Math Vale

📊 Statistics Steppe · Statistics

Range & Spread

Measure how spread out a data set is with the range, the quartiles and the interquartile range, and use the spread to spot outliers.

In short

  • A centre without a spread is half an answer; spread says how far apart the data is.
  • Range = largest - smallest, a single number, and it is at the mercy of the two most extreme values.
  • Q1 and Q3 are the medians of the lower and upper halves, and IQR = Q3 - Q1 measures the width of the middle half.
  • The 1.5 x IQR rule builds fences at Q1 - 1.5 x IQR and Q3 + 1.5 x IQR, and it uses the IQR, never the range.

An average is only half the story

Two ferry crossings both average 20 minutes. One always takes between 19 and 21; the other takes anything from 5 to 40. The average hides the difference that actually matters to a traveller with a boat to catch.

Spread measures how far apart the data is, and it is the second half of every honest summary. Give a centre without a spread and you have told half the truth.

The two spreads in this skill are the range and the interquartile range. Both are distances, so both are subtractions, and both come out as a single number in the units of the data — never as a pair of values.

The range

range = largest value - smallest value

For 15, 28, 39, 47, 67, 103, 138: range = 138 - 15 = 123.

The range is quick and easy to picture, and it has one serious weakness: it depends entirely on the two most extreme values, which are exactly the two most likely to be freakish. Add one unusual measurement and the range can double while the rest of the data has not moved at all.

That is why "the range is 15 to 138" is not an answer to "what is the range?". The range is 123. The pair of numbers describes where the data sits; the range describes how wide it is.

Quartiles and the IQR

The quartiles cut the ordered data into four parts. Q2 is the median. Q1 is the median of the lower half and Q3 the median of the upper half.

The method used here — the one most school texts use — is: order the data, find the median, and then leave the median out of both halves when the count is odd.

For 10, 52, 62, 83, 86, 106, 110, 145, 152, 153, 154 (11 values): median = 106, the 6th value. Lower half = 10, 52, 62, 83, 86, so Q1 = 62. Upper half = 110, 145, 152, 153, 154, so Q3 = 152.

IQR = Q3 - Q1 = 152 - 62 = 90.

The IQR measures the width of the middle half of the data. It throws away the top quarter and the bottom quarter, which is precisely why one wild value cannot move it — and why it is the spread to quote whenever the median is the centre you quoted.

The 1.5 x IQR rule

The IQR also gives a working definition of "unusually far out". Build a fence one and a half interquartile ranges beyond each quartile:

  • lower boundary = Q1 - 1.5 x IQR
  • upper boundary = Q3 + 1.5 x IQR

Anything outside those boundaries is flagged as an outlier.

With Q1 = 62, Q3 = 152 and IQR = 90: 1.5 x 90 = 135, so the boundaries are 62 - 135 = -73 and 152 + 135 = 287. Nothing in that data set is an outlier.

The rule uses the IQR, never the range — using the range would let the extreme values help decide whether they are themselves extreme, which is circular. And "flagged as an outlier" is not the same as "throw it away": an outlier is a value worth looking at, and sometimes it is the most interesting reading you have.

Worked examples

Example 1

Find the interquartile range of 16, 50, 54, 60, 72, 75, 86, 106, 116, 145, 155.

  1. The data is already in order, and there are 11 values, so the median is the 6th: 75.
  2. Lower half (leaving the median out): 16, 50, 54, 60, 72. Its median is the 3rd of those five, so Q1 = 54.
  3. Upper half: 86, 106, 116, 145, 155. Its median is the 3rd of those five, so Q3 = 116.
  4. IQR = Q3 - Q1 = 116 - 54 = 62.
  5. Compare with the range, 155 - 16 = 139. The IQR is much smaller because it ignores the extremes at both ends.

Example 2

A data set has Q1 = 24 and Q3 = 40. Is a reading of 70 an outlier by the 1.5 x IQR rule?

  1. First the IQR: 40 - 24 = 16.
  2. 1.5 x IQR = 1.5 x 16 = 24.
  3. Upper boundary = Q3 + 1.5 x IQR = 40 + 24 = 64.
  4. 70 is greater than 64, so yes — 70 lies outside the fence and is flagged as an outlier.
  5. For completeness the lower boundary is 24 - 24 = 0, so any negative reading would be flagged too.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

The peat loads recorded at Survey Tower, in kg, were: 1, 18, 17, 5, 13, 8, 10 What is the range?

Answer: 17 kg

  1. In order: 1, 5, 8, 10, 13, 17, 18
  2. Largest = 18, smallest = 1.
  3. Range = 18 - 1 = 17 kg

Problem 2

Difficulty 3 of 5

The peat loads at Survey Tower, in order, were: 8, 38, 40, 47, 51, 65, 76 What is the lower quartile (Q1)? Split the data at the median and leave the median itself out of both halves.

Answer: 38 kg

  1. Ordered data: 8, 38, 40, 47, 51, 65, 76
  2. Median = 47 (the 4th of 7 values).
  3. Lower half: 8, 38, 40. Upper half: 51, 65, 76.
  4. Q1 is the middle of that half — the 2nd value overall — so Q1 = 38 kg.

Problem 3

Difficulty 4 of 5

The peat loads at Survey Tower, in order, were: 10, 13, 30, 47, 48, 68, 83, 85, 121, 136, 152 What is the interquartile range? Split the data at the median and leave the median out of both halves.

Answer: 91 kg

  1. Median = 68.
  2. Q1 = 30 (middle of 10, 13, 30, 47, 48).
  3. Q3 = 121 (middle of 83, 85, 121, 136, 152).
  4. IQR = Q3 - Q1 = 121 - 30 = 91 kg
  5. Compare with the range, 142 — the IQR is smaller because it ignores the extremes.

Common mistakes

  • Giving the range when the interquartile range was asked for, so the extreme values creep back in.
  • Answering "15 to 138" rather than 123 — describing where the data sits instead of how wide it is.
  • Forgetting to order the data before hunting for quartiles.
  • Adding Q1 and Q3 instead of subtracting them, when spread is always a distance.
  • Using the IQR without the 1.5 multiplier when building the outlier boundaries.

What you should be able to do

  • Compute the range of a data set.
  • Find the lower and upper quartiles of an ordered data set.
  • Compute the interquartile range and say what it measures that the range does not.
  • Test a value for outlier status with the 1.5 x IQR rule.

Where this fits in the curriculum

Common Core

  • 6.SP.B.5

    Grade 6 — Summarise numerical data sets, giving quantitative measures of centre and of variability, including the interquartile range.

  • 7.SP.B.4

    Grade 7 — Use measures of centre and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

  • HSS-ID.A.1

    High school — Represent data with plots on the real number line, including dot plots, histograms and box plots.

Ontario

  • G8.D1

    Grade 8 — Manage, analyse and use data to make convincing arguments and informed decisions, in various contexts drawn from real life.

    Ontario numbers its Data expectations per grade document and the specific numbering could not be verified line by line here, so this tag names the strand's OVERALL expectation rather than guessing at a specific one.

SAT

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