Skewness

Sometimes shapes are not even. They can lean to one side.

We call the sides of a shape tails. One tail might be very long. The other tail might be short. This makes the shape look lopsided.

A shape with a long right tail is right-skewed.

A shape with a long left tail is left-skewed.

If both sides are the same, the skew is zero. This means the sides balance out. The shape is then called symmetric.

When we look at groups of data, we often see shapes.

Most shapes have two sides. We call these sides tails. Sometimes, one tail is much longer than the other. This makes the shape look lopsided. We use the word skewness to measure this.

A shape with a long tail on the right is right-skewed.

A shape with a long tail on the left is left-skewed.

If the tails are the same, the skewness is zero. This means the sides balance out. We call a balanced shape symmetric. A symmetric shape always has zero skewness.

However, a zero skewness does not always mean a shape is symmetric.

Sometimes, one tail is very long and thin. The other tail might be short but fat. These can balance out to reach a zero skewness. Because of this, you must look at the whole shape. Do not rely on the skewness number alone to judge symmetry. You can also look at the mean and the median. The mean is the average value. The median is the middle value. In many cases, these two numbers change based on the skewness.

When we look at groups of numbers, we often see a shape. This shape shows how the numbers are spread out. Most shapes have two sides that we call tails.

Sometimes, one tail is much longer than the other. This makes the shape look lopsided or uneven. We use the word skewness to measure this unevenness. It helps us understand the balance of the data. Knowing the skewness tells us a lot about the data's shape.

There are different ways a shape can be skewed. A right-skewed distribution has a long tail on the right side.

This means most of the data is on the left. A left-skewed distribution has a long tail on the left side.

Even though the tail is on the left, the curve might look like it leans right. If the tails are equal, the skewness is zero. This happens in a symmetric shape where both sides match.

However, a zero skewness does not always mean the shape is perfectly symmetric. One tail might be long and thin, while the other is short and fat.

Math experts use different ways to calculate this value. One famous way is called Fisher's moment coefficient of skewness. This is also known as Pearson's moment coefficient of skewness.

It uses the mean and the standard deviation to find the answer. There are also simpler ways to measure it. Karl Pearson suggested some simpler methods long ago. Another way is called Bowley's measure, which was used in 1901. George Udny Yule also created a measure in 1912. These different tools help people study different kinds of data.

Skewness can change how we see the middle of the data. We often look at the mean and the median. The mean is the average value of the group. The median is the middle value in a list.

Many textbooks say the mean follows the long tail. For example, they say the mean is to the right in a right-skewed group. But this rule of thumb can fail quite often. In some cases, like US household data, the rule does not work.

In those cases, the mean actually sits in the heavy left tail.

Understanding skewness is very useful in the real world. It helps people see if data is different from a normal distribution. A normal distribution is a perfectly symmetric shape with zero skewness. Scientists use skewness to find approximate probabilities in finance. It can also help with tests like D'Agostino's K-squared test. This test checks if data fits a normal shape. By looking at the tails, we can better predict what happens next. It turns a simple list of numbers into a meaningful picture.

Skewness is a statistical measure used to describe the asymmetry of a probability distribution. In statistics, we often study how a real-valued random variable is spread out around its mean, which is the average value. While many people assume data is always balanced, it often is not. Skewness provides specific insights into the shape of these distributions by looking at their tails.

By measuring skewness, researchers can understand if the data leans more heavily toward one side than the other.

To understand how skewness works, you must look at the tails of a distribution. The tails are the tapering sides of the curve where values become less frequent. A distribution is considered left-skewed, or left-tailed, when the tail on the left side is longer. Even though the tail is on the left, the main mass of the data is concentrated on the right. This often makes the curve appear to lean toward the right. Conversely, a right-skewed distribution has a longer tail on the right side. In this case, the mass of the distribution is concentrated on the left.

These tails are the primary visual way to determine which kind of skewness a distribution possesses.

There are several different types of skewness values that can be observed. A positive skewness value indicates a right-tailed distribution. A negative skewness value indicates a left-tailed distribution. A skewness of zero means that the tails on both sides of the mean balance out overall. While a symmetric distribution always has zero skewness, an asymmetric distribution can also have a zero value. This happens if one tail is very long and thin, while the other tail is short but fat.

Because of this, relying only on the skewness number to judge symmetry can be risky.

Mathematically, skewness can be defined in several ways. One common method is Fisher's moment coefficient of skewness, also known as Pearson's moment coefficient of skewness. This is defined as the third standardized moment. It uses the mean and the standard deviation to calculate the result.

Other researchers use different formulas, such as the nonparametric skew. In older nonparametric definitions, skewness was defined by the relationship between the mean and the median. However, modern definitions do not always match these older rules. For example, a distribution can be skewed even if the mean is not on the side of the long tail.

Many textbooks teach a common rule of thumb regarding the mean and the median. They suggest that in a right-skewed distribution, the mean will be to the right of the median. In a left-skewed distribution, the mean should be to the left of the median. However, this rule fails with surprising frequency. It often fails in multimodal distributions or when one tail is long but the other is heavy. A notable example involves the distribution of adult residents across US households. In that specific case, the skew is to the right, yet the mean actually sits in the heavier left tail.

This contradicts the standard textbook interpretation.

History shows that many mathematicians have contributed to these measurements. Karl Pearson suggested several simpler skewness statistics. These include Pearson's first skewness coefficient, which uses the mode, and his second coefficient, which uses the median. Other researchers developed quantile-based measures. Bowley created a measure of skewness in 1901, which is also called Yule's coefficient. George Udny Yule later contributed to this in 1912. These various methods allow statisticians to choose the best tool for their specific type of data.

Skewness is a vital tool in many scientific and financial fields. It is used as a descriptive statistic alongside histograms and normal quantile plots. In finance, skewness helps calculate approximate probabilities and quantiles, such as value at risk. It is also used in D'Agostino's K-squared test, which is a goodness-of-fit test for normality. Many mathematical models assume a normal distribution, which always has a skewness of zero. However, real-world data is rarely perfectly symmetric. Understanding skewness allows scientists to know if deviations from the mean are likely to be positive or negative.