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Math 6 · S.ID.A.3

Mode and bimodal data

Jelly bean colors cannot be added up or put in order, yet “which color is most common?” still has an answer. What is that answer, and what does it mean when two values stand out at once?

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Uniform and bimodal shapesMath 6
Where you are 2 in Median and mode
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Center and spread13
Median and mode2
Finding the median
Mode and bimodal data

Why this exists

Asa tips a bag of jelly beans onto the table and sorts them by color. What color is a typical bean? The mean cannot answer: there is no adding red to green and sharing the total out. The median cannot answer either: colors have no order, so there is no middle color. Yet the question has a plain answer, the color that turns up most often.

Numbers have a most common value too: an answer that really was given, more often than any other. And when two values stand out at once, far apart, that is news in itself, because it often means the data holds two groups mixed together.

Definition

The frequency of a value is the number of times it appears in the data.

The mode of a data set is the value, or category, with the highest frequency.

  • If two or more values tie for the highest frequency, each of them is a mode.
  • If the data holds at least two different values and every value has the same frequency, no value stands out, and the data has no mode.
  • Data with exactly two modes is called bimodal, meaning “two modes”. The word is also used more loosely, for data whose plot shows two separate peaks, each standing well above the values around it, even when one peak is a little taller than the other. Example 6 shows such a case.

The mode is a value, never a count. If 7 shells is the answer given most often, and it was given 3 times, then the mode is 7 shells and 3 is its frequency.

Both kinds of data from Types of data: categorical, numerical, discrete, continuous have a mode: for categories it is a category, for numbers it is a number.

Intuition

On a dot plot each value is a dot stacked above its place on the line, so the height of a stack is that value’s frequency, and the mode sits under the tallest stack. Figure 1 shows how many shells each of eleven children collected on a beach walk.

0 1 2 3 4 5 6 7 8 9 10 Shells collected by one child
Figure 1One dot for each child. The tallest stack, at 7 shells, marks the mode.

On a bar graph the height of a bar is its category’s frequency, so the mode is the category under the tallest bar. In a frequency table the mode is the value written beside the largest frequency.

The mean and the median answer “where is the middle of the data?” The mode answers a different question: “which single value is most common?” Different questions often have different answers (Example 4).

When a plot has two tall stacks with an empty or nearly empty stretch between them, each stack is usually the center of its own group (Figure 3 and Example 5). Naming the shapes a plot can take is the work of Uniform and bimodal shapes; this lesson is about what the peaks say.

Results

The mode is always a value that appears in the data. The mean and the median need not be.

The mode is the only one of the three centers that works for categories with no order, such as colors or kinds of shell.

A data set can have one mode, several modes, or no mode.

The mode depends only on the frequencies. How far apart the values are never enters it, so the mode can sit at the very end of the data, far from the mean and the median.

Two separate peaks often signal two groups mixed together. Then a single middle can fall in the gap between the peaks, where few or no values lie, and the two peaks describe the data better than any one center.

Why it is true

The mode is picked from the values that appear, so it is always one of them. A mean shares out a sum and a median is the middle of a sorted list: a sum needs numbers and a middle needs an order. Colors have neither, but counting the beans of each color needs only telling one color from another, so data of any kind has frequencies.

A tie at the top gives several modes, and two or more different values all sharing one frequency leave nothing on top.

Counting asks “is this value the same as that one?”, never “how far apart are they?” So the list 0, 0, 0, 1, 6 and the list 0, 0, 0, 10, 60 have the same mode, 0, though one runs to 6 and the other to 60.

If the data mixes two groups, such as young children and their grown-ups, each group crowds around its own typical value, so each makes its own peak. The mean is the balance point of all the values, so with two clusters it lands between them, in the gap.

Worked examples

Example 1

Shells on a beach walk

Eleven children on a beach walk counted the shells each one collected:

4,\ 7,\ 3,\ 7,\ 5,\ 9,\ 7,\ 2,\ 5,\ 6,\ 4

Find the mode.

Solution

Sorting puts equal values side by side, which makes them easy to count:

2,\ 3,\ 4,\ 4,\ 5,\ 5,\ 6,\ 7,\ 7,\ 7,\ 9

The values 4 and 5 each appear twice, 7 appears three times, and every other value once. The highest frequency is 3, and it belongs to 7, so the mode is 7 shells: more children collected 7 shells than any other number. Figure 1 shows the same data, with the tallest stack at 7.

Check: the frequencies 1 + 1 + 2 + 2 + 1 + 3 + 1 = 11 account for every child once.

Example 2

Colors in a bag of jelly beans

Asa sorted a bag of jelly beans by color (Figure 2). Find the mode, and explain why the bag has no mean color and no median color.

0 2 4 6 8 10 12 9 red 6 green 11 yellow 7 orange 4 purple 3 white Number of beans Color
Figure 2Asa’s bag of jelly beans, sorted by color.
Solution

The tallest bar is yellow, at 11 beans; the next tallest is red, at 9. So the mode is yellow. The answer is a color: 11 is the frequency of yellow, not the mode.

There is no mean color because a mean adds the values, and red plus green is not anything. There is no median color because the colors have no natural order. Lined up red, green, yellow, orange, purple, white, or lined up in alphabetical order, the beans would put different colors in the middle, so “the middle color” has no fixed meaning. The mode needs only counting, so it is the center that works here.

The bag holds 9 + 6 + 11 + 7 + 4 + 3 = 40 beans, and yellow is 11 of them.

Example 3

No mode, two modes, one mode

Arvid measured six shells, in centimeters:

3.5,\ 4.2,\ 2.8,\ 5.1,\ 3.9,\ 4.6

(a) Find the mode. (b) He adds two more shells, of 4.2 cm and 3.5 cm. Find the mode of the eight lengths. (c) He adds one more shell, of 4.2 cm. Find the mode of the nine.

Solution

(a) Sorted, the lengths are 2.8,\ 3.5,\ 3.9,\ 4.2,\ 4.6,\ 5.1. Six different values each appear once, so all frequencies are equal and nothing stands out: there is no mode.

(b) Now the sorted list is 2.8,\ 3.5,\ 3.5,\ 3.9,\ 4.2,\ 4.2,\ 4.6,\ 5.1. Both 3.5 and 4.2 appear twice and every other length once. That is a tie at the top, so there are two modes, 3.5 cm and 4.2 cm, and the data is bimodal in the strict sense.

(c) With a third 4.2, that length appears three times and stands alone at the top. The mode is 4.2 cm.

One extra shell changed the answer each time. In a small data set the mode can appear, split or move with a single value.

Example 4

Beach visits in a frequency table

Astor asked 20 classmates how many times they went to the beach this summer.

Visits0123456
Classmates7332212

Find the mode, and compare it with the median and the mean.

Solution

The largest frequency is 7, beside 0 visits, so the mode is 0 visits.

For the median, picture the 20 answers sorted: 7 zeros fill places 1 to 7, three ones fill places 8 to 10, and three twos fill places 11 to 13. The middle pair is the 10th and 11th values, 1 and 2, so the median is

\frac{1 + 2}{2} = 1.5 \text{ visits.}

For the mean, multiply each number of visits by its frequency to get the total, as in Mean from a frequency table and weighted mean:

\begin{aligned} &0 \cdot 7 + 1 \cdot 3 + 2 \cdot 3 + 3 \cdot 2 \\ &\quad + 4 \cdot 2 + 5 \cdot 1 + 6 \cdot 2 = 40 \end{aligned}

Shared among 20 classmates, that is 40 \div 20 = 2 visits.

Three centers, three answers: mode 0, median 1.5, mean 2. The mode says the single most common answer was “not at all”, even though 20 - 7 = 13 of the 20 classmates did go. It sits at the very end of the data, because it looks only at frequencies.

Example 5

Ages on a rock-pool walk

Aurel led a rock-pool walk. Figure 3 shows the ages of the 20 people on it. Find the modes, the median and the mean, and say what the data suggests.

0 5 10 15 20 25 30 35 40 Age in years
Figure 3Two clusters, a peak in each, and an empty gap between them.
Solution

Read from the plot, the sorted ages are 4, 5, 5, 6, 6, 6, 6, 7, 7, 8 in the left cluster and 31, 33, 34, 34, 35, 35, 35, 35, 36, 38 in the right one. The stacks at 6 and at 35 each hold 4 dots, and no other stack holds more than 2. So there are two modes, 6 years and 35 years, and the data is bimodal.

The median is the mean of the 10th and 11th sorted values, the last of the left cluster and the first of the right:

\frac{8 + 31}{2} = 19.5 \text{ years.}

The left cluster adds to 60 and the right to 346, so the mean is

\frac{60 + 346}{20} = \frac{406}{20} = 20.3 \text{ years.}

Both centers land near 20, in the gap where nobody is. “A typical person on the walk is about 20” describes no one there. The two modes tell the real story: a group of children around 6 and a group of grown-ups around 35, most likely children with their parents or carers. When two peaks sit this far apart, report each group.

Example 6

Two humps but one mode

Armand emptied a bucket of shells and measured each one to the nearest centimeter (Figure 4). Find the mode. Is there anything else in the plot worth reporting?

0 1 2 3 4 5 6 7 8 9 10 11 Length to the nearest centimeter
Figure 4One mode, at 3 cm, but two humps.
Solution

Reading the stacks from 1 cm to 10 cm gives frequencies 2, 6, 9, 4, 1, 2, 5, 7, 3, 1, which add to 40. The highest is 9, at 3 cm, and no other length ties it, so there is exactly one mode, 3 cm. In the strict sense the data is not bimodal.

But the stacks fall from 9 to just 1 at 5 cm, then climb again to 7 at 8 cm before falling. That is a second peak, separated from the first by a dip, which is the looser sense of bimodal from the Definition. Two humps like this often mean two groups, and when Armand sorts the bucket by kind he finds mostly periwinkles among the small shells and whelks among the large ones.

Reporting only “the mode is 3 cm” would hide the whelks entirely, though the upper hump from 6 cm to 10 cm holds 2 + 5 + 7 + 3 + 1 = 18 of the 40 shells.

Boundaries

  • The mode needs repeats. Careful measurements rarely repeat exactly, and when every value appears once there is no mode. Rounding, as Armand did, creates repeats, but then the mode depends on the rounding chosen: the same shells rounded to the nearest centimeter and to the nearest half centimeter can have different modes.
  • The mode sees only frequencies. It says nothing about spread or about where the other values lie, and it can sit at an end of the data (Example 4).
  • In a small data set the mode is fragile. One value can create a mode, split it or move it (Example 3).
  • Two peaks suggest two groups; they do not prove it. The groups have to be found in how the data was collected, such as children and grown-ups, or two kinds of shell. A small data set can show two bumps by chance.
  • Most common is not most. In Asa’s bag yellow is the mode, yet 40 - 11 = 29 of the 40 beans are not yellow.

Where this goes next

The mode is one of three centers; Mean as balance point and Finding the median build the other two, and Mean, median and skew: choosing the center weighs which center to trust when they disagree. Peaks are the first thing to notice about the Shape of a distribution, and in Uniform and bimodal shapes a plot with two peaks gets its name. In Histograms, where values are grouped into bins, the tallest bar marks the most common bin rather than a single value.

Exercises

The mode of a list

Find the mode or modes, or say that there is no mode.

  1. 3, 8, 5, 8, 2, 6, 8, 5
  2. 12, 15, 11, 14, 13
  3. 0.5, 1.5, 1.0, 0.5, 2.0, 1.5, 2.5
  4. Astor counted the shells he found on each of nine walks: 14, 9, 11, 9, 14, 12, 9, 13, 10.
  5. Seven friends each ate part of a bag of jelly beans. The fractions of a bag they ate were \displaystyle \frac{3}{4}, \frac{1}{2}, \frac{3}{4}, \frac{5}{8}, \frac{1}{2}, \frac{3}{4}, \frac{7}{8}.
  6. Six jars each hold exactly 25 jelly beans. What is the mode of the six counts?

Categories

Exercises 8 and 9 use Figure 5.

0 2 4 6 8 10 12 14 8 cockle 12 mussel 5 whelk 12 scallop 7 limpet Number of shells Kind of shell
Figure 5The shells Arvid’s class found on the beach, sorted by kind.
  1. Asa pulls ten jelly beans from a jar: red, green, red, white, yellow, green, red, purple, green, red. Find the mode.
  2. Find the mode of the kinds of shell Arvid’s class found.
  3. Arvid’s class goes back to the beach and finds 5 more cockles. What is the mode now?

Frequency tables

  1. Aurel counted the families on his rock-pool walks by size. Families of 2, 3, 4, 5 and 6 people had frequencies 5, 9, 9, 4 and 1. Find the mode or modes.
  2. One more family of 4 people joins the count in Exercise 10. Find the mode now.
  3. Asa counted the green jelly beans in each of 30 small bags. Bags with 0, 1, 2, 3, 4 and 5 green beans had frequencies 6, 3, 6, 5, 6 and 4. Find the mode or modes.

Two peaks

  1. The ages of the people on an afternoon rock-pool walk were 5, 34, 6, 7, 31, 6, 36, 5, 33, 6, 35, 8, 34, 7, 34, 4. (a) Find the mode or modes. (b) How many people were aged from 10 to 30?
  2. Two machines fill bags of jelly beans. In 26 bags, the counts 48, 49, 50, 51, 55, 56, 57 and 58 beans had frequencies 1, 3, 6, 2, 2, 5, 6 and 1. Find the modes, and suggest a reason the data has two peaks.
  3. Armand weighed 31 shells from another bucket, to the nearest gram. The weights 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 grams had frequencies 1, 4, 7, 3, 1, 0, 2, 5, 6 and 2. Find the mode. Is there a second peak, and where?

Explain

  1. Explain why a set of jelly bean colors has a mode but no mean and no median.
  2. A frequency table of shells has a column “kind of shell” and a column “number of shells”. Explain why the mode is always read from the first column, never from the second.
  3. Can the mode be larger than every other value in a data set? Give an example or explain why not.
  4. Write a list of six whole numbers that has no mode, such that changing just one number gives exactly two modes. Then explain why a list of six different numbers could never do this.
  5. For the ages in Exercise 13 the mean is about 18. Explain why that is a poor summary, and what to report instead.
  6. A sweet shop asks two questions: “Which color of jelly bean should we order most of?” and “About how many beans does a bag hold?” Which center suits each question, and why?
Answers
  1. Mode 8. Sorted: 2, 3, 5, 5, 6, 8, 8, 8. The 8 appears three times, the 5 twice.
  2. No mode. Five different values each appear once.
  3. Modes 0.5 and 1.5. Sorted: 0.5, 0.5, 1.0, 1.5, 1.5, 2.0, 2.5. Both appear twice, the rest once.
  4. Mode 9 shells. Sorted: 9, 9, 9, 10, 11, 12, 13, 14, 14. The 9 appears three times, the 14 twice.
  5. Mode \displaystyle \frac{3}{4} of a bag. It appears three times, \displaystyle \frac{1}{2} twice, the others once.
  6. Mode 25 beans. It is the only value, so it is the most frequent; “no mode” needs at least two different values with equal frequencies. Some books call this case “no mode”; this lesson follows its Definition.
  7. Mode red. Red appears 4 times, green 3 times, and white, yellow and purple once each.
  8. Two modes, mussel and scallop, with 12 shells each. The next tallest bar is cockle, at 8.
  9. Mode cockle. Cockles now number 8 + 5 = 13, more than the 12 mussels or the 12 scallops.
  10. Modes 3 and 4 people, with 9 families each.
  11. Mode 4 people. Families of 4 now number 9 + 1 = 10, above the 9 families of 3.
  12. Modes 0, 2 and 4 green beans, each in 6 bags. Check: 6 + 3 + 6 + 5 + 6 + 4 = 30.
  13. (a) Modes 6 and 34 years, three times each. Sorted: 4, 5, 5, 6, 6, 6, 7, 7, 8, 31, 33, 34, 34, 34, 35, 36. (b) 0 people: nobody is between 8 and 31. The walk held children and grown-ups.
  14. Modes 50 and 57 beans, 6 bags each. The counts fall into two clusters, 48 to 51 and 55 to 58, with no bags from 52 to 54. A likely reason: each machine fills to its own amount, one near 50 beans and the other near 57.
  15. Mode 4 grams, with 7 shells. Yes: after falling to 0 at 7 grams the frequencies climb to 6 at 10 grams, a second peak. Check: 1 + 4 + 7 + 3 + 1 + 0 + 2 + 5 + 6 + 2 = 31. The bucket probably holds two kinds of shell.
  16. A mean adds the values and a median needs them in order. Colors cannot be added and have no natural order, so neither exists. The mode needs only counting the beans of each color, which always works.
  17. The mode answers “which kind of shell is most common?”, so its answer is a kind. The second column holds frequencies: they are compared to choose the kind, but a frequency says how many, not which.
  18. Yes. In 1, 2, 9, 9 the mode is 9, larger than both other values. The mode depends only on frequencies, not on where a value sits.
  19. For example 1, 1, 2, 2, 3, 3: every value appears twice, so there is no mode. Change one 3 to 4 to get 1, 1, 2, 2, 3, 4, where 1 and 2 appear twice and the rest once: modes 1 and 2. With six different numbers, changing one to equal another makes exactly one pair, so exactly one mode; changing it to a new value keeps all six different, so no mode. Two modes never appear.
  20. The mean is 291 \div 16, about 18.2, but nobody on the walk is between 8 and 31, so it describes no one. Report the two groups instead: 9 children aged 4 to 8, most often 6, and 7 grown-ups aged 31 to 36, most often 34.
  21. The color question needs the mode: color is a category, the mode is its only center, and the most common color is the one to order most of. The bag question is about a number, so the mean or the median gives a typical count; the mode may sit at an end of the data or come with a tie.
kg1128 · updated 2026-10-13 · CC-BY-SA-4.0