Question

The Housing Price data files gives you sales of all single-family houses around Iowa State University, in Ames IA. Assume that sales price is a continuous distribution. Transform sales price, the continuous distribution, to a standard normal distribution. Comment on your findings. Is it symmetrical or skewed?

Answer 1

To transform sales prices to a standard normal distribution, calculate the mean and standard deviation of the original sales prices, then convert each price to a Z-score. A histogram of the Z-scores can help determine if the distribution is symmetrical or skewed, with real estate prices often being right-skewed.

Step-by-step explanation:

To transform sales prices, which are a continuous distribution, to a standard normal distribution, we use the concept of normalization. This requires calculating the mean (average) and the standard deviation of the sales prices. Here are the steps to perform this transformation:

1. Compute the mean (average) sales price, denoted as μ (mu).
2. Compute the standard deviation of the sales prices, denoted as σ (sigma).
3. For each sales price X, subtract the mean from the price and then divide by the standard deviation to get the standardized value Z:
   Z = (X - μ) / σ

Once we have converted the sales price data into Z-scores, we end up with a standard normal distribution, which should have a mean of 0 and a standard deviation of 1.

To determine if the transformed distribution is symmetrical or skewed, we can construct a histogram of the Z-scores and look for the shape of the distribution. If the histogram resembles a bell curve with both sides mirroring each other, the distribution is symmetrical. However, if one tail is longer or fatter than the other, then the distribution is skewed.

Typically, the distribution of house prices tends to be right-skewed because there are a few very high values that pull the mean to the right, but this observation should be verified with the actual histogram of the Z-scores.