squared gaps: 16 + 0 + 16 = 32.
- 32 ÷ 2 = 16, and the square root of 16 is 4.
Answer: 4
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.
practice
Median
worked example
Four job values are $150, $250, $270, and $1,170. What is the median job value?
Answer: $260.00
- Sort them: $150, $250, $270, $1,170.
- With an even count, average the middle two: ($250 + $270) ÷ 2 = $260.
Mean
worked example
Five job values are $60, $70, $220, $260, and $390. What is the mean job value?
Answer: $200.00
- Total $1,000 ÷ 5 jobs = $200.
Weighted average
worked example
Eight jobs average $460 and seven jobs average $130. What is the average across all fifteen jobs?
Answer: $306.00
- Totals first: 8 × $460 = $3,680 and 7 × $130 = $910.
- $3,680 + $910 = $4,590; ÷ 15 jobs = $306.
Standard deviation of three values
worked example
Three jobs took 21, 35, and 49 hours. What is the sample standard deviation, in hours, dividing by n − 1?
Answer: 14 hours
- Mean 35. Squared gaps from the mean: 196 + 0 + 196 = 392.
- Divide by n − 1 = 2: 196. Square root: 14 hours.