Illustrating Different Statistical Methods
Illustrating Different Statistical Methods
Learn how samples support inference, how confidence and significance are interpreted, and how hypothesis tests, t-tests, chi-square, regression, and correlation reveal relationships in data.
Inferential Statistics & Statistical Tests
Inferential statistics uses evidence and reasoning from data to reach conclusions. Unlike descriptive analysis, which describes the data at hand, inferential methods can use a sample to make estimates or test claims about a larger population.
Population
The complete group of records that meets the criterion being studied. A population is not limited to people; it can be products, transactions, website visits, test scores, or other records.
Sample
A subset of a larger population. Sampling is valuable when collecting information from every member of the population is difficult or impossible.
Choose the Right Test
A statistical test must match the characteristics of the data. A method designed for normally distributed data should not be relied upon when its assumptions are not satisfied.
Statistical tests help determine whether an observed difference or relationship is sufficiently supported by the data rather than simply reporting that two values look different.
Sampling Methods
Different sampling approaches suit different data structures and research needs. Tap each card to review the method.
Simple random sampling is straightforward. Stratified sampling helps preserve representation of groups. Systematic sampling can efficiently inspect large ordered datasets, such as checking every tenth product on an assembly line.
Confidence Intervals & P-Values
Confidence intervals describe uncertainty around an estimate, while p-values are used in hypothesis testing to evaluate how compatible observed results are with the null hypothesis.
Confidence Interval
A range used to express the uncertainty of an estimate. The source commonly uses a 95% confidence level.
estimate ± margin of errorAlpha
For a 95% confidence level, the example uses α = 0.05. For 97%, the corresponding value shown is 0.03.
P-Value
The material uses p < .05 as a common significance threshold, while noting that significance levels can vary by study.
Worked confidence-interval example
81
6.2562
74.74
87.25
The source example uses n = 14 and a 95% confidence level. When the entire population is measured rather than a sample, the material notes that a confidence interval is unnecessary for sampling uncertainty.
Hypothesis Testing & Errors
A research question can be expressed as competing null and alternative hypotheses, then evaluated using an appropriate statistical test.
Null Hypothesis
Assumes no relationship exists between the variables. In the study-hours example: the five extra study hours made no difference to the higher scores.
Alternative Hypothesis
Assumes a relationship exists. In the example: the five extra study hours are related to the higher scores.
Type I Error
Reject a null hypothesis that is actually true — a false positive.
Type II Error
Fail to reject/accept the null when it is actually false — represented in the source as a false negative.
False positives and false negatives can have serious consequences. The material illustrates this with medical testing: unnecessary treatment can follow a false positive, while a false negative can leave a genuinely sick person untreated.
T-Tests & Variables
A t-test compares means. The source describes its use when the dependent variable is normally distributed.
One-Sample T-Test
Compares a sample mean with a specified value, such as asking whether subscribers spend more than last year's annual average.
Two-Sample T-Test
Compares the means of two groups, such as placebo versus treatment groups or two classrooms.
Dependent Variable
The outcome being measured. In the classroom example, this is the test score.
Independent Variable
The characteristic that differs between groups. In the example, the classroom/study-hours condition distinguishes the groups.
Chi-Square Tests
Chi-square compares observed results with expected results and is useful with categorical data. Two common forms are the test of independence and goodness of fit.
Preparedness × Test Result
| Student Preparedness | Fail | Pass | Total |
|---|---|---|---|
| Very Prepared | 9 | 17 | 26 |
| Somewhat Prepared | 11 | 40 | 51 |
| Not Prepared | 12 | 11 | 23 |
| Total | 32 | 68 | 100 |
χ² = 6.9338 and p = 0.031213. With the example threshold of .05, the result is reported as significant.
Regression & Correlation Explorer
Regression estimates relationships between a dependent variable and one or more independent variables. Correlation describes statistical association; it does not by itself establish causation.
Excel Statistical Tools & Key Takeaways
Excel provides statistical functions and the Analysis ToolPak for sampling, descriptive statistics, regression, and other analyses. Later versions also include Analyze Data for suggested analyses and questions about selected data.
Match the statistical method to the question and data: t-tests compare means, chi-square handles categorical observed-versus-expected patterns, regression estimates predictor/outcome relationships, and correlation measures association.
Correlation does not establish causation. An apparent relationship between attendance and test scores, for example, does not prove that attendance alone explains the scores.
Knowledge Check
Answer all 20 questions. Each question gives immediate feedback after your first selection.