Math Vale

📊 Statistics Steppe · Statistics

Histograms, Box Plots & Shape

Read a dot plot, a histogram and a box plot, build the five-number summary a box plot is drawn from, describe the shape of a distribution as symmetric or skewed, and compare two distributions by centre and spread.

In short

  • A dot plot keeps every reading, a histogram keeps the shape, and a box plot keeps five numbers — they are three zoom levels on one data set.
  • The five-number summary is minimum, Q1, median, Q3, maximum, and the four sections of a box plot each hold a quarter of the readings.
  • Find quartiles by splitting the ordered data at the median and leaving the median out of both halves, exactly as Range & Spread does.
  • Skew is named after the tail, not the peak: bars piled at the low end with a tail running high is skewed to the right.
  • A symmetric distribution is best summarised by the mean; a skewed one by the median and the IQR, because a long tail drags the mean away from the crowd.
  • Comparing two distributions means comparing both centre and spread, and the two comparisons can point at different groups.

Three pictures of one data set

Take one set of readings — say the times, in minutes, of twenty ferry crossings. There are three standard ways to draw it, and each one keeps something the others throw away.

  • A dot plot puts one dot above the value it happened at. Nothing is lost: you can read every single reading back off the plot. That only works while the data set is small and the values are whole numbers.
  • A histogram groups the readings into equal intervals called bins and draws a bar for each. You can no longer see the individual readings — only how many landed in each bin — but the overall shape leaps out.
  • A box plot goes further still and keeps just five numbers. It shows almost nothing about individual readings, and everything about where the middle of the data sits and how far it stretches.

So the three are not rivals. They are three zoom levels on the same data, and the right one depends on the question. "Which value came up most often?" is a dot-plot question. "What shape is this?" is a histogram question. "Which of these two groups is more spread out?" is a box-plot question.

The five-number summary

A box plot is drawn from five numbers, and every one of them is a position in the ordered data:

  • the minimum, the smallest reading,
  • Q1, the lower quartile,
  • the median, the middle reading,
  • Q3, the upper quartile,
  • the maximum, the largest reading.

To find them, put the data in order, take the median first, then split the list there and leave the median itself out of both halves. Q1 is the median of the lower half and Q3 is the median of the upper half. That is the same rule used in Range & Spread, and the two skills mark the same data the same way.

The five numbers cut the data into four sections — the lower whisker, the two halves of the box, the upper whisker — and each section holds a quarter of the readings. That is what makes a box plot so quick to read: 50% of the data is inside the box, 25% sits above Q3, and 75% sits above Q1.

Two spreads come straight out of the five numbers. The range is maximum - minimum, the whole stretch. The interquartile range is Q3 - Q1, the width of the box alone, which is the spread of the middle half and ignores both extremes.

Shape: symmetric, skewed, uniform, bimodal

Look at a histogram from a distance and stop reading the numbers. What is left is the shape, and there are four names worth knowing.

  • Symmetric: the bars either side of the middle mirror each other, so neither side trails further than the other.
  • Skewed right: the bars pile up at the low end and a long thin tail stretches away towards the high values.
  • Skewed left: the mirror image — the pile is at the high end and the tail runs down towards the low values.
  • Uniform: every bar is about the same height. Bimodal: two separate peaks, usually a sign that two different groups have been mixed into one data set.

The trap is in the naming. Skew is named after the tail, not the peak. A distribution with its tall bars on the left and a tail crawling off to the right is skewed *right*, even though everything eye-catching about it is on the left. Say the direction of the tail out loud before you answer, every time.

Shape is worth naming because it is a claim about the data that anyone can check, and because it decides which summary of centre to trust.

Shape chooses the centre

The mean is a balance point: every reading pulls on it, and a reading far out in a tail pulls hard. The median only cares about position in the order, so a far-out reading counts once, like every other reading.

That difference is the whole rule.

  • In a symmetric distribution the mean and the median land in the same place. Use the mean — it uses every reading and nothing is dragging it off centre.
  • In a skewed distribution the mean is pulled out towards the tail, away from where most of the data actually sits. Use the median, and report the IQR as the spread beside it.

A single extreme reading — an outlier — does the same thing on a smaller scale. The 1.5 x IQR rule flags one: anything above Q3 + 1.5 x IQR or below Q1 - 1.5 x IQR is worth a second look. It may be a recording slip, or it may be the most interesting reading you have. Either way, the median and the IQR barely notice it, while the mean and the range move a long way.

This is why a house-price report almost always quotes a median: prices are strongly skewed right, and a handful of enormous ones would drag the mean somewhere no house actually costs.

Comparing two distributions

Once one distribution can be described, two can be compared, and a good comparison always says two things: where the centre is, and how wide the spread is.

Put the two box plots side by side and ask, in this order:

  • Centre. Which median is higher? That answers "which group is typically bigger".
  • Spread. Which box is wider — that is, which IQR is larger? That answers "which group is less consistent".
  • Shape and outliers. Is either whisker stretched far out on its own?

Those two answers do not have to point the same way. One camp can have the higher median and the tighter box: typically quicker *and* more reliable. Another can be typically slower but very consistent. Reporting only the centre hides half of what happened.

Where this leads: the IQR is a spread built out of positions, and the next one you meet, the standard deviation, is built out of distances from the mean instead. Pair it with the symmetric bell shape called the normal curve and you can say what share of the data lies within one, two or three standard deviations of the centre — the same kind of statement a box plot makes with quarters.

Worked examples

Example 1

Find the five-number summary and the interquartile range of: 14, 7, 21, 9, 15, 4, 12.

  1. Put the readings in order first: 4, 7, 9, 12, 14, 15, 21.
  2. There are 7 readings, so the median is the 4th: median = 12.
  3. Split at the median and leave it out of both halves. Lower half: 4, 7, 9. Upper half: 14, 15, 21.
  4. Q1 is the middle of the lower half, so Q1 = 7. Q3 is the middle of the upper half, so Q3 = 15.
  5. Five-number summary: minimum 4, Q1 7, median 12, Q3 15, maximum 21.
  6. IQR = Q3 - Q1 = 15 - 7 = 8, while the range is 21 - 4 = 17. The IQR is smaller because it ignores both extremes.

Example 2

A histogram of waiting times, in minutes, has bars 0-9: 20, 10-19: 12, 20-29: 10, 30-39: 5, 40-49: 3. Describe the shape, say which measure of centre suits it, and find what percentage of the waits were under 20 minutes.

  1. Read the bar heights from the low interval to the high one: 20, 12, 10, 5, 3.
  2. They pile up at the low end and trail away to the high end, so the long tail runs towards the high values: the distribution is skewed to the right.
  3. A long right tail pulls the mean up above most of the data, so the median is the better centre to report here.
  4. For the percentage, the bars under 20 minutes are 0-9 and 10-19: 20 + 12 = 32 waits.
  5. The total is 20 + 12 + 10 + 5 + 3 = 50 waits, and 32 out of 50 is the same share as 64 out of 100.
  6. So 64% of the waits were under 20 minutes.

Example 3

Two ferries were timed, in minutes. Reed Crossing: minimum 18, Q1 22, median 24, Q3 26, maximum 31. Bog Crossing: minimum 9, Q1 15, median 21, Q3 33, maximum 41. Compare them, and check Bog Crossing for an outlier.

  1. Centre first: the medians are 24 and 21, so Reed Crossing is typically the slower of the two.
  2. Spread next: IQR for Reed is 26 - 22 = 4, and for Bog it is 33 - 15 = 18.
  3. Bog Crossing is more than four times as spread out, so Reed Crossing is far the more consistent — the higher median and the tighter box belong to different ferries.
  4. For the outlier check on Bog Crossing: 1.5 x IQR = 1.5 x 18 = 27, so the upper boundary is Q3 + 27 = 33 + 27 = 60.
  5. Its maximum, 41, is below 60, so nothing at the top counts as an outlier — that long upper whisker is ordinary spread, not one stray reading.

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

A dot plot at Coin Ferry records the crossings made each hour. Reading it value by value: 0 (4 dots), 1 (2 dots), 2 (6 dots), 3 (5 dots) How many hours were recorded altogether?

Answer: 17 hours

  1. The plot reads 0 (4 dots), 1 (2 dots), 2 (6 dots), 3 (5 dots).
  2. Total = 4 + 2 + 6 + 5 = 17
  3. So the answer is 17.

Problem 2

Difficulty 3 of 5

Every one of the ropes at Scatter Fields had its length measured, in m, and the results were drawn as a histogram. The bars read: 20-24: 9, 25-29: 12, 30-34: 7, 35-39: 10, 40-44: 13 How many ropes measured under 25 m?

Answer: 9 ropes

  1. Bars that qualify: 20-24.
  2. Only that one bar counts, so 9 readings qualify.
  3. So the answer is 9.

Problem 3

Difficulty 4 of 5

The lengths of 11 ropes at Scatter Fields, in m and in order, were: 10, 13, 30, 47, 48, 68, 83, 85, 121, 136, 152 Which line is the five-number summary? Split the data at the median and leave the median itself out of both halves.

  1. minimum 10, Q1 30, median 72, Q3 121, maximum 152
  2. minimum 10, Q1 13, median 68, Q3 136, maximum 152
  3. minimum 30, Q1 10, median 68, Q3 152, maximum 121
  4. minimum 10, Q1 30, median 68, Q3 121, maximum 152
  5. minimum 10, Q1 38.5, median 68, Q3 103, maximum 152

Answer: D. minimum 10, Q1 30, median 68, Q3 121, maximum 152

  1. Ordered data: 10, 13, 30, 47, 48, 68, 83, 85, 121, 136, 152.
  2. Minimum = 10, maximum = 152, median = 68 (the 6th of 11).
  3. Lower half 10, 13, 30, 47, 48 has middle 30, so Q1 = 30.
  4. Upper half 83, 85, 121, 136, 152 has middle 121, so Q3 = 121.
  5. Five-number summary: minimum 10, Q1 30, median 68, Q3 121, maximum 152.

Common mistakes

  • Counting a bin boundary in the wrong interval: putting a reading of 30 into the bar labelled 20-29 rather than into 30-39.
  • Leaving the median inside both halves when hunting for the quartiles, which pulls Q1 and Q3 in towards the middle and gives the wrong box.
  • Reading skew from the peak instead of the tail, and calling a distribution "skewed left" because its tall bars are on the left.
  • Confusing the range with the IQR: the range is maximum - minimum, while the IQR is Q3 - Q1, the width of the box alone.
  • Reading the height of the tallest stack on a dot plot as the mode. The mode is the value written underneath that stack, not how many dots it has.
  • Answering a comparison with one number: saying which group has the higher median and never mentioning that the other group is far more spread out.
  • Assuming the whiskers of a box plot hold a few readings because they are thin. Each whisker holds a full quarter of the data.

What you should be able to do

  • Read counts and totals from a dot plot or a histogram.
  • Find the five-number summary of a data set and read a box plot back into it.
  • Describe the shape of a distribution and say which summary of centre suits it.
  • Compare two distributions by their centres and their spreads.

Where this fits in the curriculum

Common Core

  • HSS-ID.A.1

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

  • HSS-ID.A.2

    High school — Use statistics appropriate to the shape of the data distribution to compare centre and spread of two or more data sets.

  • HSS-ID.A.3

    High school — Interpret differences in shape, centre and spread in the context of the data sets, accounting for possible effects of outliers.

  • 6.SP.B.4

    Grade 6 — Display numerical data in plots on a number line, including dot plots, histograms and box plots.

Ontario

  • MTH1W.D1

    Grade 9 de-streamed — Describe the collection and use of data, and represent and analyse data involving one and two variables.

    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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