• Provide an example of a scenario where the measures of central tendency are skewed as a result of outliers. How can such a situation be identified and addressed? Next, provide an example of what information SS provides us about a set of data. Under what circumstances will the value of SS equal 0 and is it possible for SS to be negative?
• What is a sampling distribution? What does the knowledge of σ contribute to a researcher’s understanding of the theoretical sampling distribution (regarding its characteristics and shape)? Contrast a t distribution from that of the standard normal distribution. In what ways might N affect the CI?
This situation can be identified by visually inspecting the data using a histogram or a box plot. A histogram will show a long tail extending from one side, indicating skewness, while a box plot will show the outlier as a single point far from the rest of the data. To address this, researchers can use the median as the primary measure of central tendency or use statistical methods to trim or remove the outlier before calculating the mean.
Sum of Squares (SS)
The Sum of Squares (SS) provides information about the total variability or dispersion within a set of data. It is calculated by summing the squared differences between each data point and the mean of the data set. A larger SS value indicates greater variability, meaning the data points are more spread out from the mean.
The value of SS will equal 0 when all data points in the set are identical to the mean. In this case, each difference from the mean is 0, and the sum of their squares is also 0. It is not possible for SS to be negative because the calculation involves squaring each difference, which will always result in a positive value.
Sampling Distribution and Statistical Distributions
A sampling distribution is a probability distribution of a statistic (e.g., the mean) that is obtained by drawing all possible samples of a specific size from a population. It describes the behavior of a sample statistic across different samples.
Knowledge of the population standard deviation (σ) provides crucial information about the theoretical sampling distribution of the mean.
Sample Answer
Outliers and Measures of Central Tendency
An example of a scenario where measures of central tendency are skewed by outliers is in real estate prices. Imagine a small neighborhood where 9 houses sell for an average price of $300,000, but one luxury mansion sells for $5 million.
The mean (average) would be heavily skewed by the mansion: ($300,000×9)+$5,000,000/10=$770,000. The mean suggests the typical house price is much higher than it actually is.
The median is the middle value, which would be one of the $300,000 houses. The median, therefore, provides a more accurate representation of the typical home price.