Types of Skewed Distributions

Oct 13, 2025 by miodeya.com 54 views

Introduction

In statistics, a distribution is a set of values that a variable can take. Distributions can be described in various ways, including their shape, central tendency, and variability. One way to describe the shape of a distribution is to determine whether it is skewed or symmetrical. In this article, we will explore the concept of skewed and symmetrical distributions and learn how to draw histograms to visualize these distributions.

What are Skewed and Symmetrical Distributions?

A distribution is said to be skewed if it is not symmetrical about its mean. Skewed distributions have a longer tail on one side than the other, which can be either positively or negatively skewed. A positively skewed distribution has a longer tail on the right side, while a negatively skewed distribution has a longer tail on the left side.

On the other hand, a symmetrical distribution is one that is perfectly balanced about its mean. The left and right sides of the distribution are mirror images of each other, with the mean at the center.

Types of Skewed Distributions

There are two types of skewed distributions: positively skewed and negatively skewed.

Positively Skewed Distribution

A positively skewed distribution is one that has a longer tail on the right side. This type of distribution is often seen in real-world data, such as income or wealth, where there are a few individuals who have a very high income or wealth.

Example of a Positively Skewed Distribution

Suppose we have a dataset of exam scores, with a few students scoring very high. The distribution of scores would be positively skewed, with a longer tail on the right side.

Negatively Skewed Distribution

A negatively skewed distribution is one that has a longer tail on the left side. This type of distribution is often seen in data that has a floor effect, such as exam scores, where there is a minimum score that can be achieved.

Example of a Negatively Skewed Distribution

Suppose we have a dataset of exam scores, with a few students scoring very low. The distribution of scores would be negatively skewed, with a longer tail on the left side.

Types of Symmetrical Distributions

There are two types of symmetrical distributions: normal and uniform.

Normal Distribution

A normal distribution is a type of symmetrical distribution that is perfectly balanced about its mean. The normal distribution is often referred to as the "bell curve" due to its shape.

Example of a Normal Distribution

Suppose we have a dataset of exam scores, with a mean score of 70 and a standard deviation of 10. The distribution of scores would be normal, with a symmetrical bell shape.

Uniform Distribution

A uniform distribution is a type of symmetrical distribution where every value within a certain range has an equal probability of occurring.

Example of a Uniform Distribution

Suppose we have a dataset of exam scores, with a minimum score of 0 and a maximum score of 100. The distribution of scores would be uniform, with every score within the range having an equal probability of occurring.

Drawing Histograms

A histogram is a graphical representation of a distribution that shows the frequency of each value. Histograms are a useful tool for visualizing skewed and symmetrical distributions.

Drawing a Histogram of a Negatively Skewed Distribution

To draw a histogram of a negatively skewed distribution, we need to create a graph with the x-axis representing the values and the y-axis representing the frequency. We then plot the frequency of each value on the graph, with the x-axis representing the values and the y-axis representing the frequency.

Example of a Histogram of a Negatively Skewed Distribution

Suppose we have a dataset of exam scores, with a few students scoring very low. The histogram of the scores would show a longer tail on the left side, indicating a negatively skewed distribution.

Drawing a Histogram of a Symmetrical Distribution

To draw a histogram of a symmetrical distribution, we need to create a graph with the x-axis representing the values and the y-axis representing the frequency. We then plot the frequency of each value on the graph, with the x-axis representing the values and the y-axis representing the frequency.

Example of a Histogram of a Symmetrical Distribution

Suppose we have a dataset of exam scores, with a mean score of 70 and a standard deviation of 10. The histogram of the scores would show a symmetrical bell shape, indicating a normal distribution.

Drawing a Histogram of a Positively Skewed Distribution

To draw a histogram of a positively skewed distribution, we need to create a graph with the x-axis representing the values and the y-axis representing the frequency. We then plot the frequency of each value on the graph, with the x-axis representing the values and the y-axis representing the frequency.

Example of a Histogram of a Positively Skewed Distribution

Suppose we have a dataset of exam scores, with a few students scoring very high. The histogram of the scores would show a longer tail on the right side, indicating a positively skewed distribution.

Conclusion

In conclusion, skewed and symmetrical distributions are two types of distributions that can be described in various ways. Skewed distributions have a longer tail on one side than the other, while symmetrical distributions are perfectly balanced about their mean. Histograms are a useful tool for visualizing these distributions, and can be used to identify whether a distribution is skewed or symmetrical.

References

[1] Moore, D. S., & McCabe, G. P. (2012). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Johnson, R. A., & Wichern, D. W. (2007). Applied multivariate statistical analysis. Prentice Hall.
[3] Kotz, S., & Johnson, N. L. (2002). Encyclopedia of statistical sciences. Wiley.

Glossary

Skewed distribution: A distribution that is not symmetrical about its mean.
Symmetrical distribution: A distribution that is perfectly balanced about its mean.
Histogram: A graphical representation of a distribution that shows the frequency of each value.
Normal distribution: A type of symmetrical distribution that is perfectly balanced about its mean.
Uniform distribution: A type of symmetrical distribution where every value within a certain range has an equal probability of occurring.
Frequently Asked Questions: Skewed and Symmetrical Distributions ================================================================

Q: What is the difference between a skewed and symmetrical distribution?

A: A skewed distribution is one that is not symmetrical about its mean, while a symmetrical distribution is perfectly balanced about its mean.

Q: What are the two types of skewed distributions?

A: The two types of skewed distributions are positively skewed and negatively skewed. A positively skewed distribution has a longer tail on the right side, while a negatively skewed distribution has a longer tail on the left side.

Q: What is a normal distribution?

A: A normal distribution is a type of symmetrical distribution that is perfectly balanced about its mean. It is often referred to as the "bell curve" due to its shape.

Q: What is a uniform distribution?

A: A uniform distribution is a type of symmetrical distribution where every value within a certain range has an equal probability of occurring.

Q: How do I determine whether a distribution is skewed or symmetrical?

A: You can determine whether a distribution is skewed or symmetrical by creating a histogram of the data. If the histogram shows a longer tail on one side, the distribution is skewed. If the histogram shows a symmetrical bell shape, the distribution is symmetrical.

Q: What is the importance of understanding skewed and symmetrical distributions?

A: Understanding skewed and symmetrical distributions is important because it can help you to identify patterns and trends in data. It can also help you to make informed decisions based on the data.

Q: How do I draw a histogram of a skewed distribution?

A: To draw a histogram of a skewed distribution, you need to create a graph with the x-axis representing the values and the y-axis representing the frequency. You then plot the frequency of each value on the graph, with the x-axis representing the values and the y-axis representing the frequency.

Q: How do I draw a histogram of a symmetrical distribution?

A: To draw a histogram of a symmetrical distribution, you need to create a graph with the x-axis representing the values and the y-axis representing the frequency. You then plot the frequency of each value on the graph, with the x-axis representing the values and the y-axis representing the frequency.

Q: What are some common examples of skewed distributions?

A: Some common examples of skewed distributions include income or wealth, where there are a few individuals who have a very high income or wealth. Another example is exam scores, where there are a few students who score very high or very low.

Q: What are some common examples of symmetrical distributions?

A: Some common examples of symmetrical distributions include exam scores, where the mean score is around 70 and the standard deviation is around 10. Another example is the distribution of heights in a population, where the mean height is around 5 feet 9 inches and the standard deviation is around 2 inches.

Q: How do I calculate the mean and standard deviation of a distribution?

A: To calculate the mean and standard deviation of a distribution, you need to use the following formulas:

Mean: (sum of all values) / (number of values)
Standard deviation: sqrt((sum of squared differences from the mean) / (number of values - 1))

Q: What is the difference between a mean and a median?

A: The mean is the average value of a distribution, while the median is the middle value of a distribution. The mean is sensitive to outliers, while the median is not.

Q: What is the difference between a standard deviation and a variance?

A: The standard deviation is the square root of the variance. The variance is the average of the squared differences from the mean, while the standard deviation is the square root of the variance.

Q: How do I use a histogram to identify patterns and trends in data?

A: To use a histogram to identify patterns and trends in data, you need to look for the following:

Skewness: If the histogram shows a longer tail on one side, the distribution is skewed.
Symmetry: If the histogram shows a symmetrical bell shape, the distribution is symmetrical.
Outliers: If the histogram shows a few values that are far away from the mean, these are outliers.
Patterns: If the histogram shows a pattern, such as a peak or a trough, this can indicate a trend in the data.

Q: What are some common applications of skewed and symmetrical distributions?

A: Skewed and symmetrical distributions are used in a variety of applications, including:

Finance: To model the distribution of stock prices or returns.
Insurance: To model the distribution of claims or losses.
Medicine: To model the distribution of disease outcomes or treatment effects.
Social sciences: To model the distribution of attitudes or behaviors.

Q: How do I choose the right distribution for my data?

A: To choose the right distribution for your data, you need to consider the following:

Shape: Is the distribution skewed or symmetrical?
Location: Is the distribution centered around a specific value?
Scale: Is the distribution spread out or concentrated?
Outliers: Are there any outliers in the data?

By considering these factors, you can choose the right distribution for your data and make informed decisions based on the data.