A population parameter is a numerical value representing a specific characteristic of an entire population, such as the true mean or correlation. Because researchers typically cannot measure every individual within a population, they must rely on sample statistics to estimate these corresponding population-level values.

Population Parameter

A statistical range calculated around a sample statistic designed to contain the true population parameter a specified percentage of the time, most commonly $$95\%$$. Some researchers advocate using confidence intervals as an alternative to traditional null hypothesis testing because they are often more intuitive to interpret. Furthermore, a confidence interval inherently provides the data needed to perform a null hypothesis test; if a hypothetical population mean falls completely outside the calculated interval, the sample mean is considered statistically significantly different from that hypothetical value.

Confidence Interval

There is a fundamental distinction between probabilities and statistics. Probabilities are theoretical quantities that describe the underlying data generating process, such as the inherent fairness of a coin. In contrast, statistics are empirical quantities computed directly from observed data, such as the observed frequency of a specific outcome in a sample.

Probability vs. Statistics

A statistical estimator is a specially designed function applied to an observed dataset to produce empirical estimates of underlying theoretical model parameters, such as probabilities. For example, a natural estimator for the true probability of an event is the ratio of its observed occurrences to the total number of sampling trials.

Statistical Estimator

A descriptive numerical summary, such as a mean or a correlation coefficient, that is calculated from the data of a specific subset of observed individuals is called a ____.

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.

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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

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KPU Research Methods in Psychology - 4th American Edition

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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

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 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

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

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

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.

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