Question

What does it normally mean if a bell curve is skewed to one side and how does one counteract such a problem in the experiment? Be detailed.

Answer 1

What Is Skewness?

Skewness, in statistics, is the degree of distortion from the symmetrical bell curve, or normal distribution, in a set of data. Skewness can be negative, positive, zero or undefined. A normal distribution has a skew of zero, while a lognormal distribution, for example, would exhibit some degree of right-skew.

The three probability distributions depicted below depict increasing levels of right (or positive) skewness. Distributions can also be left (negative) skewed. Skewness is used along with kurtosis to better judge the likelihood of events falling in the tails of a probability distribution.

Right skewness.

KEY TAKEAWAYS

• Skewness, in statistics, is the degree of distortion from the symmetrical bell curve in a probability distribution.
• Distributions can exhibit right (positive) skewness or left (negative) skewness to varying degree.
• Investors note skewness when judging a return distribution because it, like kurtosis, considers the extremes of the data set rather than focusing solely on the average.

Explaining Skewness

Skewness is positive if the tail on the right side of the distribution is longer or fatter than the tail on the left side. The mean and median of positively skewed data will be greater than the mode. Skewness is negative if the tail of the left side of the distribution is longer or fatter than the tail on the right side. The mean and median of negatively skewed data will be less than the mode. If the data graph symmetrically, the distribution has zero skewness, regardless of how long or fat the tails are.

There are several ways to measure skewness. Pearson’s first and second coefficients of skewness are two common ones. Pearson’s first coefficient of skewness, or Pearson mode skewness, subtracts the mode from the mean and divides the difference by the standard deviation. Pearson’s second coefficient of skewness, or Pearson median skewness, subtracts the median from the mean, multiplies the difference by three and divides the product by the standard deviation.

The formulae for Pearson's skewness are:

Pearsons Skewness Formulas. Investopedia

where:

• Sk₁ is Pearson's first coefficient of skewness and Sk₂ the second;
• s is the standard deviation for the sample;
• x̄ is the mean value;
• Mo is the modal (mode) value; and
• Md is the median value.

Pearson’s first coefficient of skewness is useful if the data exhibit a strong mode. If the data have a weak mode or multiple modes, Pearson’s second coefficient may be preferable, as it does not rely on mode as a measure of central tendency.

Example

Investors note skewness when judging a return distribution because it, like kurtosis, considers the extremes of the data set rather than focusing solely on the average. Short- and medium-term investors in particular need to look at extremes because they are less likely to hold a position long enough to be confident that the average will work itself out.

Investors commonly use standard deviation to predict future returns, but standard deviation assumes a normal distribution. As few return distributions come close to normal, skewness is a better measure on which to base performance predictions. This is due to skewness risk.

Skewness risk is the increased risk of turning up a data point of high skewness in a skewed distribution. Many financial models that attempt to predict the future performance of an asset assume a normal distribution, in which measures of central tendency are equal. If the data are skewed, this kind of model will always underestimate skewness risk in its predictions. The more skewed the data, the less accurate this financial model will be.