Reading a histogram
Thirty books, sorted by length into bars. What can the bars tell you about the books, and what have they hidden?
Where you are 4 in Histograms
When you reach for it
Ayla counted the pages in every book on her class shelf. A list of thirty page counts is hard to take in; a histogram shows numerical data like this as a picture (Types of data: categorical, numerical, discrete, continuous): the number line is cut into bins of equal width that run end to end, and on each bin stands a bar as tall as the number of values that land in it, the bin’s count or frequency.
You reach for this skill whenever data come to you as a histogram: to say how many values lie in a stretch of the number line, how many there are in all, or where the median lies. One bar is read as on a bar graph (Reading a bar graph): follow its top across to the scale. What is new is that the axis along the bottom is a number line, and that the separate values have gone. A bar says how many, never which.
The procedure
What one bar says. In Ayla’s histogram (Example 1) the bar on the bin from 200 to 300 pages reaches 10, so 10 books have a page count in that bin. Whether they have 210 pages or 290, the bar does not say: inside a bin, the values could be anywhere.
A value on an edge. The number 300 ends one bin and starts the next, so a book of exactly 300 pages needs a rule. In this course a value on an edge goes in the bar to its right. The bin “200 to 300” holds every page count x with 200 \le x < 300, and a book of exactly 300 pages is counted in the 300 to 400 bar. So every value belongs to exactly one bin.
Bars add. The bins touch and do not overlap, so no value is counted twice and none is missed, and counts may be added. All the bars together give the number of values, which is usually not printed and has to be found by adding. A stretch that starts and ends on edges is a set of whole bars: “at least 300 pages” is the 300 to 400 bar and every bar to its right.
A stretch that cuts a bar. If a stretch starts or ends inside a bin, that bar cannot be split, because its values could sit anywhere in it. The count is then known only between two numbers: the bars lying wholly inside the stretch give the smallest it could be, and adding every bar the stretch cuts gives the largest.
What the bars hide. Grouping throws the separate values away. Two different lists with the same count in each bin draw the same histogram (Example 3), so a histogram cannot give the exact largest or smallest value, the exact mean or the exact median. It can say which bar holds the median, the middle value of the sorted list or the mean of the middle two when the count is even (Finding the median). Picture the values sorted: the first bar holds the first few, the next bar the next few. Adding the bars from the left, a running total, shows which bar holds the middle place. If the two middle values fall in different bars, the histogram may not say which bar holds the median.
Shape at a glance. The bars also show where values crowd together and where they thin out; the words for that come in Shape of a distribution, Symmetric and skewed distributions and Uniform and bimodal shapes.
Histograms are not bar graphs. Both draw bars as tall as a count, and there the likeness ends.
| Bar graph | Histogram | |
|---|---|---|
| Data | categories (a kind, a color) | numbers (a length, a count) |
| Along the bottom | category names | a number line |
| Order of the bars | any order | fixed by the number line |
| Bars | standing apart | touching |
| Width of a bar | means nothing | the width of the bin |
| A space between bars | only separates | an empty bin: no values there |
Worked examples
Reading Ayla’s shelf
The histogram shows the page counts of the books on Ayla’s class shelf. (All the data in this lesson is made up.) (a) How many books have from 200 up to 300 pages? (b) How many books are on the shelf? (c) How many have at least 300 pages? (d) How many have fewer than 200 pages, and what fraction of the shelf is that?
(a) The bar on the bin from 200 to 300 reaches the line for 10, so 10 books.
(b) Every book is in exactly one bar, so the number of books is the sum of all the bars:
(c) “At least 300” starts on the edge 300, so it takes the whole bars from 300 to 600. A book of exactly 300 pages sits in the first of these by the edge rule, so it is counted: 6 + 3 + 2 = 11 books.
(d) The two bars left of 200: 2 + 7 = 9 books, which is \displaystyle \frac{9}{30} = \frac{3}{10} of the shelf. Check: 9 below 200, 10 from 200 to 300 and 11 from 300 up make 9 + 10 + 11 = 30.
Azra’s butterflies: an edge, an empty bin and a cut bar
Azra measured the wingspans of butterflies in a meadow, in millimeters. (a) How many did she measure? (b) How many have a wingspan from 40 up to 60 mm? (c) One has a wingspan of exactly 60 mm. In which bar is it counted? (d) What does the empty bin from 70 to 80 mm say? (e) How many have a wingspan of more than 55 mm?
(a) 2 + 5 + 7 + 4 + 0 + 2 = 20 butterflies. The empty bin adds 0.
(b) The stretch starts and ends on edges, so it is two whole bars: 5 + 7 = 12 butterflies.
(c) The number 60 is the edge between the 50 to 60 bar and the 60 to 70 bar. A value on an edge goes right, so this butterfly is one of the 4 in the 60 to 70 bar.
(d) No wingspan is at least 70 mm and less than 80 mm. The empty bin keeps its place on the number line and shows a gap: the two widest butterflies, at 80 mm or more, are more than 10 mm wider than all the others, which are under 70 mm.
(e) The number 55 lies inside the 50 to 60 bin, so this cannot be read exactly. The bars from 60 up hold 4 + 0 + 2 = 6 butterflies, all wider than 55 mm. The 7 in the 50 to 60 bar could all be wider than 55 mm, or none, or some. So the count is at least 6 and at most 6 + 7 = 13, and the histogram cannot narrow it further.
Two lists, one histogram
Basia and Benedikt each measured eight tadpoles. Their lengths in millimeters, sorted:
| Lengths (mm) | |
|---|---|
| Basia | 10, 12, 15, 16, 17, 20, 21, 25 |
| Benedikt | 14, 14, 15, 19, 19, 24, 24, 29 |
(a) Show that, in bins 5 mm wide from 10 to 30, both lists draw the same histogram. (b) Find each median and mean. (c) With only the histogram, what can be said about the median?
Solution(a) The edge rule sends 15, 20 and 25 into the bar to their right.
| Bin (mm) | Basia | Benedikt |
|---|---|---|
| 10 to 15 | 10, 12 | 14, 14 |
| 15 to 20 | 15, 16, 17 | 15, 19, 19 |
| 20 to 25 | 20, 21 | 24, 24 |
| 25 to 30 | 25 | 29 |
Both put 2, 3, 2 and 1 values in the bins, so both draw these bars.
(b) With eight values the median is the mean of the fourth and fifth: Basia’s is \displaystyle \frac{16 + 17}{2} = 16.5 mm and Benedikt’s is \displaystyle \frac{19 + 19}{2} = 19 mm. For the means:
Basia’s mean is \displaystyle \frac{136}{8} = 17 mm and Benedikt’s is \displaystyle \frac{158}{8} = 19.75 mm.
(c) One picture, two medians and two means: the histogram cannot give either exactly, nor the longest tadpole (25 or 29 mm). It can say which bar holds the median. The running total is 2, then 2 + 3 = 5, so the second bar holds the third, fourth and fifth values. The fourth and fifth both lie in the 15 to 20 bar, and so does their mean. Both medians, 16.5 and 19, do.
Which bar holds the median?
Find which bar holds the median (a) for Ayla’s shelf in Example 1. (b) Ayla’s 16 books at home put 3, 5, 6 and 2 books in the bins 100 to 200, 200 to 300, 300 to 400 and 400 to 500 pages. What can be said about their median?
Solution(a) With 30 books the median is the mean of the 15th and 16th page counts in sorted order. Running totals are 2, 9, 19: the third bar holds books 10 to 19. Both the 15th and 16th lie in the 200 to 300 bar, so the median is at least 200 and less than 300 pages.
(b) The median of 16 values is the mean of the 8th and 9th. Running totals are 3, 8: the 8th book is the last in the 200 to 300 bar and the 9th is the first in the 300 to 400 bar. The median is the mean of one count from 200 up to 300 and one from 300 up to 400, so the smallest it could be and the number it must stay below are
It may land in either bar, and the histogram cannot say which.
Bar graph or histogram?
Ayla also sorted the same 30 books by kind. (a) Which of her two graphs is a histogram? (b) Could the bars of each be drawn in another order? (c) What does a space between bars mean in each? (d) Is there a middle kind of book?
(a) The page-count graph of Example 1: its data are numbers, on a number line from 0 to 600. Kinds of book are categories, so their graph is a bar graph. Both count the same books: 10 + 8 + 9 + 3 = 30.
(b) The kinds have no order of their own; alphabetical, tallest first or any order says the same. The bins cannot move: 200 to 300 lies between 100 to 200 and 300 to 400 on the number line, and moving it would make the number line false.
(c) In the bar graph the spaces only keep the categories apart. In the histogram every page count from 0 up to 600 belongs to some bin, so the bars touch; a space would be numbers belonging to no bin. A space in a histogram is an empty bin, as in Example 2, and says no values lie there.
(d) No. Categories have no order of size, so there is no median kind. The most common kind, adventure with 10 books, can still be named.
A smudged bar
Baptiste drew a histogram of the lengths of his 24 tadpoles, in bins 4 mm wide from 10 to 30, and one bar got smudged. (a) How many tadpoles are in the smudged bar? Then decide whether each statement is true, false, or cannot be told. (b) More than half of the tadpoles are at least 18 mm long. (c) The median is in the 18 to 22 bar. (d) The longest tadpole is 29 mm long. (e) The longest tadpole is more than 10 mm longer than the shortest.
(a) The readable bars hold 3 + 8 + 5 + 2 = 18, so the smudged bar holds 24 - 18 = 6 tadpoles.
(b) True. “At least 18” is three whole bars, 8 + 5 + 2 = 15, and 15 is more than half of 24, which is 12.
(c) True. The median of 24 values is the mean of the 12th and 13th. Running totals are 3, 3 + 6 = 9, 9 + 8 = 17, so both lie in the third bar, 18 to 22. This needed the count from (a).
(d) Cannot be told. The longest is in the 26 to 30 bar, so it is at least 26 mm, but it could be any length in that bar.
(e) True, without a single exact value. The shortest is in the 10 to 14 bar, so it is less than 14 mm; the longest is at least 26 mm. Their difference is more than 26 - 14 = 12 mm.
Boundaries
- Numerical data only. A bin is a stretch of a number line. Categories go in a bar graph.
- Equal bins. Reading a taller bar as a more crowded stretch needs every bin the same width; a bin twice as wide collects more values just by being wide. Choosing bins is Making a histogram and choosing the bin width.
- Nothing inside a bar. A count is exact only for a stretch that starts and ends on bin edges. Otherwise the histogram gives a smallest and a largest possible count.
- The edge rule has to be known. “At least 300” depends on where a value of exactly 300 was put. A histogram that does not state its rule leaves values on edges uncertain.
- Counts, not shares. Histograms of two groups of different sizes cannot be compared bar for bar. That needs shares of the total, the Relative frequency histograms.
Where this goes next
Making a histogram and choosing the bin width runs this lesson backwards, from a list of values to bins and bars. A dot plot keeps every value; a histogram gives them up to show the shape of a large data set, and Choosing a display for a question weighs the two. The shape itself is described in Shape of a distribution, and the running totals used here, drawn as a graph, become a Cumulative frequency graphs and percentiles.
Exercises
Azra’s second meadow
- How many butterflies does the histogram show?
- How many have a wingspan from 45 up to 55 mm?
- How many have a wingspan of less than 50 mm?
- How many have a wingspan of at least 55 mm? What fraction of the butterflies is that?
- One butterfly has a wingspan of exactly 50 mm. In which bar is it counted? In which bar is a wingspan of 49.5 mm counted?
Benedikt’s bookshop shelf
- How many books are on the shelf?
- How many books have at least 200 pages?
- How many books have fewer than 150 pages?
- What fraction of the books have from 250 up to 350 pages?
What the bars cannot say
- On the meadow histogram, how many butterflies have a wingspan of more than 52 mm? Give the smallest and the largest the count could be.
- Can the shelf histogram say how many pages the longest book has? What can it say?
- Two lists of five wingspans in millimeters, sorted, are P: 41, 44, 52, 55, 58 and Q: 48, 49, 50, 50, 59. Show that both draw the same histogram with the bins 40 to 50 and 50 to 60 mm. Then find each list’s median and mean.
- Which bar of the meadow histogram holds the median wingspan?
- Which bar of the shelf histogram holds the median page count? Give the smallest the median could be and a number it must be less than.
- Ten tadpoles put 2, 3 and 5 values in the bins 10 to 15, 15 to 20 and 20 to 25 mm. Explain why the histogram cannot give the mean. Then find the smallest the mean could be and a number it must be less than.
Bar graph or histogram
- Say whether each data set belongs in a histogram or a bar graph, and why: (a) the number of pages in each of 50 books; (b) the color of each of 30 butterflies; (c) the length in millimeters of each of 40 tadpoles; (d) the kind of pet each of 25 people owns.
- The bars of a bar graph of butterfly colors can be redrawn tallest first. Could the bars of a histogram of wingspans? Explain.
Explaining
- Explain why the bars of a histogram always add to the number of values. What would go wrong if a value on an edge were counted in both bars it touches?
- A histogram has an empty bin between two bars. Explain why the empty bin cannot be dropped and the bars on either side pushed together.
Answers
- 25: 1 + 4 + 6 + 8 + 4 + 2 = 25.
- 14: the 45 to 50 and 50 to 55 bars, 6 + 8 = 14.
- 11: the three bars left of 50, 1 + 4 + 6 = 11.
- 6, from 4 + 2 = 6, which is \displaystyle \frac{6}{25} of the butterflies.
- 50 mm is on an edge, so it goes right, into the 50 to 55 bar. 49.5 mm is in the 45 to 50 bar.
- 40: 3 + 6 + 11 + 9 + 7 + 4 = 40.
- 31: the four bars from 200 up, 11 + 9 + 7 + 4 = 31.
- 3: only the 100 to 150 bar.
- \displaystyle \frac{2}{5}: those two bars hold 9 + 7 = 16 books, and \displaystyle \frac{16}{40} = \frac{2}{5}.
- At least 6 and at most 14. The bars from 55 up hold 4 + 2 = 6, all wider than 52 mm; the 8 in the 50 to 55 bar may or may not be, so at most 6 + 8 = 14.
- No. The longest book is in the 350 to 400 bar, so it has at least 350 pages, but it could have any page count in that bar.
- Each puts 2 values in 40 to 50 and 3 in 50 to 60 (Q’s two 50s go right by the edge rule). Medians 52 and 50 mm, the third values. Means \displaystyle \frac{250}{5} = 50 and \displaystyle \frac{256}{5} = 51.2 mm.
- The 50 to 55 bar. The median of 25 values is the 13th, and running totals 1, 5, 11, 19 put it in the fourth bar.
- Either the 200 to 250 or the 250 to 300 bar. The median of 40 values is the mean of the 20th and 21st; running totals 3, 9, 20 put the 20th at the end of the 200 to 250 bar and the 21st in the next. It is at least \displaystyle \frac{200 + 250}{2} = 225 and less than \displaystyle \frac{250 + 300}{2} = 275 pages.
- The mean needs the total of the values, and the bars hide them. Each value is at least its bin’s left edge and less than its right edge, so the mean is at least \displaystyle \frac{2 \times 10 + 3 \times 15 + 5 \times 20}{10} = 16.5 mm and less than \displaystyle \frac{2 \times 15 + 3 \times 20 + 5 \times 25}{10} = 21.5 mm.
- (a) Histogram: page counts are numbers. (b) Bar graph: colors are categories. (c) Histogram: lengths are numbers. (d) Bar graph: kinds of pet are categories.
- No. Each histogram bar stands on a stretch of the number line, and the stretches come in a fixed order. Putting 50 to 60 before 40 to 50 would make the axis no longer a number line. Colors have no order, so their bars can move.
- The bins run end to end without overlapping, and the edge rule puts a value on an edge in just one of them, so each value is counted once. Counting an edge value in both bars would count it twice, and the bars would add to more than the number of values.
- The empty bin is a stretch of the number line where no values lie. Dropping it would put two bins that are not neighbors side by side: the number line would skip a stretch without showing it, and the gap in the data would vanish.