An example of a unimodal distribution can be observed in a histogram plotting the self-esteem scores of a sample of students, where a single, distinct peak emerges near the middle of the graph with tails tapering off in both directions.

Example of a Unimodal Distribution

A typical distribution of self-esteem scores features a distinct peak situated near the middle, with tails that taper off in either direction from this peak. It serves as an example of a unimodal distribution simply because it possesses one distinct peak.

Unimodal Distribution of Self-Esteem Scores

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

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.

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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

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

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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

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

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 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.

Measures of Dispersion

What does the distribution of a variable describe in a dataset?

A symmetrical distribution is a specific shape of data distribution where the left and right halves are mirror images of each other. In such a distribution, the scores spread out evenly on both sides of a central peak.

Symmetrical Distribution

A bimodal distribution is a specific shape of data distribution characterized by two distinct peaks, representing two scores or categories that occur most frequently. While less common than unimodal distributions in psychological research, they can emerge when different subsets of a sample respond divergently.

Bimodal Distribution

The shape of a distribution affects the relationship between the mean, median, and mode. In a unimodal and symmetrical distribution, all three measures are very close to each other at the peak. In a bimodal distribution, the mode aligns with the tallest peak, while the mean and median fall between the peaks. In a skewed distribution, the mean is pulled in the direction of the longer tail (the skew), causing it to differ from the median. For highly skewed data, the median is generally preferred because the mean becomes an inaccurate representation of the distribution's center.

Impact of Distribution Shape on Central Tendency

When describing the visual pattern of a quantitative variable's distribution on a histogram, which of the following characteristics are primarily used to define its shape?

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