An example of variability can be observed when comparing two distributions that share the same central tendency (e.g., a mean, median, and mode of $$10$$) but differ in their spread. In a distribution with relatively low variability, all scores are clustered close to the center. Conversely, in a distribution with relatively high variability, the scores are spread across a much greater range.

Example of Variability in Distributions

The range is a straightforward measure of variability calculated as the difference between the highest and lowest scores in a distribution.

Range

The variance, symbolized as $$SD^2$$, is a measure of variability defined as the mean of the squared differences between each score and the distribution's mean. While variance is a foundational measure in inferential statistics, its square root—the standard deviation—is typically used in descriptive statistics.

Variance

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

What do measures of dispersion primarily describe within a dataset?

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.

Google

Descriptive statistics are a set of quantitative techniques used to summarize, organize, and visually display the data collected from a research sample. Important examples include measures of central tendency (such as the mean, median, and mode), which describe average scores, and measures of dispersion (such as the range, standard deviation, and variance), which indicate how spread apart the scores are.

Descriptive Statistics

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

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

KPU Research Methods in Psychology - 4th American Edition

A numerical index, commonly represented as Pearson's $$r$$, that measures the direction and strength of a linear relationship between two quantitative variables. The coefficient is bounded between $$-1.00$$ and $$+1.00$$, where values further from zero indicate a tighter, more predictable association, and values near zero indicate a negligible relationship. In research, this coefficient frequently serves as an effect size measure to evaluate how strongly variables are connected in the population.

Correlation Coefficient

A sample statistic is a descriptive numerical summary—such as a mean or a correlation coefficient—that is calculated from the data of a specific sample. Researchers compute these statistics to summarize their observed data and to make inferences about the corresponding, unknown values in the broader population.

Sample Statistic

In experimental research studies where an independent variable is manipulated to create multiple groups and a quantitative dependent variable is measured, researchers typically compute descriptive statistics like means and standard deviations. The means from each experimental condition are calculated separately and then compared to determine if they differ from one another.

Descriptive Statistics in Experimental Research

In non-experimental research, descriptive statistics often involve computing simple percentages to describe the proportion of individuals engaging in a certain behavior or holding a specific belief. However, researchers more commonly use this method to compute a correlation coefficient to assess the statistical relationship between two naturally occurring variables.

Descriptive Statistics in Non-Experimental Research

Inferential statistics are a set of mathematical techniques used by researchers to determine whether a statistical result or relationship observed in a study's sample reflects something that is genuinely true of the broader population.

Inferential Statistics

Before conducting inferential statistics, researchers must thoroughly understand their data at a descriptive level to clearly see 'what happened' in their study. Even though inferential statistics are required for formal reporting, descriptive statistics alone can provide strong preliminary evidence about whether a treatment worked or if variables are unrelated.

Understanding Descriptive Statistics

A scatterplot is a graph used to display the relationship between two quantitative variables, particularly when the independent variable on the $$x$$-axis possesses a wide range of levels. In this type of visualization, each individual's scores are represented by a single point plotted at the intersection of the values on the $$x$$ and $$y$$ axes, with no lines connecting the data points.

Scatterplot

Tables are efficient tools used to present large amounts of data, such as means and standard deviations, particularly in complex research designs. A well-constructed table should be simple, self-interpretable, and focus on highlighting major trends rather than merely repeating statistical details that are already covered in the text.

Tables in Research Reports

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

In psychological research, statistical relationships between variables generally take two basic forms: differences between distinct groups or conditions, and correlations between quantitative variables. Researchers utilize descriptive statistics to examine and communicate these relationships in greater detail.

Forms of Statistical Relationship

A widely adopted statistical metric used to quantify the effect size, or the standardized difference, between two independent group means. It is calculated by taking the absolute difference between the two sample means and dividing it by the standard deviation. Because it measures the actual magnitude of an experimental effect, researchers often report it alongside null hypothesis tests to provide a clearer picture of relationship strength.

Cohen's $$d$$

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

A bar graph is a visual representation generally used to present and compare the mean scores for two or more groups or conditions. Researchers conventionally use a bar graph when the independent variable plotted on the $$x$$-axis is categorical.

Bar Graph

A line graph is used to display data when the independent variable plotted on the $$x$$-axis is quantitative or measured in a continuous manner, such as time. Line graphs are also used to present correlations between quantitative variables when the independent variable has, or is organized into, a relatively small number of distinct levels. Each point on the graph represents the mean score on the dependent variable for participants at one level of the independent variable.

Line Graph

What is the primary purpose of using descriptive statistics in psychological research?

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

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.

Learn Before

Related

Learn After