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

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.

Sensitivity of the Mean to Outliers

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?

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.

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1. The dependent variable must be continuous (interval/ratio).
2. The observations are independent of one another.
3. The dependent variable should be approximately normally distributed.
4. The dependent variable should not contain any outliers.

Assumptions

The distribution of a variable describes how individual scores or values are spread across the different levels or categories of that specific variable within a dataset.

Variable Distribution

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

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

KPU Research Methods in Psychology - 4th American Edition

Dive into Deep Learning

The one sample t-test requires the sample data to be numeric and continuous, as it is based on the normal distribution. Continuous data can take on any value within a range (income, height, weight, etc.). The opposite of continuous data is discrete data, which can only take on a few values (Low, Medium, High, etc.). Occasionally, discrete data can be used to approximate a continuous scale, such as with Likert-type scales.

Level of Measurement

Independence of observations is usually not testable, but can be reasonably assumed if the data collection process was random without replacement. In our example, we would want to select laptop computers at random, compared to using any systematic pattern. This ensures minimal risk of collecting a biased sample that would yield inaccurate results.

Independence

Bell-shaped distribution, often referred to as Gaussian. To test the assumption of normality, a variety of methods are available, but the simplest is to inspect the data visually using a histogram or a Q-Q scatterplot. Real-world data are almost never perfectly normal, so this assumption can be considered reasonably met if the shape looks approximately symmetric and bell-shaped. The data in the example figure below is approximately normally distributed.

Normal Distribution

Outlier

A frequency table is a tabular method used to display the distribution of a variable. It organizes data by listing the possible values of the variable in one column and the frequency—the number of times each score occurs—in a second column. For numerical data, the values are typically ordered from highest at the top to lowest at the bottom, and only the range of scores actually present in the dataset is included. This format allows researchers to quickly observe important features of the distribution, such as the range of scores, the most and least common values, and any extreme outliers.

Frequency Table

A unimodal distribution is a specific shape of a distribution that features exactly one distinct peak. In a histogram, this appears as a single prominent high point from which the tails taper off.

Unimodal Distribution

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

A variable's distribution can be illustrated using both quantitative and categorical variables. For instance, in a sample of 100 students, the distribution for the quantitative variable 'number of siblings' might show that 10 students have no siblings, 30 have one, and 40 have two. Similarly, the distribution for the categorical variable 'sex' in the same sample might indicate that 44 students have a score of 'male' and 56 have a score of 'female'.

Examples of a Variable Distribution

The shape of a quantitative variable's distribution describes the visual pattern formed when its scores are plotted, such as in a histogram. A distribution's shape is characterized by the number of distinct peaks it contains—such as being unimodal or bimodal—and whether its tails fall symmetrically or are skewed to one side.

Distribution Shape

The percentile rank of an individual score describes its relative location within a distribution. It is defined as the percentage of all scores in the distribution that are strictly lower than that specific score. This measure provides a standardized way to compare an individual's performance against the rest of the group and is frequently used to report results on standardized tests of ability or achievement.

Percentile Rank

Measures of central tendency are descriptive statistics used to describe the typical, average, or center scores within a distribution. Common examples of these measures include the mean, median, and mode.

Measures of Central Tendency

Measures of dispersion are descriptive statistics that describe how spread apart or dispersed the scores in a distribution are. Common examples of these measures include the range, standard deviation, and variance.

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