Symmetrical Distribution

Published Sep 8, 2024

Definition of Symmetrical Distribution

A symmetrical distribution is a type of probability distribution in which values are evenly distributed around the mean. When divided in half, the left and right sides of the distribution mirror each other. In a perfectly symmetrical distribution, the mean, median, and mode are all the same value. This characteristic makes symmetrical distributions easy to identify and analyze in statistics and economics.

Example

Consider the heights of a group of people where most people are of average height with equal numbers of taller and shorter individuals. If plotted on a graph, the distribution of heights will form a bell-shaped curve that is symmetric about the average height.

Another common example is the normal distribution, also known as the Gaussian distribution. Picture a class of students taking a standardized test. The scores likely follow a symmetrical distribution if the test is well-designed. Most students will score around the average mark, with the number of students scoring extremely high or low decreasing symmetrically as you move away from the mean.

Why Symmetrical Distributions Matter

Symmetrical distributions are significant in economics and statistics for several reasons:

• Ease of Analysis: Symmetrical distributions simplify the calculation of statistical measures such as variance, standard deviation, and confidence intervals.
• Predictive Value: They are often used in making economic forecasts and in quality control processes because they allow for consistent and reliable predictions.
• Central Tendency: The alignment of mean, median, and mode makes it easier to assess central tendency and understand the spread and skewness of data.

Frequently Asked Questions (FAQ)

What is an example of a symmetrical distribution in real life?

A real-life example of a symmetrical distribution is the distribution of IQ scores within a population. IQ scores are designed to follow a normal distribution, which is symmetric around the mean of 100. This means there are as many people with IQ scores above the average as there are below it.

Can all distributions be symmetrical?

No, not all distributions are symmetrical. Many real-world datasets are skewed, meaning they are asymmetric. For example, income distribution within a society is often right-skewed, with a small number of people earning very high incomes while the majority earn lower incomes. These types of distributions require different statistical treatments and analysis.

How do you test if a distribution is symmetrical?

To test if a distribution is symmetrical, you can use statistical tools and visual inspections:

1. Visual Inspection: Plotting the data on a histogram or a probability plot can provide a visual clue. If the plot forms a mirror image around the central axis (the mean), the distribution is likely symmetrical.
2. Statistical Measures: Compare the mean, median, and mode. If these values are equal or very close, the distribution might be symmetric.
3. Skewness Coefficient: Calculate the skewness of the dataset. A skewness value close to zero indicates a symmetrical distribution.
4. Quantile-Quantile (Q-Q) Plot: This plot compares the quantiles of the data with the quantiles of a normal distribution. If the points lie on the line y=x, the distribution is symmetrical.

Are there any economic models that assume a symmetrical distribution?

Yes, several economic models and theories assume symmetrical distributions for simplicity and analytical tractability. For instance, the efficient market hypothesis often assumes that stock returns are normally distributed. This assumption helps in deriving and validating financial models like the Black-Scholes option pricing model. Additionally, symmetric distributions are frequently used in macroeconomic models to analyze economic indicators such as GDP growth rates and inflation.