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Uniform and bimodal shapes

Some plots stand level like a wall and some rise in two humps with a dip between. What do those two shapes say about where the data came from?

You need
Mode and bimodal dataMath 6
Where you are 3 in Shape of a distribution
Statistics7
Center and spread13
Shape of a distribution3
Symmetric and skewed distributions
Uniform and bimodal shapes
Mean, median and skew: choosing the center

Why this exists

A dot plot or a histogram shows a whole data set at once, and the first thing the eye takes in is its outline: where it is tall, where it is short, where there is nothing. Naming that outline turns a picture into a sentence someone else can use.

Two outlines carry a story. A plot that stands level often comes from fair chance, such as a spinner or the month a person happens to be born in. A plot with two humps often comes from two groups mixed together, such as two breeds of cow in one herd. The name tells you the next question: “is anything favored here?” for the first, “what are the two groups?” for the second.

Definition

A distribution is the values a data set takes together with how often each occurs. Its shape is the outline of its dot plot or histogram.

A peak is a stack or bar clearly taller than the stacks or bars beside it. A difference of a dot or two in a small data set does not make a peak.

A distribution is uniform when its stacks or bars are roughly level across the whole range: every value, or every bin, is about equally common, and no peak stands out.

A distribution is bimodal when it has two clear peaks with a dip or a gap between them. The peaks need not be equally tall. This is the looser of the two meanings of “bimodal” in Mode and bimodal data: it describes the plot, not an exact tie between two modes.

A plot with a single peak is called unimodal; whether it is symmetric or has a long tail on one side is the subject of Symmetric and skewed distributions.

Intuition

Figure 1 shows the birth months of the 24 people in Fedor’s class.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Birth month
Figure 1Birth months in Fedor’s class. No month stands out.

The tops of the stacks wobble between 1 and 3 dots, but nothing towers over the rest and no stretch of the year is empty. That is a uniform shape: a wall with a slightly uneven top.

Now picture a herd of two breeds, one smaller and one larger, and the liters of milk each cow gives in a day. The smaller cows crowd around their own typical amount and the larger cows around theirs. On one plot that makes two humps with a dip where neither breed usually lands: a bimodal shape, two hills with a valley between.

The two shapes ask for different reports. For a uniform shape, give the range and say every part of it is about as common as every other. For a bimodal shape, describe each hump, and if you can, split the data into its two groups and look at each one by itself.

Results

In a uniform distribution no single value is typical. Values near the ends of the range are as common as values in the middle.

Fair chance gives a roughly uniform shape, rarely an exactly level one. Equal chances make the counts come out close to equal, with a small wobble above and below the level.

A bimodal shape is a signal to look for two groups. Splitting the data by group usually turns one two-humped plot into two one-humped plots, each with its own center.

A single center of a bimodal distribution can land in the dip, where it describes few or none of the values (Examples 3 and 4). Report each hump instead.

Why it is true

On a fair spinner every outcome is as likely as every other, so over many spins no outcome has a reason to collect more results, and the counts sit near one level. Each spin is still decided by chance, so some counts land a little above that level and some a little below.

In a mixed herd each breed has its own typical amount, and most cows of a breed give amounts close to it, so each breed builds its own hump. When the two typical amounts are far apart compared with the spread inside each breed, the humps sit side by side with a dip between. Splitting by breed removes one hump from each plot.

Worked examples

Example 1

Birth months in one class

Use Figure 1. What shape do Fedor’s class birth months have? Where would the stacks sit if they were perfectly level?

Solution

The stacks for January to December are 2, 3, 2, 2, 1, 2, 3, 2, 2, 2, 1, 2, which add to 24. Shared evenly over 12 months, level stacks would each hold

24 \div 12 = 2 \text{ people.}

Every stack is within one dot of that level: February and July hold 3, May and November hold 1, and the other eight months hold 2. No stack stands clearly above its neighbors, so there is no peak and the shape is uniform. Birthdays in this class are spread about evenly through the year.

Example 2

Forty spins of a fair spinner

Eudora spins a fair spinner with five equal sections, numbered 1 to 5, forty times. The numbers come up 8, 9, 7, 8, 8 times. Describe the shape. Is the number 2 a peak?

Solution

Forty spins shared evenly over five numbers give 40 \div 5 = 8 each. Every count is within one of 8, so a dot plot of the spins is roughly level: a uniform shape, as a fair spinner should give.

The number 2 came up most often, so it is the mode in the strict sense of Mode and bimodal data. It is not a peak: it stands one spin above the level, the size of wobble chance produces.

Example 3

Milk from a mixed herd

Fabian’s herd has cows of two breeds. Figure 2 shows the milk each of his 40 cows gave on one day. A bin holds the amounts from its left edge up to, but not including, its right edge. Describe the shape, and find which bin holds the median.

14 16 18 20 22 24 26 28 30 32 34 36 38 0 2 4 6 8 Number of cows Daily milk of Fabian's 40 cows Milk in one day (liters)
Figure 2Two humps with an empty bin between them.
Solution

The bars rise to a peak of 7 cows at 18 to 20 liters, fall to an empty bin at 26 to 28 liters, then rise to a second peak of 8 cows at 32 to 34 liters. Two clear peaks with a gap between: the shape is bimodal, most likely one hump per breed.

The median of 40 values is the mean of the 20th and 21st in sorted order. Adding the bars from the left gives running totals

1,\ 5,\ 12,\ 17,\ 19,\ 20,\ 20,\ 22,\ \ldots

So the 20th cow is the last in the 24 to 26 bar and the 21st is the first in the 28 to 30 bar. The median is therefore at least

\frac{24 + 28}{2} = 26

and less than \displaystyle \frac{26 + 30}{2} = 28. It lies in the bin from 26 to 28 liters, the one bin with no cow at all. “A typical cow gives about 27 liters” would describe none of Fabian’s cows. The honest summary names the humps: 20 cows give less than 26 liters, most often 18 to 20, and the other 20 give 28 liters or more, most often 32 to 34.

Example 4

Splitting by breed

Feliks wrote down each cow’s milk on one day, in liters. His smaller-breed cows gave 19, 17, 22, 19, 20, 18, 21, 19 and his larger-breed cows gave 33, 29, 36, 33, 31, 34, 32, 33. Find the median of all 16 cows, then split by breed and describe each group.

Solution

All 16 amounts, sorted, are

\begin{aligned} &17,\ 18,\ 19,\ 19,\ 19,\ 20,\ 21,\ 22, \\ &29,\ 31,\ 32,\ 33,\ 33,\ 33,\ 34,\ 36 \end{aligned}

A dot plot would show two clusters with nothing from 23 to 28: a bimodal shape. The median of 16 values is the mean of the 8th and 9th,

\frac{22 + 29}{2} = 25.5 \text{ liters,}

in the empty stretch between the clusters.

Split by breed. The smaller breed sorted is 17, 18, 19, 19, 19, 20, 21, 22, with median \displaystyle \frac{19 + 19}{2} = 19 liters and its tallest stack at 19. The larger breed sorted is 29, 31, 32, 33, 33, 33, 34, 36, with median \displaystyle \frac{33 + 33}{2} = 33 liters and its tallest stack at 33. Each group alone has one peak, and each median sits among its own cows.

The useful report: smaller-breed cows give about 19 liters a day, larger-breed cows about 33. The single figure 25.5 hid both. Splitting worked because Feliks recorded each cow’s breed.

Example 5

Two rushes at a bakery

Fennel recorded the time each of the 60 customers came into her bakery one day (Figure 3). A customer who arrives exactly on the hour is counted in the bar that starts at that hour. Describe the shape. How many customers came from 10{:}00 up to 12{:}00? In which bar is a customer who came at exactly 9{:}00?

7 8 9 10 11 12 13 14 15 0 2 4 6 8 10 12 14 16 Number of customers Arrival times of Fennel's 60 customers Hour of arrival (24-hour clock)
Figure 3A morning rush and a lunchtime rush.
Solution

The bars climb to 14 customers from 8{:}00 to 9{:}00, drop to 3 from 10{:}00 to 11{:}00, climb again to 15 from 12{:}00 to 13{:}00, and fade toward closing. Two clear peaks with a deep dip between: the shape is bimodal, a breakfast crowd and a lunch crowd.

From 10{:}00 up to 12{:}00 covers two bars, so

3 + 7 = 10 \text{ customers.}

A customer at exactly 9{:}00 sits on the edge between two bars, and the rule puts an edge value in the bar that starts there: the 9 to 10 bar.

The shape answers Fennel’s real question, when to have extra help: in the two peak hours, which together bring 14 + 15 = 29 of the 60 customers. One “typical arrival time” would point at the quiet middle of the day.

Example 6

Predicting a shape before collecting data

Predict the shape of each distribution. (a) Eudora glances at a clock at 120 random moments, writes down the minute shown, from 0 to 59, and groups the minutes into six bins of 10. (b) The day of the month, 1 to 31, on which each of many people was born, ignoring leap years.

Solution

(a) The minute hand spends one minute on each of the 60 minutes, so a random moment is equally likely to show any of them. Level bins would hold 120 \div 6 = 20 glances each: expect a roughly uniform histogram, wobbling a little around 20.

(b) Every month has days 1 to 28, so those days have equal chances and their stacks should be roughly level. But a year that is not a leap year has a 29th and a 30th in only 11 of its 12 months, and a 31st in only 7 months. So the stacks at 29 and 30 should be about \displaystyle \frac{11}{12} as tall as the rest, and the stack at 31 about \displaystyle \frac{7}{12} as tall: uniform from 1 to 28, then stepping down. A uniform shape needs equal chances, and the last three days do not have them.

Boundaries

  • “Roughly level” is a judgment. With few values, chance alone makes stacks uneven, so a small data set from a fair spinner can look bumpy. The more data, the more a shape can be trusted.
  • A uniform shape needs equally likely values. Days 29 to 31 in Example 6 break it, and so, by a little, do months, since months differ in length (Exercise 14).
  • The bins shape the picture. Bins too wide can merge two peaks into one flat block, and bins too narrow can break a smooth hump into ragged spikes; choosing them is part of Making a histogram and choosing the bin width.
  • Two close groups make one hump. Two peaks appear only when the groups’ typical values are far apart compared with the spread inside each. Two breeds giving similar amounts of milk would show one wide hump.
  • Two peaks suggest two groups; they do not prove it. The groups have to be found in how the data was collected, as Mode and bimodal data explains, and splitting needs a record of which value belongs to which group.

Where this goes next

Uniform and bimodal are two of the shapes in Shape of a distribution; Symmetric and skewed distributions names the one-peaked shapes, and Mean, median and skew: choosing the center asks which center to trust once a shape is known. The level shape that fair chance produces is the picture behind equally likely outcomes, and What randomness means looks at why chance gives a wobble rather than a perfect wall.

Exercises

Name the shape

Exercises 1 to 4 use Figure 4. Name the shape as uniform, bimodal or neither, and give the value under each peak, if there is one.

1 2 3 4 5 6 7 8 9 Plot A 1 2 3 4 5 6 7 8 9 Plot B 1 2 3 4 5 6 7 8 9 Plot C 1 2 3 4 5 6 7 8 9 Plot D
Figure 4Four shapes to name.
  1. Plot A.
  2. Plot B.
  3. Plot C.
  4. Plot D.

Level stacks

  1. Feliks spins a fair spinner with four equal sections 60 times. The numbers 1, 2, 3, 4 come up 14, 16, 15, 15 times. What would exactly level counts be? Is the shape uniform?
  2. Fennel spins a spinner numbered 1 to 6 thirty times. The numbers come up 2, 9, 3, 8, 2, 6 times. What would level stacks be? Is the shape uniform?
  3. The 36 members of Fedor’s choir have birth months January to December with frequencies 3, 2, 4, 3, 3, 2, 4, 3, 3, 4, 2, 3. Find the level that even stacks would reach, name the shape, and list the strict modes.

Two peaks

  1. Fedor’s herd of 30 cows gave milk on one day in 4-liter bins from 10 to 38 liters, each bin including its left edge but not its right. The bins hold 3, 8, 4, 1, 5, 7, 2 cows. (a) Name the shape and the peak bins. (b) How many cows gave less than 22 liters? (c) One cow gave exactly 22 liters. Which bin is it in?
  2. Eudora wrote down the milk of 14 cows, in liters: 21, 33, 19, 35, 22, 20, 34, 31, 18, 33, 21, 32, 20, 34. (a) Sort the list and describe the shape. (b) Find the median of all 14. (c) Split the list at the gap and find the median of each cluster.
  3. Feliks’s bakery had 40 customers on a weekday. The hour bins from 7{:}00 to 14{:}00 hold 3, 10, 5, 2, 4, 11, 5 customers. (a) Name the shape and the peak hours. (b) How many customers came from 10{:}00 up to 12{:}00? (c) In which bar is a customer who came at exactly 12{:}00?
  4. Fabian’s smaller-breed cows gave 16, 19, 20, 18, 21, 19 liters and his larger-breed cows gave 30, 34, 32, 33, 31 liters. Find the median of each breed and of all 11 cows. Does the median of all 11 describe the larger breed?
  5. The milk of 20 cows, in 2-liter bins from 16 to 32 liters, gives counts 2, 6, 3, 1, 0, 2, 5, 1. (a) Regroup into 4-liter bins. (b) Regroup into 8-liter bins. (c) Which of the three histograms show two peaks?

Predict the shape

  1. Predict the shape, with a reason. (a) The number on a fair spinner numbered 1 to 10, spun 1{,}000 times. (b) The daily milk of every cow at a show where half the cows are of a small breed and half of a much larger breed. (c) The arrival times of customers at a shop that is busy only in the early morning.
  2. Suppose 730 people were born on days spread evenly through a year of 365 days. How many were born in February, in a 30-day month, and in a 31-day month? Is the plot of their birth months exactly uniform?

Explain

  1. A dot plot over nine values has stacks of 4 dots everywhere except one stack of 5. Explain why the shape is uniform even though the data has a mode.
  2. For the 40 customers in Exercise 10, the 20th and 21st arrivals fall in the bars that start at 10{:}00 and 11{:}00. Feliks plans to add a second baker around 11{:}00, “when the typical customer comes”. What does the shape say about that plan?
  3. Explain why splitting a bimodal data set into its groups gives a better report than one center for the whole set.
  4. Write a list of 12 whole numbers from 1 to 6 whose dot plot is exactly level. Then change four of the numbers so that the dot plot becomes bimodal, and give the new stacks.
Answers
  1. Uniform, no peak. The stacks are all 3 or 4.
  2. Bimodal, peaks at 3 and 8. The stacks rise to 6 at 3, dip to 1 at 5 and 6, and rise to 6 again at 8.
  3. Neither: one peak, at 5, with 7 dots, falling away evenly on both sides. Such shapes are named in Symmetric and skewed distributions.
  4. Bimodal, peaks at 1 and 9, with 6 dots each and an empty stack at 5. A peak can sit at the end of the scale.
  5. 60 \div 4 = 15 each. Every count is within one of 15, so the shape is uniform.
  6. 30 \div 6 = 5. The counts run from 2 to 9, far from level, so the shape is not uniform. With only 30 spins this could be chance or an unfair spinner; more spins would tell.
  7. Level 36 \div 12 = 3. Every month has 2, 3 or 4, so the shape is uniform. The strict modes are March, July and October, with 4 each, but none is a peak.
  8. (a) Bimodal, peaks at 14 to 18 liters (8 cows) and 30 to 34 liters (7 cows), with a dip to 1 cow at 22 to 26. (b) 3 + 8 + 4 = 15 cows. (c) The 22 to 26 bin, since a bin includes its left edge.
  9. (a) 18, 19, 20, 20, 21, 21, 22, 31, 32, 33, 33, 34, 34, 35. Two clusters with nothing from 23 to 30: bimodal. (b) The 7th and 8th values give \displaystyle \frac{22 + 31}{2} = 26.5 liters, in the gap. (c) Lower cluster median 20 liters, upper cluster median 33 liters, the 4th of seven in each.
  10. (a) Bimodal, peaks at 8{:}00 to 9{:}00 (10 customers) and 12{:}00 to 13{:}00 (11 customers). (b) 2 + 4 = 6 customers. (c) The bar from 12{:}00 to 13{:}00, which starts at that edge.
  11. Smaller breed sorted 16, 18, 19, 19, 20, 21: median 19 liters. Larger breed sorted 30, 31, 32, 33, 34: median 32 liters. All 11 sorted: 16, 18, 19, 19, 20, 21, 30, 31, 32, 33, 34, and the 6th value is 21 liters. No: 21 is a smaller-breed amount, 9 liters below the lowest larger-breed cow.
  12. (a) 2 + 6 = 8, 3 + 1 = 4, 0 + 2 = 2, 5 + 1 = 6, so the bins hold 8, 4, 2, 6. (b) 8 + 4 = 12 and 2 + 6 = 8. (c) The 2-liter and 4-liter histograms show two peaks; two bars cannot show a dip, so the 8-liter bins hide it.
  13. (a) Roughly uniform, about 100 for each number, since every number is equally likely. (b) Likely bimodal: each breed makes its own hump, far apart. (c) One peak in the early morning, then a fall: neither uniform nor bimodal.
  14. 730 \div 365 = 2 people a day. February: 2 \cdot 28 = 56. A 30-day month: 2 \cdot 30 = 60. A 31-day month: 2 \cdot 31 = 62. Check: 56 + 4 \cdot 60 + 7 \cdot 62 = 730. Not exactly uniform, since months differ in length, but roughly uniform: the stacks run from 56 to 62.
  15. The stack of 5 is the strict mode, but it is only one dot taller than the rest, which is not clearly taller. With no peak and the stacks roughly level, the shape is uniform.
  16. The middle of the data is around 11{:}00, but the bars there hold only 2 and 4 customers. The shape is bimodal, with rushes from 8{:}00 to 9{:}00 and 12{:}00 to 13{:}00, so the second baker is needed then, not in the quiet middle.
  17. Each group crowds around its own typical value, so each group’s center describes its members well. One center for the whole set can fall in the dip and describe nobody, and it hides that there are two groups.
  18. For example 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, two dots on every value. Change both 3s to 2 and both 4s to 5: the list 1, 1, 2, 2, 2, 2, 5, 5, 5, 5, 6, 6 has stacks 2, 4, 0, 0, 4, 2, with peaks at 2 and 5 and an empty gap at 3 and 4.
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