# Symmetric vs. Skewed Distributions

## Symmetric Distribution Definition

Data is a collection of facts, often numbers such as measurements or statistics. A data set is a group of data collected and often displayed in some sort of visual representation, such as a table or graph. What is meant by the distribution of data? The **distribution of data** refers to the shape of the data set when displayed on a graph, oftentimes a bar graph. The distribution of a data set can take different shapes, including symmetric and skewed.

What is a symmetric data distribution? A **symmetric distribution** is a graphic distribution of data that looks nearly the same on both sides. It is important to note that the data does not need to be *exactly* the same on both sides to be considered symmetric, only *nearly* the same. Think of a mirror in the middle of the data distribution. Both sides should be near mirror images of one another.

Below is an example of a symmetric data distribution, shown as a symmetric bar graph. The graph shows the number of questions answered correctly by a fictional class of students on a pop quiz. The scores, in order from lowest to highest, are 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 10.

The two sides of the graph mirror, or nearly mirror, one another. The peak of the graph is right in the middle. An equal number of students scored above and below the peak.

An important thing to know about a symmetric data set is that the mean, median, and mode will all occur at the same point. The *mean*, or average, the *median*, or exact middle number, and the *mode*, or most common number, will all be the same in a symmetric data set.

Consider again the bar graph of the pop quiz results, which is shown above. The mean, median, and mode of the data set all occur at the number 5, making this a symmetric distribution.

## Skewed Distribution Definition

What is a skewed data distribution? A **skewed distribution** is not symmetric, but instead peaks on one side or the other. A distribution might be very skewed or only slightly skewed. Consider the examples below.

The graph below, in red, shows an example of a skewed distribution of data. The two sides of the graph do not mirror one another, and the peak is not in the middle. There are many more data points on the right side of the graph. This graph shows the number of questions answered correctly by a fictional class of students on an announced quiz. There are many more high-scoring students than low-scoring students. The mean, median, and mode of this data set will not occur at the same number. A graph of a distribution such as this, that has a peak on the right and a tail to the left, is said to be *skewed left.* If the peak of the data distribution is on the left side and the tail is on the right, the distribution is *skewed right*.

The graph below shows the quiz scores of a fictional class in which the quizzes were graded with the wrong answer key. This distribution is not symmetric, but it is instead skewed right. Notice also the empty spaces for scores of 7 or 8 answers correct. An area such as this, with no data points or observations, is called a **gap**.

The graph below shows the quiz scores of a fictional class on a quiz. This data distribution is neither symmetric nor skewed, but it is distributed in a more random way. Notice that this graph has two peaks: one at a score of 3 and the other at a score of 8. This means the data set has two modes. A distribution such as this is said to be **bimodal**, which means it has two modes.

### Skewed and Symmetric Distribution Examples

What are examples of distributions that might be symmetric or skewed? Any distribution can take on either shape, but some are more likely than others to be usually symmetric or usually skewed. Distributions that tend to be symmetric include standardized test scores, heights of adult men or adult women, weather data over a long period of time, and salaries. In these cases, the peak is likely to be in the middle, and the two sides of the graph are likely to be nearly symmetric. Some distributions tend to be skewed. An example of this is the number of adults living in a household. Most households will have 1, 2, or perhaps 3 adults, and few will have 6, 7, or 8 adults. Such a data distribution is likely to have a peak on the left and a tail on the right, and it will be considered skewed right.

## Graphing Data Sets

In a visual representation of a data set, the distribution can easily be seen. Common visual representations used to display data sets and their distributions are bar graphs, histograms, dot plots, line plots, and box and whisker plots.

Bar graphs and histograms are very similar, but a bar graph shows one value per bar, while a histogram shows a range of values per bar. The graphs shown at the beginning of the lesson are bar graphs, and the one shown below is a histogram. The histogram shows a distribution of iris petals of varying lengths. Each interval is marked as 1 centimeter, and there are 2 bars in each interval. The first bar in the first interval shows the number of petals between 0 and 0.49 centimeters, and the second bar shows the number of petals between 0.5 and 0.99 centimeters. This is how a histogram is able to show a range of values.

A dot plot uses points rather than bars to show the distribution of data. A random dot plot is shown below. A line plot uses x's rather than dots.

A box and whisker plot uses a *box* and a *whisker* to show each side of the distribution.

### Graphing Symmetrical Data Sets

How does a person graph a symmetrical data set? Here is an example: This data set shows the high temperatures, in degrees Fahrenheit, of the fictional city of Park City, each day for one month. The data has been arranged in numerical order: 50, 53, 53, 53, 56, 56, 57, 58, 58, 58, 59, 59, 60, 60, 60, 60, 60, 60, 61, 62, 62, 63, 63, 63, 63, 65, 65, 65, 68, 70. The data values are too spread out to make a bar graph; a histogram is a better choice. In order to make a histogram, the data values will have to be grouped. These data values can be grouped in 3's (50-51-52 / 53-54-55 / 56-57-58 / 59-61 / 62-63-64 / 65-66-67 / 68-69-70).

Looking at the distribution of data, it appears that this is a symmetric distribution. The peak is in the middle, and the two sides are almost mirror images of one another. To check this, calculate the mean, median, and mode of the data set using the ungrouped values. The mean, median, and mode of this set of data are all 60, which confirms that this is a symmetric distribution.

#### Symmetric Bar Graph

Here is another example: This data set shows the high temperatures, in degrees Fahrenheit, of the fictional city of Hill City, each day for one month. The data has been arranged in numerical order: 57, 58, 58, 58, 58, 59, 59, 59, 59, 59, 59, 60, 60, 60, 60, 60, 60, 60, 60, 61, 61, 61, 61, 61, 61, 62, 62, 62, 62, 63. The range of data values is narrow enough to allow this distribution to be graphed with a bar graph.

Looking at the distribution of data, it appears that this data set is also a symmetric distribution. The peak is in the middle, and the two sides appear to be mirror images of one another. To check this, calculate the mean, median, and mode of the data set. The mean, median, and mode of this set of data are all 60, which confirms that this is another symmetric distribution.

### Graphing Skewed Data Sets

How does a person graph a skewed data set? Here is an example: This data set shows the high temperatures, in degrees Fahrenheit, of the fictional city of Silver City, each day for one month. The data has been arranged in numerical order: 57, 58, 60, 60, 61, 61, 62, 62, 62, 63, 63, 63, 63, 64, 64, 64, 64, 64, 65, 65, 65, 65, 65, 65, 65, 66, 66, 66, 66, 66. The range of data values is narrow enough to allow this distribution to be graphed with a bar graph.

Looking at the distribution of data, it appears that this data set is a skewed distribution. The peak is on the right, and there is a tail extending to the left. The distribution of this data set is skewed left. Notice also the gap that appears at a temperature of 59, where there is no data.

Here is one more example: This data set shows the high temperatures, in degrees Fahrenheit, of the fictional city of Crystal City, each day for one month. The data has been arranged in numerical order: 57, 57, 57, 58, 58, 58, 58, 58, 58, 59, 59, 59, 59, 59, 60, 60, 60, 60, 60, 60, 61, 61, 61, 62, 62, 62, 63, 63, 64, 65, 66. The range of data values is narrow enough to allow this distribution to be graphed with a bar graph.

Looking at the distribution of data, it appears that this data set is also a skewed distribution. The peak is on the left, however, and the tail is extending to the right. The distribution of this data set is skewed right. Notice also this distribution is also bimodal. There are two modes, one at 58 and the other at 60.

## Lesson Summary

A **distribution of data** is the shape, when graphed, of a data set. Here are some important terms to know regarding the distribution of a data set:

**symmetric distribution**: A data set in which the mean, median, and mode all occur at the same point. When graphed, the two sides of this distribution will appear to be*almost*mirror images of another.**skewed distribution**: A data set which, when graphed, has a peak on one side and a tail extending to the other side. If the tail extends left, the graph is skewed left. If the tail extends right, the graph is skewed right.**gap**: A space where there is no observation or data.**bimodal**: A data set with two modes.

Data sets can be displayed in different ways, including bar graphs and histograms. Some data sets, such as height, are more likely to have a symmetric distribution. Other data sets, such as the number of adults living in a household, are more likely to have a skewed distribution.

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#### How do you tell if a bar graph is skewed right or left?

A graph with a peak on the right and a tail that extends to the *left* is said to be skewed *left*.

A graph with a peak on the left and a tail that extends to the *right* is said to be skewed *right*.

#### How do you know if data is symmetric?

A set of data is symmetric if the mean, median, and mode all occur at the same number. When graphed, the two sides of the graph will be almost mirror images of one another.

#### What is an example of a symmetrical distribution?

Standardized test scores are an example of a symmetrical distribution. The mean, median, and mode of the data set will all occur at the same value.

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