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Shapes of Distribution

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Tutorial

This tutorial will cover the different shapes that distributions can take. You will learn about:

- Distribution
- Symmetric Distribution
- Skewed Distribution
- Uniform Distribution

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

**Distribution **

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

Distribution is 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.

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

**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.

Here is an example of symmetric distribution:

For 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 terribly often. Only a few distributions are actually truly symmetric. Often we get distributions that look something like this:

These distributions are close to being symmetrical, but they're not *exactly* symmetric. 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 not exactly symmetric.

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

**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.

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:

There are two ways that a distribution can be skewed: to the right and to the left:

- If the tail is on the right side of the median, it is referred to as skewed to the right or positively skewed.
- If the tail is to the left of the median, it is skewed to the left, or negatively skewed.

**Skewed Right (Positively Skewed) Distributions **

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

**Skewed Left (Negatively Skewed) Distribution **

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

These are referred to as “positive” and “negative” because right is more positive on the number line and left is more negative on the number line.

When all values are equally distributed, then the shape is referred to as being in **uniform distribution**.

**Uniform Distribution**

A distribution where all values are equally likely.

Here is an example of uniform distribution:

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.

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

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

If you rolled the die 600 times, you would expect about 100 of each. But maybe you only got 95 1s and 102 2s. 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.

**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.

Thank you and good luck!

Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS