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Shapes of Distribution

Author: Sophia

what's covered
This tutorial will cover the different shapes that distributions can take. Our discussion breaks down as follows:

Table of Contents

1. Distribution

A distribution is a way to visually show how many times a variable takes a certain value.

While distributions display the values the variable takes and how often, shape describes the data points as a whole. This tutorial will use qualifying descriptors to identify how the distribution of a data set can look when graphed.

term to know
Distribution
A display of data that shows the values the data take and how often those values occur.

2. Symmetric Distribution

A symmetric distribution will have the same mean as its median. If plotted, it will look like two mirror images on the same plot.

The figure below shows three examples of symmetric distributions.

Three symmetric distributions, presented from left to right. The first one has a peak in the middle, with a downward taper on both sides. The graph to the left of the peak is a mirror image of the one on the right. The second has a distribution in which the center is the lowest value. From the center value, the graph rises, reaches a peak, and then decreases again but not all the way down to the axis. In the third graph, the center value is neither the highest nor the lowest point. From the center value, the graph increases on each side and then decreases down to the horizontal axis.

In the graph on the far left, for example, the line in the center of the graph is the mirror line, and it represents both the mean and the median of this distribution.

Symmetrical distribution doesn't happen too often. Only a few distributions are actually truly symmetric. Often, we get distributions that look something like this:

A histogram with nine bins of varying heights. On the left, there is a flat row of three short, equal-sized bins. From the fourth bin, the height increases, and the fifth bin is the tallest, reaching a peak. The sixth bin tapers down in height along with the seventh bin. The eighth and ninth bins have the same height as the first three bins.

Although this distribution is close to being symmetrical, it is not exactly symmetric.

Note that when you say the word “symmetric,” you must mean exactly. Thus, qualifiers like approximately symmetric, roughly symmetric, or nearly symmetric are necessary to make it clear when a distribution is nearly, but not exactly, symmetric.

term to know
Symmetric Distribution
A distribution where the mean and median are the same. It will appear to have a "mirror line" at the median of the distribution.

3. Skewed Distribution

Certain distributions aren't even close to being symmetric. Many asymmetric distributions are called skewed distributions.

These distributions are characterized by a hump, which is sort of a dense grouping with lots of points at certain values and some values that only have a few occurrences. The part of the distribution with fewer occurrences is called a tail. The tail occurs to one side of the median of the distribution. These distributions look like this:

A histogram with eight bins of varying heights. The tallest column is the second from the left. To its right, the bins decrease sharply in height and then level out into a long, flat row of four short, equal-sized bins. An arrow points to these shorter bins with the label ‘tail’.

There are two ways that a distribution can be skewed.

Skewed Distributions
Right-Skewed
(Positively Skewed) 		Tail is on the right side of the median.

Right is more positive on the number line. 		A histogram with eight bins of varying heights. The tallest column is the second from the left. To its right, the bins sharply decrease in height and then level out into a long, flat row of four short, equal-sized bins at the end.
Left-Skewed
(Negatively Skewed) 		Tail is on the left side of the median.

Left is more negative on the number line. 		A histogram with eight bins of varying heights. From the left, there is a flat row of four short, equal-sized bins. From here, the bins increase sharply in height, reaching a peak at the seventh bin. The final bin on the far right drops back down to the same height as the initial equal-sized bins.

terms to know
Skewed Distribution
A distribution where the majority of values are on one side of the distribution, and there are only a few values on the other.
Right-Skewed (Positively Skewed) Distributions
A distribution where the majority of values are low, and there are only a few high values that form a "tail" to the right of the median.
Left-Skewed (Negatively Skewed) Distribution
A distribution where the majority of values are high, and there are only a few low values that form a "tail" to the left of the median.

4. Uniform Distribution

When all values are equally distributed, then the shape is referred to as being in uniform distribution. Here is an example of uniform distribution:

A histogram with six bins and a horizontal axis ranging from 0 to 7. The bins 1, 2, 3, 4, 5, and 6 have the same height.

Uniform distributions are a certain kind of symmetric distribution. Imagine you put a line of symmetry between the three and four. The two sides would then be symmetric. Moreover, this is a distribution where all the values are equally distributed.

You can also use the same qualifiers for uniform distribution as are used with symmetry.

EXAMPLE

If you rolled a die six times, you might get one 6, one 5, one 4, one 3, one 2, and one 1.

Suppose you rolled the die 600 times; you would expect about 100 of each. However, perhaps you only got 95 1's and 102 2's. The distribution will look almost uniform, so we can use those words like “approximately,” “nearly,” or “almost” uniform in place of the word “exactly” uniform.

term to know
Uniform Distribution
A distribution where all values are equally likely.

5. Unimodal Distribution

Often, distributions will have a clear peak to their shape. They will peak in just one place on the distribution.

In the table below, each graph has a clear peak, so all of these are called unimodal distributions.

Unimodal Distributions
Peak in the Center A histogram with nine bins of varying heights. From the left, there is a flat row of three short, equal-sized bins. The fourth bin increases sharply in height, and the peak is reached at the fifth bin. From the sixth bin onward, there is a decrease in height. The last two bins drop back down to the same height as the initial equal-sized bins.
Peak to the Right A histogram with eight bins of varying heights. From the left, there is a flat row of four short, equal-sized bins. The fifth, sixth, and seventh bins rise steadily, with the seventh being the tallest and forming a peak, followed by the eighth bin, which is the same height as the first four bins.
Peak to the Left A histogram with eight bins of varying heights. From the left, the first bin is a short bin, followed by the second bin, which is the tallest and forms a peak. The third and fourth bins are shorter than the second. These are followed by four short bins with the same height as the first bin.

hint
The tallest bar is called the mode.

term to know
Unimodal/Single-Peaked Distribution
A distribution where one value or bin contains more data than the other values or bins.

6. Bimodal Distribution

You might have a distribution that has two distinct regions with lots of data points and a gap in the middle. When this happens, two peaks form on the distribution. These are both called modes, and a distribution like this is called bimodal distribution.

A histogram with the horizontal axis ranging from 4 to 9 and the vertical axis ranging from 0 to 60 in increments of 20. The histogram has two distinct peaks, one near 5 and another near 8, with lower bars between them. Arrows point to the two peaks with the label ‘modes’.

Technically, there's only one bin that is the mode: the very tallest bar. However, in the above graph, there are two bins that are the tallest relative to the others around them—also known as local modes.

Now, sometimes you have a distribution that appears bimodal, like the graph below:

A histogram with the horizontal axis labeled ‘HEIGHT’ and ranging from 60 to 76. The vertical axis ranges from 2 to12. There are bars corresponding to the values 62 through 66, then none for 67, followed by bars corresponding to values 68 through 72, and a single bar for 74. The break at 67 could suggest that this single histogram represents two different populations.

Even though it appears to be bimodal, upon further examination of heights, it's possible that you have two different distributions that happen to be graphed on the same set of axes (see below).

Two histograms, one representing boys’ height and one representing girls’ height. The boys’ histogram shows values at 64 as a low outlier and then bars for values 68 through 74 except 73, while the girls' histogram shows bars at values 62 through 66.

There might be some hidden variable that causes the bimodality. When viewed separately, you end up with two unimodal distributions that just happened to be graphed on the same set of axes.

term to know
Bimodal Distribution
A distribution where there are two distinct values or bins that contain more data than the others, usually separated by a gap.

7. Multimodal Distribution

Any distribution with more than two peaks is called a multimodal distribution. This distribution, for instance, has four peaks:

A histogram with four distinct peaks.

You can have the same issue with this type of distribution as you did with the bimodal distribution, in that it may be multiple distributions graphed on the same set of axes.

big idea
“Uni” means one, “bi” means two, and “modal” means the number of modes each distribution has.

term to know
Multimodal Distribution
A distribution where there are many values or bins that contain more data than other nearby bins, usually separated by gaps.

summary
Distributions, when graphed, have many descriptors that we can use to describe their shape. Symmetric distributions visually have mirror halves, and mathematically they have the same mean and median. Uniform distributions are a specific type of a symmetric distribution that are visually very flat. Skewed distributions have a hump on one side of the median and a tail on the other side of the median; if the tail is on the right side of the median, it is called skewed to the right, or positively skewed, and if the tail is to the left of the median, it is skewed to the left, or negatively skewed. Some distributions are unimodal, or single-peaked distributions. Others are bimodal, which means they are clearly double-peaked, and some are multimodal, with more than two peaks. Sometimes, a bimodal distribution is simply two unimodal distributions graphed together.

Good luck!

Source: THIS TUTORIAL WAS AUTHORED BY SOPHIA LEARNING. PLEASE SEE OUR TERMS OF USE.

Terms to Know
Bimodal Distribution

A distribution where there are two distinct values or bins that contain more data than the others, usually separated by a gap.

Distribution

A display of data that shows the values the data take and how often those values occur.

Left-Skewed (Negatively Skewed) Distribution

A distribution where the majority of values are high, and there are only a few low values that form a "tail" to the left of the median.

Multimodal Distribution

A distribution where there are many values or bins that contain more data than other nearby bins, usually separated by gaps.

Right-Skewed (Positively Skewed) Distributions

A distribution where the majority of values are low, and there are only a few high values that form a "tail" to the right of the median.

Skewed Distribution

A distribution where the majority of values are on one side of the distribution, and there are only a few values on the other.

Symmetric Distribution

A distribution where the mean and median are the same. It will appear to have a "mirror line" at the median of the distribution.

Uniform Distribution

A distribution where all values are equally likely.

Unimodal/Single-Peaked Distribution

A distribution where one value or bin contains more data than the other values or bins.

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