When a dataset contains a large number of different scores spanning a broad range, it is often more effective to summarize the data using a grouped frequency table. Instead of listing individual scores, the first column displays ranges of values (typically between $$5$$ and $$15$$ equal-width intervals), and the second column shows the frequency of scores falling into each of those ranges.

Grouped Frequency Table

Frequency tables can also be used to display the distribution of categorical variables. In this application, the first column lists the category labels rather than numerical values. While the ordering of these categories is somewhat arbitrary, they are frequently arranged from the most common category at the top to the least common at the bottom.

Frequency Tables for Categorical Variables

A practical illustration of a frequency table involves a hypothetical distribution of scores on the Rosenberg Self-Esteem Scale for a sample of $$40$$ college students. In this table, the first column lists the individual self-esteem scores descending from $$24$$ down to $$15$$. The second column indicates the frequency of each score, revealing that $$3$$ students scored $$24$$, $$5$$ students scored $$23$$, and $$10$$ students scored $$22$$. By organizing the data this way, it becomes immediately apparent that the scores range from $$15$$ to $$24$$, the most common score is $$22$$, and the least common score is $$17$$, which has a frequency of $$0$$.

Frequency Table of Self-Esteem Scores

Which of the following best describes the primary structure and purpose of a frequency table?

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.

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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

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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

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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