Skewness: Definition and Formula for Positive and Negative Skewness

What is Skewness?

Skewness is a statistical measure that describes the asymmetry of a probability distribution. It quantifies the extent to which a distribution deviates from being symmetrical or bell-shaped. Skewness is an important concept in statistics and data analysis as it provides insights into the shape and characteristics of a dataset.

A symmetrical distribution has a skewness of zero, indicating that the data is evenly distributed around the mean. Positive skewness occurs when the tail of the distribution is skewed to the right, indicating that the majority of the data is concentrated on the left side of the distribution. Negative skewness, on the other hand, occurs when the tail of the distribution is skewed to the left, indicating that the majority of the data is concentrated on the right side of the distribution.

Definition and Explanation

Skewness is a statistical measure that describes the asymmetry of a probability distribution. It measures the extent to which the data deviates from a symmetrical distribution. A symmetrical distribution has equal probabilities on both sides of the mean, while a skewed distribution has a longer tail on one side.

Skewness can be positive or negative, indicating the direction of the skew. Positive skewness means that the tail of the distribution is skewed to the right, while negative skewness means that the tail is skewed to the left.

Positive Skewness

In statistics, positive skewness refers to a distribution that is skewed to the right. This means that the tail of the distribution is longer on the right side, indicating that there are more extreme values on the right side of the distribution.

Positive skewness can occur when there are a few extremely high values in the dataset, which pull the mean and median towards the right. This results in a distribution that is “skewed” or “lopsided” towards the right side.

Formula and Interpretation

The formula for calculating skewness is:

A positive skewness value indicates that the distribution is positively skewed. The magnitude of the skewness value can also provide information about the degree of skewness. A larger positive skewness value indicates a more pronounced skewness.

Interpreting the skewness value depends on the context of the data. In some cases, positive skewness may be desirable, such as when analyzing income distribution, where a few individuals may have significantly higher incomes. However, in other cases, positive skewness may indicate a problem with the data, such as outliers or a non-normal distribution.

It is important to consider the skewness of a distribution when analyzing data, as it can affect the validity of statistical tests and the interpretation of results. Skewness is just one of the many statistical measures that can be used to describe and analyze data distributions.

Formula and Interpretation of Positive Skewness

The formula to calculate positive skewness is:

Interpreting the value of positive skewness can provide insights into the shape of the distribution. Here are some key points to consider:

1. Skewness Value

The skewness value can range from 0 to positive infinity. A positive skewness value indicates a right-skewed distribution, where the tail is longer on the right side.

2. Mean, Median, and Mode

In a positively skewed distribution, the mean is typically greater than the median, and the median is greater than the mode. This is because the tail on the right side pulls the mean towards higher values.

3. Outliers

Positive skewness is often caused by outliers or extreme values on the right side of the distribution. These outliers can significantly affect the mean and pull it towards higher values, resulting in a right-skewed distribution.

4. Data Interpretation

When analyzing data with positive skewness, it is important to consider the impact of outliers and the distribution’s shape. The presence of outliers can affect the interpretation of the mean as a measure of central tendency. Additionally, the right-skewed shape indicates that a significant proportion of the data is concentrated towards lower values, with a few extreme values on the right side.

Negative Skewness: Formula and Interpretation

In statistics, negative skewness refers to the asymmetry in a distribution where the tail on the left side of the distribution is longer or fatter than the tail on the right side. This means that the majority of the data points are concentrated on the right side of the distribution, while the left side is stretched out.

To calculate the skewness of a dataset, you can use the following formula:

Where:

The skewness value can range from negative infinity to positive infinity. A negative skewness value indicates a left-skewed distribution.

Interpreting negative skewness depends on the context of the data. In finance, for example, negative skewness in the returns of an investment indicates that the majority of the returns are positive, but there are a few extreme negative returns that drag the average down. This suggests that the investment has a higher probability of generating positive returns, but with a potential for occasional large losses.

In other fields, negative skewness may indicate a similar pattern of data, where the majority of observations are higher or larger, but there are a few smaller values that pull the average down.

Formula and Interpretation of Negative Skewness

Negative skewness is a measure of the asymmetry or lack of symmetry in a probability distribution. It indicates that the tail of the distribution is longer on the left side, or the left tail is heavier than the right tail. In other words, the distribution is skewed to the left.

The formula for calculating negative skewness is:

Where:

  • Mean is the average of all the data points in the distribution.
  • Median is the middle value of the distribution when the data points are arranged in ascending order.
  • Standard Deviation is a measure of the dispersion or spread of the data points around the mean.

A negative skewness value indicates that the distribution has a longer left tail and the majority of the data points are concentrated on the right side. This means that the distribution is negatively skewed, or left-skewed.

Interpreting negative skewness depends on the context of the data. In finance, for example, negative skewness in the returns of an investment indicates that there is a higher probability of extreme negative returns. This suggests that the investment carries more downside risk.

It is important to note that skewness is just one measure of the shape of a distribution and should be considered in conjunction with other statistical measures to fully understand the characteristics of the data.