Math 6 · 6.SP.A.2

Symmetric and skewed distributions

Two dot plots can have their tallest stack in the same place and still look nothing alike, one balanced and one leaning to a side. How do you say which way data leans, and why does so much real data lean?

Unlocks

Mean, median and skew: choosing the centerMath 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

After a large earthquake the ground goes on shaking for days, in smaller quakes called aftershocks. Enid counted the aftershocks felt in her town on each of 24 days: on most days none, one or two, on a few days many. The median says where the middle of those counts is, but not how the values are arranged: piled up at one end and thinning out toward the other, or balanced on both sides of the middle.

That arrangement is the shape of the data. It often points back at the thing measured (a day cannot have fewer than 0 aftershocks, and that floor bends the whole picture), and it decides which center describes the data fairly.

Definition

The distribution of a numerical data set is the way its values are spread along the number line: which values occur, and how often. A dot plot or a histogram draws it.

The peak is where the values crowd most, the tallest stack or bar. A tail is a stretch at one end of the distribution where the values thin out, fewer and fewer, moving away from the peak.

A distribution is symmetric when a vertical line can be drawn through the plot so that the two sides are mirror images: folded along the line, each stack lands on a stack of the same height. Real data is almost never exactly symmetric, so the working phrase is roughly symmetric: the two sides nearly match and the two tails are about the same length.
A distribution is skewed right when the values crowd toward the low end and a long tail runs out toward the high values, to the right on the number line.
A distribution is skewed left when the values crowd toward the high end and a long tail runs out toward the low values, to the left.

The name follows the tail, not the peak. A distribution skewed right has its peak on the left.

Intuition

Figure 1 shows the widths of 20 snail shells. Fold the page along the dashed line at 22 mm and the stacks on the left land almost exactly on the stacks on the right: roughly symmetric.

Figure 1Folded along the dashed line at 22 mm, the two sides nearly match.

Figure 2 shows Enid’s aftershock counts. The dots pile up over 0, 1 and 2, then string out to the right, a lone dot at a time, as far as 12. That long thin part is the tail, and it points right, so the data is skewed right.

Figure 2Enid’s aftershock counts. The dots crowd at the left and trail off to the right.

Think of the tail as a pointing finger: “skewed” names the way it points, and the bulk of the data sits at the other end.

Results

The direction of skew is the direction of the long tail. Skewed right: tail to the right, values crowded on the left. Skewed left: the reverse.

A floor close to most of the values tends to make the data skewed right, and a ceiling close to most of the values tends to make it skewed left. A floor is a number the values cannot go below, such as 0 aftershocks; a ceiling is one they cannot go above, such as a score of 20 out of 20.

A few very large values make a long right tail; a few very small ones make a long left tail.

Measuring from the other end flips the shape. Replacing each value v by t - v, for a fixed number t at least as large as every value (words wrong instead of words right), turns skewed left into skewed right and back, and leaves a symmetric shape symmetric.

Why it is true

Floor. Below the floor there are no values, so on that side the plot stops sharply. A value that differs from the crowd near the floor can only differ upward, so the stragglers line up on the high side, and stragglers thinning out are exactly a tail. A ceiling is the same argument turned round.

The floor matters only when the values crowd near it. A shell cannot be less than 0 mm wide, but grown snails’ shells are nowhere near 0 mm wide, so that floor bends nothing (Figure 1).

Flipping. Take t = 20. The rule v \to 20 - v sends 19 to 1, 18 to 2 and 10 to 10: each value lands the same distance from 10, on the other side. So every stack moves to its mirror position across 10 and keeps its height, and the whole plot is reflected left for right. A tail on the left becomes a tail on the right, and a plot that matched its mirror image still does. For any t the mirror line is at \tfrac{t}{2}.

Worked examples

Example 1

Snail shells: a roughly symmetric distribution

Eleni measured the width of the shell of each of 20 garden snails, to the nearest millimeter (Figure 1). Describe the shape of the distribution.

Solution

The tallest stack, 4 snails, sits at 22 mm, and the widths run from 17 mm to 27 mm. The peak is exactly halfway between the ends, since 22 - 17 = 5 and 27 - 22 = 5, so fold at 22 mm and compare the stacks at equal distances on each side:

Distance from 22 mm	Left side	Right side
1	3	3
2	2	3
3	2	1
4	0	0
5	1	1

The left side holds 3 + 2 + 2 + 0 + 1 = 8 snails and the right side 3 + 3 + 1 + 0 + 1 = 8, and stacks at the same distance differ by at most one snail. Both tails reach 5 mm from the peak. The distribution is roughly symmetric about 22 mm; not exactly, since 24 mm has 3 snails and 20 mm has 2.

Check: 8 + 4 + 8 = 20 snails.

Example 2

Aftershocks each day: skewed right

Describe the shape of Enid’s aftershock counts in Figure 2, and give a reason from the situation for that shape.

Solution

The tallest stack is at 1 aftershock (6 days), with 5 days at 0 and 4 days at 2, so 5 + 6 + 4 = 15 of the 24 days had 2 aftershocks or fewer. To the right the stacks shrink: 3 days at 3, 2 at 4, 1 at 5, then single days at 7, 9 and 12. The values crowd at the low end and a long tail runs right, so the distribution is skewed right.

The plot stops sharply at 0 because a day cannot have fewer than 0 aftershocks: a floor, with most of the values right next to it. A day that differs from the usual can only differ upward, and a handful of busy days make the long tail.

Check: 5 + 6 + 4 + 3 + 2 + 1 + 1 + 1 + 1 = 24 days.

Example 3

Spelling scores: the peak is on the right, the skew is left

Eoin’s class of 25 took a test of 20 spelling words; Figure 3 shows how many words each child spelled right. Describe the shape, and say why it is named for the left although the tallest stack is near the right.

Figure 3Eoin’s class. The crowd is at the right and the tail runs left.

Solution

The tallest stack is at 19 words (6 children), and 5 + 6 + 4 = 15 children scored 18, 19 or 20. Moving left from 18 the stacks shrink: 3 children at 17, 2 at 16, 2 at 15, then single children at 14, 13 and 11. The values crowd at the high end and a long thin tail runs left as far as 11, so the distribution is skewed left.

The peak says where most children are; the skew names where the stragglers are, and here they trail away to the left. The plot stops sharply at 20 because no one can spell more than 20 of 20 words right: a ceiling, with most of the scores close to it.

Check: 15 + 3 + 2 + 2 + 1 + 1 + 1 = 25 children.

Example 4

Aftershock sizes on a histogram

Ermin listed the magnitudes (a measure of how strong a quake is) of the 30 aftershocks of magnitude 2.0 or more after another earthquake, and grouped them into the histogram in Figure 4. Each bar includes its left edge and not its right edge, so an aftershock of magnitude exactly 3.0 is counted in the bar from 3.0 to 3.5. Describe the shape and give a reason for it.

Figure 4Ermin’s aftershocks, grouped by magnitude. Each bar holds its left edge but not its right.

Solution

Reading the bars from the left: 11, 8, 5, 3, 2, 1. The tallest bar is at the low end and each bar is shorter than the one before, a long tail stepping down to 5.0, so the distribution is skewed right. A histogram hides single values, but its bar heights still show where values crowd and where they thin out, which is all that shape needs.

Here 11 + 8 = 19 of the 30 aftershocks were below magnitude 3.0, and only 2 + 1 = 3 were 4.0 or above: many small aftershocks and a few large ones, and the few large values make the long right tail.

Check: 11 + 8 + 5 + 3 + 2 + 1 = 30.

Example 5

Counting mistakes instead of marks

For Eoin’s class in Example 3, count how many words each child spelled wrong. Give the frequencies and describe the shape.

Solution

Each child spelled 20 words, so a child who scored s spelled 20 - s wrong. The 4 children who scored 20 have 0 wrong, the 6 who scored 19 have 1 wrong, and so on. For 0, 1, 2, 3, 4, 5, 6, 7 and 9 words wrong the frequencies are 4, 6, 5, 3, 2, 2, 1, 1 and 1.

Now 4 + 6 + 5 = 15 children crowd at 0, 1 and 2 wrong, and the tail runs right to 9. Same children, same test, but this distribution is skewed right. The rule s \to 20 - s reflects the plot across 10, so the tail swaps sides, and the ceiling of 20 right becomes a floor of 0 wrong. A shape word only means something next to what was measured.

Check: 4 + 6 + 5 + 3 + 2 + 2 + 1 + 1 + 1 = 25 children.

Boundaries

Shape needs numbers on a number line. The bars of a bar graph of colors or kinds of shell can be put in any order, so “skewed right” means nothing there (Types of data: categorical, numerical, discrete, continuous).
A shape word needs enough values to show a pattern. With five or six values, one more value can turn a tail into a lump, so describe the values instead.
“Roughly symmetric” is a judgment, not a calculation. When the call is close, say what is there: “nearly symmetric, with a slightly longer tail on the right”.
These three words do not cover every shape. Flat and two-peaked distributions are named in Uniform and bimodal shapes, and a two-peaked one can still be symmetric (Exercise 19).
A floor or a ceiling tends to make skew; it does not force it. The values must crowd near it (Exercise 16).
On a histogram, very wide bars can swallow a tail into one bar.

Where this goes next

Shape tells you which center to trust: in a skewed distribution the long tail pulls the mean toward it while the median stays with the crowd, which Mean, median and skew: choosing the center works through. With Uniform and bimodal shapes, this lesson completes the words for the Shape of a distribution. A box plot often shows skew too, as a longer whisker or a longer half of the box on the tail side, but it hides peaks and gaps, so read shape from a dot plot or histogram when you have one. Later, in statistics, smooth curves stand in for histograms, and the most important of them is the symmetric, bell-shaped The normal distribution.

Exercises

Name the shape

Estela measured the height of the shell of each of 20 snails, to the nearest millimeter. For heights 10, 11, 12, 13, 14, 15 and 16 mm the frequencies were 1, 2, 4, 6, 4, 2 and 1. Describe the shape of the distribution.
Enid counted aftershocks on 14 more days: 0, 0, 1, 0, 2, 0, 1, 0, 0, 3, 1, 0, 6, 1. Describe the shape and give a reason from the situation.
Sixteen children took a test of 10 spelling words. Their scores were 10, 9, 10, 8, 10, 9, 7, 10, 9, 4, 10, 8, 9, 6, 10, 9. Describe the shape.
Eleni lifted 18 flowerpots and counted the snails under each. For 0, 1, 2, 3, 4, 5, 6 and 7 snails the frequencies were 2, 3, 5, 4, 2, 1, 0 and 1. Describe the shape.

Reading a plot

Eleni weighed 40 snails and drew Figure 5. Estela’s class of 22 took a test of 12 bonus spelling words, and Figure 6 shows each child’s score.

Figure 5Eleni’s snails, grouped by weight. Each bar holds its left edge but not its right.

Figure 6Estela’s class and the 12 bonus words.

Describe the shape of Figure 5.
Give new heights for the bars of Figure 5, still 40 snails in all, that would make the histogram skewed right.
Describe the shape of Figure 6 and give a reason from the situation.
For each child in Figure 6, find the number of bonus words spelled wrong. Give the frequencies and describe the shape.

Predict the shape

Say whether you expect each distribution to be roughly symmetric, skewed right or skewed left, and why.

The number of aftershocks felt each day, for every day of the year after an earthquake.
Scores on a test of 20 very hard spelling words, on which most children spelled only a few right.
The widths of the shells of 50 grown snails of one kind from one garden.
Scores on a test of 20 spelling words that the class practiced all week.

Changing the data

Every snail in Exercise 1 grows until its shell is exactly 2 mm taller. Describe the new distribution.
Remove the day with 6 aftershocks from Exercise 2. Is the data still skewed right?

Explaining

Explain why skew is named by the tail and not by the peak. What would go wrong if Figure 3 were named for the side its peak is on?
A test of 20 spelling words has a ceiling: no score can be above 20. Can its scores still be skewed right? Explain.
A data set is exactly symmetric about 10. Each value v is replaced by 20 - v. Explain why the new dot plot is exactly the same as the old one.
Explain why data measured from real snails or real earthquakes is almost never exactly symmetric, and why “roughly symmetric” is still worth saying.
Write a data set of 8 whole numbers that is exactly symmetric about 3 and has two peaks.

Answers

Symmetric about 13 mm, and exactly so, which real data seldom is. The frequencies 1, 2, 4, 6, 4, 2, 1 read the same from either end, so folding at 13 puts 12 on 14 (4 each), 11 on 15 (2 each) and 10 on 16 (1 each).
Skewed right. Sorted: 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 6. Seven days at 0, four at 1, then single days at 2, 3 and 6: crowded at 0 with a tail running right. A day cannot have fewer than 0 aftershocks, and most days sit on that floor.
Skewed left. Sorted: 4, 6, 7, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10. Six children scored 10 and five scored 9, then the stacks thin out to the left: two at 8 and single children at 7, 6 and 4. Most scores sit near the ceiling of 10.
Skewed right. The peak is at 2 snails (5 pots). On the left the plot stops at 0, two steps from the peak; on the right it thins out, 4, 2, 1, then 0 pots at 6 and 1 pot at 7, five steps from the peak. Check: 2 + 3 + 5 + 4 + 2 + 1 + 0 + 1 = 18 pots.
Roughly symmetric about 11 grams, the middle of the tallest bar. The bars left of it are 9, 5, 2 and those right of it 8, 4, 2: nearly mirror images.
One answer: 12, 10, 7, 5, 3, 2, 1 from left to right, since 12 + 10 + 7 + 5 + 3 + 2 + 1 = 40. The tallest bar is at the low end and the bars step down slowly to the right. Any heights that crowd at the left and thin out to the right will do.
Skewed left. The tallest stack is at 11 words (5 children), with 4 at 10 and 3 at 12; to the left the stacks shrink, 3 at 9, 2 at 8, 2 at 7, and single children at 6, 5 and 3. No one can score above 12, and most scores sit near that ceiling.
Words wrong = 12 - \text{score}. For 0, 1, 2, 3, 4, 5, 6, 7 and 9 words wrong the frequencies are 3, 5, 4, 3, 2, 2, 1, 1 and 1, which add to 22. Skewed right: crowded at 0 to 2 wrong, with a tail running right to 9. It is Figure 6 reflected across 6.
Skewed right. A day cannot have fewer than 0 aftershocks, most days have none or a few, and a handful of busy days make a long tail toward large counts.
Skewed right. The scores crowd near the floor of 0 and thin out upward. The ceiling of 20 is far from the crowd, so it bends nothing.
Roughly symmetric. Grown shells of one kind gather round a typical width, with no floor or ceiling near the crowd to push the values to one side (compare Figure 1).
Skewed left. Most children spell most of the words right, so the scores crowd near the ceiling of 20 and a few lower scores trail off to the left.
Still symmetric, now about 15 mm: heights 12 to 18 mm with the same frequencies 1, 2, 4, 6, 4, 2, 1. Adding 2 slides every dot 2 mm right and changes no stack (Shifting and rescaling a data set).
Yes. What is left is 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3. The tail is shorter, but the values still crowd on the floor and thin out to the right.
Every one-peaked shape has a crowd somewhere; what tells shapes apart is where the stragglers go, and that is the tail. Named for its peak, Figure 3 would be “right”, and a reader would picture a crowd on the left with a tail running right: the mirror image of the real plot.
Yes. A ceiling bends the data only when the scores crowd near it. On a very hard test the scores crowd near the floor of 0 and thin out upward, so they are skewed right.
Replacing v by 20 - v moves each value to its mirror position across 10. By symmetry, the stack at any value w is as tall as the stack at 20 - w. The new stack at w is the old stack from 20 - w, so it is as tall as the old stack at w. Every stack is unchanged.
Each value comes from a separate snail or day, and nothing makes the count at one value match the count at its mirror value exactly; one extra snail on one side breaks the match. “Roughly symmetric” still says something true: the values balance about a middle with no long tail on either side, unlike Figures 2 and 3.
One answer: 1, 1, 1, 2, 4, 5, 5, 5. About 3, the value 1 pairs with 5 (three of each, both 2 away) and 2 pairs with 4 (one of each), so the plot matches its mirror. The tallest stacks, at 1 and 5, are the two peaks.