Question

Symmetric or Skewed? Do you expect the distributions of the following variables to be symmetric or skewed? Explain. a. Size in dollars of nonsecured loans b. Size in dollars of secured loans c. Price of an 8 -ounce can of peas d. Height in inches of freshman women at your university e. Number of broken taco shells in a package of 100 shells f. Number of ticks found on each of 50 trapped cottontail rabbits

Short answer

a. Size in dollars of nonsecured loans: _____________ b. Size in dollars of secured loans: ________________ c. Price of an 8-ounce can of peas: _________________ d. Height in inches of freshman women at your university: __________________ e. Number of broken taco shells in a package of 100 shells: _________________ f. Number of ticks found on each of 50 trapped cottontail rabbits: _________________

Step-by-step solution

a. Size in dollars of nonsecured loans

We can expect this variable to be skewed because nonsecured loans are more likely to be smaller in size since they are riskier for lenders. Lenders will have many small loans and only a few large loans, causing the distribution of loan sizes to be right-skewed since there will be a long tail on the right.

b. Size in dollars of secured loans

This variable would likely be skewed as well. Secured loans might be more evenly distributed between small and large loans since they are less risky for lenders. However, since larger loans are still less common, we can expect the distribution to be right-skewed but not as much as for nonsecured loans.

c. Price of an 8-ounce can of peas

We can expect this variable to be symmetric. The price of an 8-ounce can of peas should not vary significantly, and any variations should be consistent across different stores and brands. Thus, the distribution of prices would be concentrated around a single peak, forming a symmetric bell-shaped curve.

d. Height in inches of freshman women at your university

We can expect this variable to be symmetric. Human heights usually follow a normal distribution, which is a symmetric distribution. As a result, the height in inches of freshman women at the university will likely follow a symmetrical bell-shaped curve, with a single peak representing the average height.

e. Number of broken taco shells in a package of 100 shells

This variable would likely be skewed. The number of broken taco shells would likely range from 0 to a few, but it is very unlikely to have many broken shells in a single package. This distribution will likely be right-skewed, because there will be a long tail on the right corresponding to the low probability of having many broken shells.

f. Number of ticks found on each of 50 trapped cottontail rabbits

This variable can be skewed. Ticks might be distributed unevenly among rabbits, with some having no ticks, some having a few, and others might have many. This variation in tick count will likely cause the distribution to be right-skewed, with a long tail on the right because of the low probability of a rabbit having an extremely high number of ticks.

Key concepts

Symmetric Distribution

In data distribution, a symmetric distribution is a type of distribution where the left and right sides are mirror images of each other. This often means that the data is evenly spread around the central point. A common example of a symmetric distribution is a bell curve, where most data points are concentrated around the mean.
For example, if we consider the height of freshman women at a university, it is likely to be symmetrically distributed. This means the heights that are below average are as frequent as those that are above average.

• The average or mean is at the center of the distribution.
• The data falls off evenly as you move away from the mean in both directions.
• Symmetric distributions are often easier to analyze statistically, as many statistical tests assume symmetry.

Understanding symmetric distribution is important, as it helps in identifying normal distributions and analyzing data effectively.

Skewed Distribution

A skewed distribution occurs when data is not symmetrical and tends to lean more towards one side. This can result in a long tail on one side of the distribution. Skewness tells us about the direction of this tail.
Typical causes of skewness include data that involves limits or constraints, such as income data where few individuals earn extremely high wages, skewing the distribution.

• In a left-skewed distribution, the tail is longer on the left side, with most of the data concentrated on the higher end.
• In a right-skewed distribution, the tail is longer on the right side, with most of the data concentrated on the lower end.

Understanding the skewness of data helps us in selecting the appropriate statistical techniques to analyze the data and in making more informed conclusions about the quantitative patterns present.

Normal Distribution

The normal distribution is a type of symmetric distribution that is very often found in nature. It's also known as the Gaussian distribution. It forms a perfectly symmetrical bell-shaped curve and is defined by its mean and standard deviation. The properties of the normal distribution make it an essential concept in statistics.

• Most of the observations are clustered around the mean.
• Approximately 68% of data falls within one standard deviation of the mean.
• About 95% of data falls within two standard deviations.
• The normal distribution is used in many statistical analyses and hypothesis testing.

An example is the distribution of heights of people, which often approximates a normal distribution, making it a useful model for data analysis.

Right-Skewed Distribution

A right-skewed distribution, also known as positively skewed, is when the tail of the distribution stretches more towards the right side. This is a common type of distribution in datasets where many occurrences happen at the lower end, with fewer and fewer occurrences happening as you move to a higher value.
Examples of right-skewed distributions include income levels, where more people earn lower wages than higher ones, causing a longer tail on the right.

• The mean of the dataset is usually greater than the median.
• The mode is typically less than the median.
• Right-skewed data might require transformations for certain statistical analyses.

Understanding right-skewed distributions helps analysts recognize biases and plan interventions or analyses accordingly.