Question

1.56 distributions and appropriate statistics (part ii): for each of the following, state whether you expect the distribution to be symmetric, right skewed, or left skewed. also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or iqr. explain your reasoning. (a) housing prices in a country where 25% of the houses cost below $350,000, 50% of the houses cost below $450,000, 75% of the houses cost below $1,000,000 and there are a meaningful number of houses that cost more than $6,000,000. the distribution is expected to be: left skewed symmetric right skewed a typical observation is best represented by the: median mean the variability in the observations is best measured by the: standard deviation iqr (b) housing prices in a country where 25% of the houses cost below $300,000, 50% of the houses cost below $600,000, 75% of the houses cost below $900,000 and very few houses that cost more than $1,200,000. the distribution is expected to be: left skewed

Explanation:

Step1: Analyze skewness for (a)

Since a meaningful number of houses cost more than $6,000,000 while 50% cost below $450,000, there are large - valued outliers on the right. So the distribution is right - skewed.

Step2: Choose measure of central tendency for (a)

In a right - skewed distribution, the mean is pulled towards the outliers. The median is less affected by outliers. So the median best represents a typical observation.

Step3: Choose measure of variability for (a)

In a right - skewed distribution with outliers, the standard deviation is affected by outliers. The inter - quartile range (IQR) is more robust. So the IQR best represents the variability.

Step4: Analyze skewness for (b)

25% of houses cost below $300,000, 50% below $600,000, 75% below $900,000 and very few cost more than $1,200,000. The data seems to be relatively evenly spread and there are no extreme outliers pulling the distribution in one direction. So the distribution is symmetric.

Step5: Choose measure of central tendency for (b)

In a symmetric distribution, the mean and median are equal (or very close). The mean can represent a typical observation.

Step6: Choose measure of variability for (b)

In a symmetric distribution without extreme outliers, the standard deviation is an appropriate measure of variability.

Answer:

(a)
Right skewed
Median
IQR
(b)
Symmetric
Mean
Standard deviation