Right-Skewed

Definition

Right-skewed, also known as positively skewed, is a statistical term that describes a probability distribution where the tail on the right side of the distribution is longer or more drawn out compared to the left side. This asymmetry results in the mean being greater than the median, indicating that the data has a higher concentration of values on the left side of the distribution.

5 Must Know Facts For Your Next Test

1. In a right-skewed distribution, the mean is typically greater than the median, and the mode is often the smallest value in the distribution.
2. Right-skewed distributions are common in real-world data, such as income levels, stock returns, and waiting times, where a small number of very high values pull the mean to the right.
3. The chi-square distribution, which is used in hypothesis testing, is a right-skewed distribution, with the degree of skewness decreasing as the number of degrees of freedom increases.
4. The F-distribution, which is used in the analysis of variance (ANOVA), is also a right-skewed distribution, with the degree of skewness depending on the degrees of freedom in the numerator and denominator.
5. Identifying the skewness of a distribution is important in statistical analysis, as it can inform the choice of appropriate statistical tests and the interpretation of results.

Review Questions

• Explain how the relationship between the mean, median, and mode is affected in a right-skewed distribution.
  • In a right-skewed distribution, the mean is typically greater than the median, which is greater than the mode. This is because the longer tail on the right side of the distribution pulls the mean towards the higher values, while the median and mode are less affected by the extreme values. The concentration of data points on the left side of the distribution results in the mode being the smallest value in the distribution.
• Describe the implications of a right-skewed distribution in the context of the chi-square distribution.
  • The chi-square distribution, which is used in hypothesis testing, is a right-skewed distribution. This means that the distribution has a longer tail on the right side, with a higher concentration of values on the left. As the number of degrees of freedom increases, the degree of skewness in the chi-square distribution decreases, making it more symmetric. The right-skewed nature of the chi-square distribution affects the interpretation of test statistics and the corresponding p-values, as the distribution is not centered around zero like a normal distribution.
• Analyze the relationship between the right-skewed nature of the F-distribution and its application in ANOVA.
  • The F-distribution, which is used in the analysis of variance (ANOVA), is also a right-skewed distribution. The degree of skewness in the F-distribution depends on the degrees of freedom in the numerator and denominator. A right-skewed F-distribution indicates that the majority of the values are concentrated on the left side of the distribution, with a longer tail on the right. This skewness affects the interpretation of the F-statistic in ANOVA, as the test statistic is compared to the critical values of the F-distribution to determine the significance of the observed differences between groups. The right-skewed nature of the F-distribution must be taken into account when making inferences about the underlying population parameters.

Related terms

Skewness:

Skewness is a measure of the asymmetry of a probability distribution. It quantifies the degree and direction of skew in a dataset.

Median:

The median is the middle value in a sorted dataset, dividing the data into two equal halves.

Mode:

The mode is the value that appears most frequently in a dataset.