A real estate agent is marketing a neighborhood of 10 homes. Nine of the homes are valued at approximately $200,000 each, while one newly built mansion is valued at $3,200,000. The agent advertises the 'average home value' in the neighborhood as $500,000. Which statement best evaluates the agent's use of this figure?

Consider a dataset of four reaction times: $$200$$, $$250$$, $$280$$, and $$250$$ milliseconds (ms), which has a mean of $$245$$ ms. If an outlier score of $$5,000$$ ms is added—perhaps due to participant inattention—the mean drastically increases to $$1,445$$ ms. This new mean is larger than $$80\%$$ of the scores in the dataset, demonstrating how a single extreme outlier can pull the mean so far that it no longer accurately represents the typical behavior in the distribution.

Example of the Mean's Sensitivity to Outliers

In a highly skewed distribution, how does the presence of extreme scores typically affect the mean?

A key characteristic of the mean is its high sensitivity to extreme scores, or outliers. In highly skewed distributions, the mean is pulled away from the median in the direction of the skew (the longer tail). Because extreme scores heavily influence its value, the mean may cease to be an accurate representation of the typical score in such datasets, leading researchers to often prefer the median for skewed distributions.

Google

A skewed distribution is an asymmetrical shape of a data distribution where scores cluster heavily toward one end of the scale, creating a prominent peak, while a relatively long tail trails off toward the opposite end.

Skewed Distribution

An outlier is an extreme score that falls significantly above or below the rest of the scores within a distribution. While these extreme values can represent genuine cases on the variable being measured—such as a single clinically depressed individual in an otherwise happy sample—they can also result from equipment malfunctions, participant misunderstandings, or researcher errors.

Outlier

The median is a measure of central tendency that identifies the exact middle score of a distribution when all values are arranged from lowest to highest. It divides the distribution perfectly in half, meaning that $$50\%$$ of the scores are less than the median and $$50\%$$ are greater. If a distribution contains an even number of scores, the median is calculated as the value exactly halfway between the two middlemost scores.

Median

The mean is a measure of central tendency representing the average of a distribution, symbolized as $$M$$. It is calculated by finding the sum of all the scores and dividing by the total number of scores. The mean is widely used because it provides a reliable indication of central tendency, is easy to understand, and has properties that are valuable for inferential statistics.

Mean

Spielman, R. M., Jenkins, W. J., & Lovett, M. D. (2020, April 22). Psychology (2nd ed.). OpenStax. https://openstax.org/books/psychology-2e/pages/1-introduction

OpenStax Psychology (2nd ed.) Textbook

https://kpu.pressbooks.pub/psychmethods4e/

KPU Research Methods in Psychology - 4th American Edition

A positively skewed distribution is an asymmetrical shape where the peak of the scores is shifted toward the lower end of its range, resulting in a relatively long positive tail that trails off toward the higher values.

Positively Skewed Distribution

Sensitivity of the Mean to Outliers

A negatively skewed distribution is an asymmetrical shape of a data distribution where the majority of scores are clustered toward the higher end of the scale, creating a prominent peak on the right, while a relatively long tail trails off toward the lower end of the scale on the left.

Negatively Skewed Distribution

Which of the following best describes the shape of a skewed distribution?

When preliminary analyses reveal extreme outliers resulting from data entry errors, misunderstandings, or a lack of participant effort, researchers may justify excluding these anomalous data points. However, strict methodological standards require researchers to apply exclusion criteria consistently, retain the original data without permanently deleting it, and transparently report both the number of exclusions and the specific reasons for their removal.

Excluding Outliers

Extreme outliers may sometimes represent honest and accurate estimates rather than errors. To manage these valid but extreme scores, researchers can either employ resistant statistics like the median, or they can compare analyses conducted both with and without the outliers. If the results differ substantially between the two analyses, best practice dictates reporting both sets of results and thoroughly discussing the discrepancies.

Handling Valid Extreme Outliers

An example of an outlier might involve a single clinically depressed individual whose score on the Beck Depression Inventory falls drastically outside the typical range when measured within a sample of otherwise happy and high-functioning peers.

Example of an Outlier

While outliers are generally understood as extreme values within a dataset, $$z$$ scores provide a formal, quantitative method for identifying them. Commonly, outliers are defined as scores that fall more than three standard deviations away from the mean. In standardized terms, any score with a $$z$$ score strictly less than $$-3.00$$ or strictly greater than $$+3.00$$ is classified as an outlier.

Defining Outliers using z Scores

In a sample composed almost entirely of happy and high-functioning peers, a single clinically depressed person scoring very high on the Beck Depression Inventory would represent an outlier. This occurs because their score represents a truly extreme value relative to the rest of the group.

Outlier in Beck Depression Inventory Scores

An outlier might represent a misunderstanding or lack of effort by a participant. For example, in a task where most participants take only a few seconds to respond, a reaction time of $$3$$ minutes is an outlier. This suggests the participant was not paying attention or did not understand the task.

Reaction Time Outlier Example

An outlier might represent a truly extreme but accurate response rather than an error. For instance, in a university survey where most participants reported having had fewer than $$15$$ sexual partners, a few participants reported extreme scores of $$60$$ or $$70$$. These extreme scores could be honest and accurate estimates rather than intentional exaggerations.

Sexual Partners Survey Outlier Example

Although the range is easy to calculate, it can be highly misleading when outliers are present. A single extreme score can drastically increase the range, creating the false impression that the overall distribution has high variability when most scores are actually grouped closely together.

Impact of Outliers on the Range

$$z$$ scores provide a quantitative method for defining outliers within a distribution. Commonly, an outlier is defined as any score that is more than three standard deviations away from the mean, which corresponds to a $$z$$ score less than $$-3.00$$ or greater than $$+3.00$$.

Identifying Outliers Using z Scores

When handling outliers, researchers must decide how to manage them based on their cause. Data entry errors can be corrected. If an outlier stems from misunderstanding or lack of effort, the response may be excluded using consistent criteria. Importantly, excluded data should not be permanently deleted but set aside for potential future review. When presenting results, researchers must clearly report how many participants or responses were excluded and detail the specific criteria used. For valid extreme responses, researchers can use robust statistics like the median or analyze the data both with and without the outliers.

Handling Outliers

What term is used to describe an extreme score that falls significantly above or below the rest of the scores within a distribution?

To find the median, scores must first be arranged from lowest to highest. For the set of seven scores $$8, 4, 12, 14, 3, 2, 3$$, the ordered sequence is $$2, 3, 3, 4, 8, 12, 14$$. The middle score is $$4$$, which is the median. If an eighth score of $$15$$ is added, the new ordered set is $$2, 3, 3, 4, 8, 12, 14, 15$$. The two middle scores are $$4$$ and $$8$$, so the median is the value halfway between them, which is $$6$$.

Example of Finding the Median

To find the median of a set of seven scores ($$8$$, $$4$$, $$12$$, $$14$$, $$3$$, $$2$$, $$3$$), the scores are rearranged from lowest to highest ($$2$$, $$3$$, $$3$$, $$4$$, $$8$$, $$12$$, $$14$$), making the middle score $$4$$. If an even number of scores exists, such as by adding a score of $$15$$ to the set, there are two scores in the middle of the distribution ($$4$$ and $$8$$), and the median is the value halfway between them, which is $$6$$.

Example of Calculating the Median

What does the median represent in a distribution of scores?

Compared to other measures of central tendency, the mean possesses statistical properties that make it uniquely useful for conducting advanced inferential statistics. However, researchers must consider its sensitivity to outliers before relying on it exclusively.

Usefulness of the Mean in Statistical Analyses

The $$t$$-test is the most frequently used null hypothesis test for evaluating the statistical relationship between two means. There are three primary variations of this procedure—the one-sample $$t$$-test, the dependent-samples $$t$$-test, and the independent-samples $$t$$-test—each designed to accommodate slightly different research designs.

t-Test

The mathematical formula for the mean of a distribution is: $$M = \frac{\Sigma X}{N}$$. In this equation, $$M$$ represents the mean, $$\Sigma$$ (the Greek letter sigma) is the summation sign indicating to add all values of the variable $$X$$, and $$N$$ represents the total number of scores in the distribution.

Statistical Mean Formula

A hypothetical population mean, symbolized mathematically as $$\mu_0$$, serves as a specific, interesting standard of comparison in statistical research. It represents an expected baseline value for an entire population, against which researchers can compare an actually observed sample mean ($$M$$) to draw statistical conclusions.

Hypothetical Population Mean

The mathematical formula for the mean (symbolized as $$M$$) is $$M = \Sigma X / N$$. In this equation, $$\Sigma$$ (the Greek letter sigma) indicates the summation across all values of the variable $$X$$, and $$N$$ represents the total number of scores in the distribution.

Formula for the Mean

A $$z$$ score indicates how many standard deviations a particular raw score is above or below the mean of its distribution. It standardizes the distance of a score from the mean, providing a consistent way to describe a score's relative location.

z Score

The standard deviation is the most widely used measure of variability, representing the average distance that individual scores within a distribution deviate from the mean.

Standard Deviation

Which measure of central tendency is calculated by finding the sum of all scores in a distribution and dividing that sum by the total number of scores?

Learn Before

Related

Learn After