91Ó°ÊÓ

1.42 Distributions and appropriate statistics. For each of the following, describe 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. (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 $$\$ 6,000,000$$ (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 $$\$ 1,200,000$$ (c) Number of alcoholic drinks consumed by college students in a given week. (d) Annual salaries of the employees at a Fortune 500 company.

Step by step solution

01

Analyze Distribution (a)

For part (a), most houses cost below the median price of $450,000, with a substantial number above $6,000,000. This suggests a right skewed distribution because a high percentage of extremely high values pulls the tail to the right. The median, which is $450,000, is a better representation of central tendency, as the mean would be affected by the high-priced houses. Variability is best represented by the IQR, since it is less affected by extreme values.

02

Analyze Distribution (b)

For part (b), housing prices primarily fall under $900,000, with a few exceeding $1,200,000. This implies a distribution relatively more symmetric compared to (a) but still slightly right skewed due to the higher values up to $1,200,000. The median, being $600,000, effectively represents a typical observation, and variability is once again better captured by the IQR due to the slight skewness.

03

Analyze Distribution (c)

For part (c), the number of drinks consumed typically varies widely, often with fewer students consuming a large number of drinks, leading to a right skewed distribution. Here, the median serves as a better measure of central tendency than the mean, as it is not influenced by students who consume exceptionally large amounts. The IQR should be used to describe variability, as it is more robust to outliers.

04

Analyze Distribution (d)

For part (d), in most companies, including Fortune 500 ones, a few high-level employees earn significantly more than others, creating a right skewed distribution of salaries. The median is a more robust measure for central tendency, and the IQR is preferred for describing spread, as both are less influenced by very high salaries.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distribution

Understanding distribution is crucial in statistics. It describes how data points are spread across different values. In simpler terms, distribution tells us the shape of the data. There are three common types of distribution:

• Symmetric Distribution: This occurs when data is evenly spread on both sides of the center point. The mean and median are equal in this case.
• Right-Skewed Distribution: Also known as positively skewed distribution, this happens when there are a few very high values stretching the tail on the right side. Most data are clustered on the left.
• Left-Skewed Distribution: This is the opposite of right skewness, where the fewer very low values stretch the tail on the left side, and most data are on the right.

Each distribution type affects how we interpret other statistics, like central tendency and variability.

Central Tendency

Central tendency represents a typical value within a set of data. It gives us an idea of where most data points are "centered." There are three main measures of central tendency:

• Mean: This is the average of all data points. It's calculated by adding up all the values and dividing by the total number of observations.
• Median: This is the middle value when all data points are ordered from smallest to largest. It’s more resistant to outliers and skewed data.
• Mode: This is the most frequently occurring value in the dataset.

When deciding which measure to use, the choice between mean and median often depends on the distribution. For skewed distributions, the median is typically a better representation since it is not affected by extreme values.

Skewness

Skewness is all about the asymmetry of the distribution. It indicates how data is deviating from the normal distribution. Skewness can be:

• Zero: A perfectly symmetrical distribution with equal data spread on both sides of the mean.
• Positive/Figure-Right: This indicates right skewness. The tail on the right side is longer, showing a few higher values compared to most data.
• Negative/Figure-Left: This suggests left skewness, with a few lower values extending the tail on the left side.

Understanding skewness is important as it helps determine the appropriate statistical measures and representations to use for the data set, such as choosing between the mean or median. Analyzing skewness provides deeper insights into data behavior and prediction modeling.

Variability

Variability, also known as spread, tells us how much the data points differ from each other. It measures the dispersion within a dataset. The two common measures of variability are:

• Standard Deviation: This indicates how much individual data points deviate from the mean. It's useful for symmetrical distributions where variability is blanched.
• Interquartile Range (IQR): This measures the range within the middle 50% of the data. It is calculated as the difference between the first quartile (Q1) and third quartile (Q3). IQR is particularly helpful in skewed distributions as it is robust to outliers.

Choosing the right measure of variability depends heavily on the data's distribution. In skewed distributions, IQR is often favored, whereas, for symmetric data, standard deviation provides better insights.