Introduction to Statistics: Statistical and Critical Thinking

Data
- Data is defined as collections of observations. These observations can include measurements, genders, survey responses, or other recorded characteristics.
Statistics
- Statistics is the science of several interconnected processes: planning studies and experiments; obtaining data; and organizing, summarizing, presenting, analyzing, and interpreting those data to draw conclusions based on them.
Population
- A population refers to the complete collection of all measurements or data being considered. It is typically the entire group that is the subject of statistical inferences.
Census versus Sample
- Census: The collection of data from every single member of a population.
- Sample: A subcollection of members selected from a population.
Example: Social Media Postings and Job Disqualification
- In a survey conducted by The Society for Human Resource Management, $410$ human resource professionals were surveyed.
- Of these, $148$ professionals stated that job candidates were disqualified because of information found in social media postings.
- Population: All human resource professionals.
- Sample: The specific $410$ human resource professionals included in the survey.
- Objective: Use the sample data to draw conclusions about the entire population of human resource professionals.

Context
- Determine what the data represent and identify the specific goal of the study.
- Case Study: Shoe Print Lengths and Heights of Males
  - Forensic scientists measure shoe prints at crime scenes to estimate a criminal's height.
  - Goal: Determine if a relationship exists between shoe print length and male height.
  - Hypothesis: Males with larger shoe print lengths tend to be taller.
  - Reasoning for Specificity: The study uses only male data because males commit $84\%$ of burglaries.
  - Sample Data ( $n = 8$ ):
    - Shoe Print ( $\text{cm}$ ): $27.6$ , $29.7$ , $29.7$ , $31.0$ , $31.3$ , $31.4$ , $31.8$ , $34.5$
    - Height ( $\text{cm}$ ): $172.7$ , $175.3$ , $177.8$ , $175.3$ , $180.3$ , $182.3$ , $177.8$ , $193.7$
Source of the Data
- Investigate if the data comes from a source with a special interest or pressure to produce favorable results. For the shoe print study, the source (Data Set 9 "Foot and Height" in Appendix B) is considered reputable.
Sampling Method
- Evaluate if the collection method was unbiased. Random selection is a sound sampling method, whereas methods like voluntary response are biased.

Graph and Explore the Data
- Analysis begins with appropriate visual representations and exploration.
- Identify outliers (numbers significantly far from the rest of the data).
- Determine important summary statistics, such as the mean and standard deviation.
- Analyze the distribution of the data.
- Account for missing data or subjects who refused to respond.
Apply Statistical Methods
- Use technology to obtain results. Sound statistical analysis requires common sense and adherence to methods rather than just complex manual calculations.

Statistical Significance
- Statistical significance is reached when the likelihood of an event occurring by random chance is $5\%$ or less.
- Example: Obtaining $98$ girls in $100$ random births is statistically significant because such an extreme result is highly unlikely to happen by chance.
- Example: Obtaining $52$ girls in $100$ random births is not statistically significant because it can easily happen by chance.
Practical Significance
- A finding may be statistically significant but not practically significant if the treatment or result does not make a large enough difference to justify its use or cost.
- Case Study: Atkins Weight Loss Program
  - In a trial of $21$ subjects, the mean weight loss after one year was $2.1\,\text{kg}$ ( $4.6\,\text{lb}$ ).
  - According to the study in the Journal of the American Medical Association (Volume 93, Number 1), this loss is statistically significant.
  - However, the loss of only $2.1\,\text{kg}$ after one year of effort, cost, and time may lack practical significance for many dieters.

Also known as a Self-Selected Sample, this occurs when respondents decide for themselves whether to be included.
Flaws: These samples are seriously biased because people with strong opinions are more likely to participate.
Common Examples:
- Internet polls where online users decide to click and respond.
- Mail-in polls where recipients choose to reply.
- Telephone call-in polls (e.g., radio or TV announcements requesting calls).
Comparison Example: United Nations (UN) Headquarters
- Nightline Poll: Viewers called in to express opinions on whether the UN should leave the US. $67\%$ of $186,000$ voluntary respondents wanted the UN to move.
- Random Survey: A separate, independent survey of $500$ randomly selected respondents found only $38\%$ wanted the UN to move.
- Finding: The small random sample ( $n = 500$ ) is far more reliable than the large voluntary sample ( $n = 186,000$ ) due to the superior sampling method.

Misleading Conclusions: Statements should be clear even to those without statistical training. Avoid ambiguous results.
Reported vs. Measured Data: It is better to take physical measurements than to ask subjects to report their own data (e.g., asking for weight vs. using a scale).
Loaded Questions: Results can be misleading if questions are not worded neutrally.
Order of Questions: The sequence of questions in a survey can unintentionally influence the responses.
Nonresponse: Occurs when subjects refuse to respond or are unavailable. This can lead to bias.
Low Response Rates: A very low rate of response decreases reliability and increases the likelihood of bias among the few who did respond.
Percentages: References to percentages exceeding $100\%$ are often unjustified. Remember that $100\%$ represents the entirety of a quantity.