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D1.6 · Describe common distribution shapes

Learn to describe common distribution shapes through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Data Management

Grade 11 Mathematics for College Technology — MBF3C, D1.6

A set of data can be shown in a graph so that its overall pattern is easier to see. That pattern is called the distribution’s shape. For example, the daily number of customers at a small shop may be low on most days but very high on a few busy days. A graph can show this pattern at a glance. In this lesson, you will learn to describe common distribution shapes by noticing where values gather, how the two sides compare, and whether one or more peaks appear.

What you will learn

Describe a distribution as symmetric, skewed, uniform, or having more than one peak.
Use the shape of a dot plot or histogram to support a description.
Distinguish the direction of a skew from the direction of the longer tail.

1. A quick bridge: read the data display

A distribution describes how data values are spread out and how often values occur. A dot plot shows each data value with a dot above a number line. If a value occurs more than once, its dots are stacked. A histogram groups values into intervals, called bins, and uses bars to show how many values fall in each interval. In either display, taller stacks or bars mean that more values occur in that part of the scale.

Before naming a shape, read the horizontal scale from low values to high values. Then look for where most of the data sits, how the pattern changes on either side, and whether a small number of values extend farther in one direction. A tail is the thinner, stretched-out part of a distribution. It points toward values that occur less often and lie farther from the main group.

A peak is a place where the display is relatively high. A distribution may have one main peak, two clear peaks, or no strong peak. A peak does not have to be a single value; several neighbouring values can form a high area.

Start with the scale, then notice the main group and the tails.
The direction of a tail is named by the values it reaches toward.

2. Describe the common shapes

A symmetric distribution has a similar pattern on both sides of its centre. The two sides do not need to match perfectly, especially with a small data set. The key idea is that neither side stretches much farther than the other, and the overall outline looks balanced.

A skewed distribution is not balanced. It has a main group of values on one side and a longer, thinner tail on the other. A right-skewed distribution has its tail toward the larger values on the right. A left-skewed distribution has its tail toward the smaller values on the left. Name the skew by the tail, not by the side where most values gather.

A roughly uniform distribution has values occurring at similar frequencies across much of the displayed range. Its outline is fairly level rather than having one dominant peak. Real data may not be perfectly even, so describe it as roughly uniform when the bars or stacks are generally similar in height.

A bimodal distribution has two noticeable peaks separated by a lower area. The word means that the display has two modes, or high points. The two peaks may suggest that the data includes two common groups, but the shape alone does not explain why those groups occur. If there are more than two clear peaks, describe the display as having multiple peaks rather than forcing it into a one-peak shape.

Symmetric: the sides look broadly balanced.
Right-skewed: the longer tail reaches toward larger values.
Left-skewed: the longer tail reaches toward smaller values.
Roughly uniform: frequencies are fairly even across the range.
Bimodal: two noticeable peaks have a lower area between them.

3. Use evidence, not just a label

A useful description names the shape and points to visible evidence. For example, instead of saying only “skewed,” say, “The distribution is right-skewed because most values are grouped at the low end and a thinner tail extends toward higher values.” This links the label to the display.

Do not decide a distribution is symmetric just because its highest bar is near the middle. Check both sides. Likewise, a gap or an unusually high bar may affect the appearance, but it does not automatically make the distribution bimodal. Look for two clear clusters of high frequencies with a lower area between them.

A distribution is a summary of a data set, not a claim about what caused the pattern. For example, a bimodal display of travel times may show two common ranges. Without more information, the graph does not tell you why the ranges differ.

Support each shape label with an observation about the display.
Describe what the graph shows; do not claim a cause that the graph cannot establish.

4. Guided examples and application

In the first example, compare the low and high ends of a histogram to decide which way the tail extends. In the second, look for two high areas separated by a lower one. In both cases, the explanation uses the visible pattern rather than a calculation.

Read from low values to high values before naming a skew.
For a possible bimodal shape, look for two distinct high areas.

Shape clues at a glance

Shape	What to look for
Symmetric	The two sides have a broadly balanced outline.
Right-skewed	A thinner tail stretches toward larger values.
Left-skewed	A thinner tail stretches toward smaller values.
Roughly uniform	Frequencies are fairly similar across much of the range.
Bimodal	Two noticeable peaks have a lower area between them.

Worked example

A tail toward larger values

A histogram of the number of customer calls received by a repair shop each day has its tallest bars between 10 and 20 calls. The bars become shorter as the number of calls increases, and a few days extend out to 60 calls. What shape best describes the distribution?

Find the main group
The tallest bars show where the values gather most often. Here, most days are in the lower range of 10 to 20 calls.
Follow the thinner end
The smaller bars continue from the main group toward larger call counts, reaching as far as 60. That stretched-out part is the tail.
Name the shape
Because the tail points toward larger values, the distribution is right-skewed. The name refers to the tail’s direction, not to where most days occur.

Answer: The distribution is right-skewed: most days have fewer calls, and a thinner tail extends toward higher call counts.

Check: The high values lie to the right on the usual low-to-high number line, so the tail points right.

Worked example

Two common groups of travel time

A dot plot of bus travel times has one tall group near 15 minutes and another tall group near 35 minutes. There are noticeably fewer trips near 25 minutes. Describe the shape.

Count the high areas
The dot plot has two distinct tall groups: one near 15 minutes and one near 35 minutes. Each group forms a peak.
Check between the peaks
There are fewer trips near 25 minutes, so the area between the two peaks is lower. This supports calling the peaks distinct rather than treating the display as one broad high area.
Describe the distribution
A distribution with two noticeable peaks and a lower area between them is bimodal. The shape does not, by itself, tell us why the trips fall into two common time ranges.

Answer: The distribution is bimodal because it has two noticeable peaks, near 15 and 35 minutes, with fewer trips between them.

Check: There are two high groups, not one balanced peak or an evenly level pattern.

Common mistakes and how to avoid them

Calling a distribution right-skewed because most values are on the right.

Correction: Find the tail. A right-skewed distribution has a tail toward larger values, even if most values are grouped on the left.

Calling any graph with two tall bars bimodal.

Correction: Look for two distinct high areas with a lower area between them. Two neighbouring tall bars may be part of one broad peak.

Expecting a symmetric or uniform distribution to look perfect.

Correction: Describe the overall pattern. Real data can have small differences and still be broadly balanced or roughly even.

Lesson summary

A distribution’s shape describes how its values are spread and how often they occur.
A symmetric distribution looks broadly balanced; a roughly uniform one is fairly level across its range.
A skewed distribution has a longer tail. Name it by the direction of that tail.
A bimodal distribution has two noticeable peaks with a lower area between them.
Use visible evidence to support a shape description, and do not infer a cause from shape alone.

Check your understanding

Question 1

A dot plot has most of its values near the high end, while a thin tail stretches toward smaller values. Which description fits?

Left-skewed
Right-skewed
Roughly uniform
Bimodal

Show answer and explanation

Left-skewed

The tail reaches toward smaller values on the left, so the distribution is left-skewed.

Question 2

A histogram has bars of similar height across most of its range, with no dominant peak. Which shape is the best description?

Symmetric
Roughly uniform
Bimodal
Right-skewed

Show answer and explanation

Roughly uniform

Similar frequencies across much of the range are the main clue for a roughly uniform shape.

Question 3

A distribution has two clear high groups separated by a lower area. What is the most suitable description?

Symmetric
Left-skewed
Bimodal
Roughly uniform

Show answer and explanation

Bimodal

Two noticeable peaks with a lower area between them describe a bimodal distribution.

Key terms

Distribution: A description of how data values are spread out and how often they occur.
Bin: An interval used to group values in a histogram.
Tail: The thinner, stretched-out part of a distribution.
Peak: A relatively high area in a data display where values occur often.
Bimodal: Having two noticeable peaks.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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