Skewness

Definition

Skewness is a measure of the asymmetry or lack of symmetry in the distribution of a dataset. It describes the extent to which a probability distribution or a data set deviates from a symmetric, bell-shaped, or normal distribution.

5 Must Know Facts For Your Next Test

1. Skewness can be positive, negative, or zero, indicating the direction and degree of asymmetry in the data.
2. Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values, while negative skewness indicates a distribution with an asymmetric tail extending toward more negative values.
3. Skewness can affect the relationship between the mean, median, and mode, with the mean being pulled toward the longer tail of the distribution.
4. Measures of central tendency, such as the mean and median, can be used to assess the degree of skewness in a dataset.
5. Estimating the binomial distribution with the normal distribution requires considering the skewness of the binomial distribution, as the normal approximation is most accurate when the binomial distribution is symmetric or only slightly skewed.

Review Questions

• Explain how skewness relates to the measures of central tendency (mean, median, and mode) in a dataset.
  • Skewness can affect the relationship between the mean, median, and mode of a dataset. In a positively skewed distribution, the mean is pulled toward the longer right tail, resulting in the mean being greater than the median, which is greater than the mode. In a negatively skewed distribution, the mean is pulled toward the longer left tail, resulting in the mean being less than the median, which is less than the mode. When the distribution is symmetric, the mean, median, and mode are all equal.
• Describe the role of skewness in estimating the binomial distribution with the normal distribution.
  • When estimating the binomial distribution with the normal distribution, the skewness of the binomial distribution must be considered. The normal approximation is most accurate when the binomial distribution is symmetric or only slightly skewed. As the binomial distribution becomes more skewed, the normal approximation becomes less accurate, and alternative methods, such as the Poisson approximation, may be more appropriate.
• Analyze how skewness can be used to assess the spread and variability of a dataset.
  • $$\text{Skewness} = \frac{E[(X - \mu)^3]}{\sigma^3}$$ where $X$ is the random variable, $\mu$ is the mean, and $\sigma$ is the standard deviation. Skewness provides information about the asymmetry of the distribution, which can be used to understand the spread and variability of the data. Positive skewness indicates a longer right tail, while negative skewness indicates a longer left tail. The degree of skewness can be used to assess how the data is distributed and the potential impact on measures of central tendency and dispersion.

Related terms

Measures of Central Tendency: Statistical measures that describe the central or typical value in a dataset, such as the mean, median, and mode.

Measures of Dispersion: Statistical measures that describe the spread or variability of a dataset, such as the range, variance, and standard deviation.

Normal Distribution:

A symmetric, bell-shaped probability distribution where the mean, median, and mode are all equal, and the data is evenly distributed around the central value.