Descriptive Statistics and Inferential Statistics

Descriptive and inferential statistics are two central areas of statistics. Exploratory data analysis is a closely related approach used to discover patterns and possible relationships. Descriptive statistics summarize the data at hand. Inferential statistics use sample data to estimate population characteristics or test claims about a population while accounting for sampling uncertainty.

Overview

A common goal in statistics is to learn about a population. In most cases, it is not practical to collect data from every member of the population, so a sample is taken. To support reliable conclusions, the sample should represent the population of interest. This sample is then analyzed using descriptive statistics, which helps summarize key characteristics, such as the mean and the variability within the sample.

Describing the sample alone does not tell us how closely its results reflect the population. That is the role of inferential statistics. Inferential methods use sample data to estimate unknown population parameters and quantify the uncertainty in those estimates.

Inferential statistics therefore draws conclusions that extend beyond the observed data. Common methods include confidence intervals and hypothesis tests such as the t-test or analysis of variance.

Descriptive statistics

After collecting data, useful first steps are to visualize the data, calculate suitable summary measures, and examine the distribution. These are tasks of descriptive statistics.

Thus, the goal of descriptive statistics is to gain an overview of the distribution of data sets. Descriptive statistics help to describe and illustrate data sets.

Descriptive statistics describes only the observed data. By itself, it does not justify conclusions about a larger population or results at another time. Such conclusions require inferential statistics and an appropriate study or sampling design. The main areas of descriptive statistics can be summarized as follows:

The appropriate summary depends on the research question and the variable's level of measurement. Common summaries include:

• Measures of location: Mean, median, mode, sum
• Measures of dispersion: Standard deviation, variance, range
• Tables: Absolute, relative and cumulative frequencies
• Charts: Histograms, bar charts, box plots, scatter plots, scatterplot matrix

The first group consists of measures of location, such as the mean, median, and mode. They describe the center or a typical value of a data set. Which measure is most informative depends on the distribution and level of measurement.

The second group are measures of dispersion. They provide information about how spread out the values are. Some measures, such as the standard deviation, describe variability around the mean; others, such as the range or interquartile range, describe spread in different ways. A common example is the standard deviation.

Which measures of location and dispersion are suitable depends on the variable's distribution and level of measurement. Common levels are metric, ordinal, and nominal.

Finally, a large area of descriptive statistics is diagrams such as the bar chart, the pie chart, or the histogram.

Descriptive Statistics Example

Suppose a random sample of 10 male basketball players is selected and their heights are measured in meters.

Once you have copied the data into the table of the Online Statistics Software, click on "Descriptive Statistics" in the calculator and select the variable "Body height".

numiqo will now give you the following table of descriptive statistics (relevant measures of dispersion and location) for the players' heights.

Inferential Statistics

In contrast to descriptive statistics, inferential statistics uses sample data to estimate characteristics of a population or test claims about that population. Because surveying an entire population is often impractical, researchers collect a sample—for example, 1,000 people selected from all Canadian citizens. The reliability of conclusions depends on how the sample was selected, the study design, the assumptions of the method, and the amount of sampling uncertainty.

The appropriate method depends on the question and type of data. Tests for group differences include the t-test and analysis of variance (ANOVA). The chi-square (χ²) test can assess associations between categorical variables. Methods for examining relationships between numerical variables include correlation analysis and regression.

In the Hypothesis Test Calculator you can calculate many inferential tests directly in your browser.

Inferential Statistics Example

In the example above, descriptive statistics summarizes the observed sample of 10 basketball players. To draw a conclusion about the population from this sample, inferential statistics is needed. For example, we might ask whether the population mean height of male basketball players differs from a known reference mean for adult men. A one-sample t-test can compare the sample mean with that reference value, provided its assumptions are reasonable.

We might also ask whether basketball players are taller than football players. For this purpose, independent samples of basketball and football players could be selected. Their mean heights can then be compared using an independent t-test. Now a statement can be made about whether the population mean heights differ, subject to the quality of the samples and the test assumptions.

A difference observed in the samples may partly reflect random sampling variation. Therefore, the result should be reported with its estimated effect, uncertainty—for example, a confidence interval—and the hypothesis-test result, rather than as a certain conclusion.

Example of descriptive and inferential statistics

Testing a New Medication's Effectiveness

A pharmaceutical company has developed a new medication to lower blood pressure. To determine whether the medication is effective, researchers conduct a study with 200 patients randomly assigned to a medication group or a control group.

Descriptive Statistics

Average Blood Pressure Reduction: The researchers calculate the average reduction in each group. Suppose the mean reduction is 15 mmHg in the medication group and 10 mmHg in the control group.

Standard Deviation: They also calculate the standard deviation to describe the variability in blood pressure reduction within each group. Suppose it is 5 mmHg in the medication group. This value describes the typical distance of the observed reductions from their mean; its interpretation should be considered together with the shape of the distribution.

Frequency Distribution: They create a histogram to show how many patients experienced different levels of blood pressure reduction in each group, providing a visual summary of the observed responses.

These descriptive statistics give the company an understanding of the average response and variability in blood pressure reduction within the sample.

Inferential Statistics

Hypothesis Testing: To test whether the medication causes a greater reduction than an alternative treatment or placebo, patients should ideally be randomly assigned to treatment and control groups. A suitable test can then compare the groups. A small p-value indicates that the observed difference would be unusual under the null hypothesis; it does not by itself prove that the medication is effective or show whether the effect is clinically important.

Confidence Interval: The researchers also calculate a 95% confidence interval for the difference in population mean reductions. For instance, an estimated difference of 5 mmHg might have a confidence interval from 3 to 7 mmHg. This interval gives a range of effect sizes that are reasonably compatible with the data and the statistical model. In repeated sampling, 95% of intervals calculated in this way would contain the true population mean difference.

Generalizing to the Population: Based on the results of the t-test and the confidence interval, the researchers can draw a cautious conclusion about the population represented by the study participants. Generalization to a broader population also depends on the sampling method, study design, adherence, missing data, and other potential sources of bias.

This example illustrates how descriptive statistics summarize sample data, while inferential statistics help researchers draw appropriately qualified conclusions about the population represented by the study.