Asymmetrical Distribution

TL;DR

• A distribution in which values are concentrated more on one side of the mean, producing a lopsided histogram.
• Two forms: positive skew (tail extends right) and negative skew (tail extends left).
• Skewness affects choice of central tendency (mean vs median) and can change how dispersion measures (like standard deviation) should be interpreted.

Definition

Asymmetrical distribution, also known as skewed distribution, is a type of probability distribution where the values are not evenly distributed around the central value or mean; the distribution is not symmetrical, with values concentrated more on one side of the mean than the other.

Explanation

• Visually, an asymmetrical distribution appears lopsided in a histogram and lacks the mirror symmetry of a symmetrical distribution.
• Positive skew: values are concentrated more on the right side of the mean, with the tail extending to the right. A few much higher values pull the mean to the right and make the distribution skewed.
• Negative skew: values are concentrated more on the left side of the mean, with the tail extending to the left. A few much lower values pull the mean to the left and make the distribution skewed.
• Skewness affects measures of central tendency and dispersion: the mean, median, and mode may not coincide in a skewed distribution, and standard deviation may not accurately reflect dispersion when the distribution is skewed.
• The presence of skewness does not automatically mean the data is not normal; many distributions can be skewed and still be considered normal if a few extreme values pull the mean to one side.

Examples

Positive skew example

• Income of a group of people, where a few people may have extremely high incomes while the majority have lower incomes.

Negative skew example

• Height of a group of people, where a few people may be very short while the majority are taller.

Use cases

• Statistical analysis and decision making where the choice of central tendency matters (e.g., deciding between mean and median).
• Interpreting dispersion measures such as standard deviation when the distribution is skewed.

Notes or pitfalls

• Mean, median, and mode may differ in a skewed distribution; choosing the appropriate measure of central tendency depends on the skewness.
• Standard deviation may not accurately reflect dispersion in a skewed distribution.
• Skewness does not necessarily imply the data are not normal; a normal distribution can appear skewed if a few extreme values pull the mean to one side.

Related terms

• Skewed distribution
• Positive skew
• Negative skew
• Mean
• Median
• Mode
• Standard deviation
• Variance
• Histogram