doing math

courses › Probability & Statistics

Spread

level 22 course

Range, quartiles, and standard deviation.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 22.

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Builds on: Averages (not open yet) · Exponents & roots (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

Spread says how far data wanders from its centre. The range is largest minus smallest. The interquartile range (IQR) is the spread of the middle half: Q3 minus Q1, the medians of the upper and lower halves.

The standard deviation (SD) is the typical distance from the mean. Adding the same number to every value moves the mean but no distances, so the spread stays put. Multiplying every value by a positive k multiplies both the centre and the spread by k.

Techniques

Halves, then their middles

The interquartile range, using the median of each half.

  1. Sort the data and split it into a lower and an upper half.
  2. Q1 is the median of the lower half, Q3 of the upper half.
  3. Subtract: Q3 minus Q1.
worked example

Example: Data: 3, 5, 8, 10, 12, 15, 18, 20. Using the median of each half, what is the interquartile range?

  1. Lower half 3, 5, 8, 10: Q1 = (5 + 8) ÷ 2 = 6.5.
  2. Upper half 12, 15, 18, 20: Q3 = (15 + 18) ÷ 2 = 16.5.
  3. 16.5 − 6.5 = 10.

Answer: 10

SD in four steps

The population standard deviation of a list.

  1. Find the mean.
  2. Square each value's distance from it.
  3. Average the squares, dividing by the count. That is the variance.
  4. Take the square root.
worked example

Example: What is the population standard deviation of 5, 11, 11, 13?

  1. Mean: 40 ÷ 4 = 10.
  2. Squared distances: 25, 1, 1, 9, adding to 36.
  3. 36 ÷ 4 = 9, and the square root of 9 is 3.

Answer: 3

Shift moves, scale stretches

Every value gets the same number added, or is multiplied by the same number.

  1. Adding c: the mean and median go up by c. The range and SD stay the same.
  2. Multiplying by a positive k: all four are multiplied by k.
worked example

Example: A data set has mean 20 and standard deviation 4. Every value is multiplied by 3, then 5 is added. What is the new standard deviation?

  1. Multiplying by 3: 4 × 3 = 12.
  2. Adding 5 moves no gaps, so it stays 12.

Answer: 12

Tips by skill

  • TipRange & IQR: Sort first. Range: largest minus smallest. IQR: the median of the upper half minus the median of the lower half.
  • TipShift & scale: Adding moves the centre and leaves the spread alone. Multiplying by a positive k multiplies the centre and the spread by k.
  • TipStandard deviation: Mean, then squared distances from it, then their average (divide by the count), then the square root.

Watch out for

  • Giving the range when the question asks for the interquartile range.
  • Dividing by one less than the count, the sample formula, when the question asks for the population SD.
  • Saying the SD goes up when the same number is added to every value. Only the centre moves.

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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