Introduction to Statistics

Statistical and Critical Thinking

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make informed decisions. Statistical thinking involves not only performing calculations but also applying critical reasoning to understand and interpret results.

• Statistics: The science of planning studies and experiments; obtaining data; and organizing, summarizing, presenting, analyzing, and interpreting those data to draw conclusions.

• Statistical Thinking: Requires critical thinking and the ability to make sense of results, not just executing calculations.

• Key Process: Conducting a statistical study involves three main steps: prepare, analyze, and conclude.

Types of Data

Data are collections of observations, such as measurements, genders, or survey responses. Understanding the types of data is fundamental to statistical analysis.

• Data: Collections of observations (e.g., measurements, survey responses).

• Population: The complete collection of all measurements or data being considered. It is the group about which we want to make inferences.

• Census: Data collected from every member of a population.

• Sample: A subcollection of members selected from a population.

Example: In a survey of 410 human resource professionals, 148 said job candidates were disqualified due to social media postings. Here, the population is all human resource professionals, and the sample is the 410 surveyed.

Collecting Sample Data

Collecting data properly is essential for valid statistical conclusions. The method of sampling can greatly affect the reliability of results.

• Random Sampling: Individuals are randomly selected, which helps ensure unbiased results.

• Voluntary Response Sample: Also known as self-selected sample, respondents decide whether to participate. This method is prone to bias and should not be used to make conclusions about a population.

Examples of Voluntary Response Samples:

• Internet polls

• Mail-in polls

• Telephone call-in polls

Example: A television poll with 186,000 volunteer respondents showed 67% wanted the UN moved out of the US, while a random sample of 500 showed only 38%. The random sample is more reliable due to superior sampling method.

The Statistical Study Process

Prepare

Preparation involves understanding the context, identifying the source of data, and ensuring the sampling method is sound. For example, forensic scientists may use shoe print lengths to estimate height, and the data source and sampling method must be reputable and random.

• Context: What is the goal of the study?

• Source of Data: Is the source reputable?

• Sampling Method: Was the sample randomly selected?

Analyze

Analysis begins with graphical exploration and continues with statistical methods. Good analysis requires common sense and attention to sound methods, not just computational skills.

• Graph and Explore: Use appropriate graphs to visualize data.

• Apply Statistical Methods: Use statistical tools to analyze data.

Conclude

The final step is to draw conclusions, distinguishing between statistical significance and practical significance.

• Statistical Significance: Achieved if the likelihood of an event occurring by chance is 5% or less.

• Practical Significance: Even if a result is statistically significant, it may not be meaningful in practice.

Example: In a weight loss study, a mean loss of 2.1 kg is statistically significant, but may not be practically significant for dieters.

Analyzing Data: Potential Pitfalls

Common Pitfalls in Data Analysis

Several issues can undermine the validity of statistical conclusions:

• Misleading Conclusions: Conclusions should be clear and understandable.

• Sample Data Reported Instead of Measured: Measurements should be taken directly, not self-reported.

• Loaded Questions: Poorly worded survey questions can bias results.

• Order of Questions: The sequence of questions can unintentionally influence responses.

• Nonresponse: Occurs when individuals refuse or are unavailable to respond.

• Low Response Rates: Decreases reliability and increases bias.

• Misleading Percentages: Percentages exceeding 100% are often unjustified.

Summary Table: Census vs. Sample

Summary Table: Statistical Significance vs. Practical Significance

Key Formulas

• Probability of an event:

• Sample Mean: