D. Statistical Analysis · One-Variable Data Analysis

The Shape of a Distribution: Symmetry, Skewness and Modes

D1.3 D1.5About 65 minutes

Learning goals

I can describe the shape of a distribution as symmetric, uniform, skewed right, skewed left, bimodal or U-shaped.
I can predict how the mean and median compare from the shape, and predict the shape from the mean and median.
I can explain real-world reasons that a variable has a particular shape.
I can choose the measure of centre that best describes a distribution of a given shape.

Words to know

distribution
The pattern of values a variable takes and how often it takes them, shown by a frequency table or graph.

symmetric
A distribution whose left and right halves are approximately mirror images. The mean and median are about equal.

skewed right (positively skewed)
A distribution with a long tail on the right (a few large values). Usually mean > median.

skewed left (negatively skewed)
A distribution with a long tail on the left (a few small values). Usually mean < median.

modal class
The class with the highest frequency in a grouped distribution. One clear peak is unimodal; two clear peaks is bimodal.

tail
The thin part of a distribution far from the centre, where values are rare.

Every distribution has a personality

Picture three histograms: the marks on an easy quiz, household incomes in Ontario, and the heights of everyone at a Grade 1 class's family picnic. The first piles up near 100% with a few low marks trailing off to the left. The second piles up at moderate incomes with a long tail of very high incomes stretching to the right. The third has two peaks: one for the six-year-olds and one for the adults.

The shape of a distribution is the first thing a statistician looks at, because it tells you which numbers to trust. A mean is a fine summary of a symmetric distribution and a poor one for a skewed one. The curriculum asks you to "describe the characteristics of a one-variable data set", and shape is one of the four big characteristics, along with centre, spread and unusual values.

Six common shapes. The tail is the long, thin side: a skewed-right distribution has its tail on the right, even though its peak is on the left.

The shape vocabulary

Symmetric: the two halves are mirror images. A mound-shaped (bell-shaped) symmetric distribution is very common for measurements such as heights of adults of one sex or masses of manufactured products. The normal distribution (lesson 11) is the ideal mound.
Uniform: all classes have about the same frequency, like the results of rolling one fair die many times.
Skewed right (positive skew): a long tail to the right. Common when a variable has a floor at zero but no ceiling: incomes, house prices, waiting times, part-time work hours.
Skewed left (negative skew): a long tail to the left. Common when there is a ceiling: marks on an easy test (no one can score above 100%), or the age at which people retire.
Bimodal: two peaks, often a sign that the data mixes two different groups (children and adults; weekday and weekend sales). Split the groups and analyse them separately.
U-shaped: peaks at both ends and few values in the middle, such as responses to a strongly divisive survey question.

If you are naming a skew, name the side of the tail, not the side of the peak.

Remember the skew by the tail

Think of a skewed distribution as a whale: the body is the peak and the long, thin tail points in the direction of the skew. Tail to the right → skewed right.

Shape, mean and median

The median depends only on the middle of the ordered list. The mean uses every value, so a few extreme values in a long tail pull the mean toward the tail.

Symmetric: mean ≈ median ≈ mode.
Skewed right: mode < median < mean.
Skewed left: mean < median < mode.

This works in reverse too: if a report says the mean income is much greater than the median income, you can infer the incomes are skewed right. (These are typical patterns, not laws: unusual distributions, especially discrete ones with few values, can break them.)

Worked example

Example 1: A skewed-right distribution

Fifty students reported their weekly part-time work hours. Classes [0, 4), [4, 8), [8, 12), [12, 16), [16, 20), [20, 24), [24, 28) have frequencies 14, 12, 10, 6, 4, 2, 2. Describe the shape, and estimate the mean and median to check the rule.

Shape: the frequencies are highest at the left and fall steadily, with a long thin tail to the right. The distribution is skewed right, with modal class [0, 4).
Mean (estimated with midpoints 2, 6, 10, ..., 26): (2·14 + 6·12 + 10·10 + 14·6 + 18·4 + 22·2 + 26·2) ÷ 50 = 452 ÷ 50 ≈ 9.0 h. (Lesson 5 explains this method.)
Median: the 25th and 26th values. The cumulative frequency is 14 after the first class and 26 after the second, so both are in [4, 8). Using an ogive-style estimate: 4 + (25 − 14)/12 × 4 ≈ 7.7 h.
Check: modal class (0 to 4 h) < median (7.7 h) < mean (9.0 h) ✓, as expected for right skew.

Skewed right. The few students who work 20 or more hours pull the mean up to about 9.0 h, while a typical student (the median) works about 7.7 h.

In a skewed-right distribution the long tail pulls the mean (about 9.0 h) to the right of the median (about 7.7 h), which is to the right of the mode class.

Worked example

Example 2: Reasoning from summary statistics

For each situation, predict the shape and explain. (a) A test has mean 61% and median 70%. (b) Commute times have mean 24 min and median 18 min. (c) The masses of 500 bags of flour have mean 1 002 g and median 1 002 g.

(a) Mean < median: a few very low marks pull the mean down. Skewed left.
(b) Mean > median: a few very long commutes pull the mean up. Skewed right (commutes have a floor at 0 but no fixed ceiling).
(c) Mean = median: consistent with a symmetric distribution, likely mound-shaped, because filling machines make small random errors in both directions.

(a) skewed left, (b) skewed right, (c) symmetric. Always check with a graph when you can: the same mean and median can come from different shapes.

Which centre should you report?

Symmetric data: the mean is the usual choice (it uses all the data and works with the standard deviation).
Skewed data or data with outliers: the median describes a typical value better. That is why Statistics Canada often reports median household income, and why careful reports on house prices give a median or a typical (benchmark) price, not just the average.
Bimodal data: neither one is a good summary. Report the two groups separately.
Categorical data: only the mode makes sense.

Just for fun

A statistician walks into a bar with a billionaire. Suddenly the mean income in the room is enormous, and the median has no idea what all the fuss is about.

Hands-on activity

Dice shapes

You need: two ordinary six-sided dice (from a board game) or a dice-rolling app, paper and a pencil.

Roll one die 60 times and make a frequency table and histogram of the results (1 to 6).
Roll two dice 60 times and record the sum (2 to 12). Make a frequency table and histogram.
Roll two dice 60 times and record the larger of the two numbers (if they match, record that number). Make a third histogram.
Describe each shape with this lesson's vocabulary, and give a reason for each shape.

What to look for: one die gives a roughly uniform shape (each face about 10 times). The sum of two dice gives a symmetric, mound-shaped (triangular) shape peaking at 7, because there are more ways to make 7 than 2 or 12. The larger of two dice is skewed left, with 6 most common, because the larger number is 6 whenever either die shows 6. With only 60 rolls your graphs will be bumpy; the overall pattern is what matters.

Your turn

A distribution has a long tail on the left. Name its skew and predict how the mean and median compare.

The ages of people at a Grade 1 class's family picnic give a histogram with two peaks. Name the shape and give a reason.

Predict the shape of the distribution of the number of goals scored per game by a hockey team over a season, and explain.

A data set has mean 48, median 48 and mode 48. Is it necessarily symmetric?

Which measure of centre would you report for house prices in a city? Why?

Is the distribution of the last digit of many randomly chosen phone numbers likely to be symmetric, skewed or uniform?

More practice: warm-up

Which side is the tail on in a positively skewed distribution?

In a skewed-right distribution, which is usually larger, the mean or the median?

What is the name for a distribution with all bars about the same height?

What shape is the sum of two dice?

What measure of centre is the only one that works for categorical data?

More practice: on your own

Frequencies for classes of reaction times are 2, 5, 9, 14, 16. Describe the shape and predict how the mean and median compare.

A news article says, "The average (mean) salary at the company is $95 000." The median is $58 000. What does this tell you, and which number better describes a typical employee?

Sketch (or describe) the shape you expect for the number of years of life remaining on a set of new car batteries rated for 5 years.

A histogram of the masses of apples from one orchard is bimodal. Suggest a cause and what to do about it.

Explain why waiting times at a doctor's office are usually skewed right.

Challenge

Pearson's coefficient of skewness is Sk = 3(mean − median) ÷ (standard deviation). For a data set with mean 54, median 60 and standard deviation 12, find Sk and interpret it.

Create a set of 5 whole numbers with mean 10 and median 6. What shape does this suggest?

A class's marks are skewed left with mean 72 and median 78. The teacher adds 5 marks to every student. What happens to the shape, mean and median?

Check your understanding

Quiz

A distribution has its peak on the left and a long tail on the right. What is its shape?

For a strongly skewed-left distribution, which order is typical?

Household incomes are usually reported with the median because

A histogram has two clear peaks. What is the most likely explanation?

Mean = $410, median = $410 for weekly grocery bills of 200 families. Which conclusion is justified?

Which variable is most likely skewed left?

For parents and teachers

Big idea (D1.3, D1.5): shape (symmetric, uniform, skewed, bimodal) decides which summaries are meaningful. A long tail pulls the mean toward it.

How to help: Look for "average" and "median" in news stories about house prices, rent or incomes, and ask your teen what the gap between them suggests about the shape.

Common difficulties:

Naming the skew after the peak instead of the tail (the most common error).
Treating "mean > median" as a proof of skew rather than a strong hint.
Reporting a mean for bimodal data instead of splitting the groups.
Forgetting to give a reason in context for a shape (floors, ceilings, mixed groups).