BackFundamental Concepts and Pitfalls in Statistics: Definitions, Data Analysis, and Working with Percentages
Study Guide - Smart Notes
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Key Definitions in Statistics
1. Data
In statistics, data refers to collections of observations, measurements, or facts gathered for analysis.
Example: Recording the weights of students in a class.
2. Population
The population is the complete collection of all measurements or data points being considered in a statistical study.
Example: The weight of everyone who attends a particular college.
Additional info: Populations are often large, making it impractical to collect data from every member.
3. Census
A census involves collecting data from every member of the population.
Example: Surveying every student at a college to record their weight.
4. Sample
A sample is a subset of the population, selected for the purpose of analysis.
Example: Asking the weight of a few people in a class instead of the entire college.
Additional info: Samples are used because it is often impractical or impossible to collect data from the entire population.
5. Voluntary Response Sample
In a voluntary response sample, respondents decide for themselves whether to participate in the study.
Examples: Internet polls, phone calls, email surveys.
Note: Voluntary response samples are often biased and unreliable because they may not represent the entire population.
6. Statistical Significance
Statistical significance occurs when a result is unlikely to have occurred by chance alone, typically when the probability is less than 5%.
Example: Getting 95 tails when flipping a coin 100 times. The expected number is about 50, so 95 is highly unlikely by chance.
Analyzing Data: Potential Pitfalls
1. Misleading Conclusions
Drawing incorrect conclusions from data, such as assuming correlation implies causation.
Example: A study finds a correlation between studying and grades, but this does not mean studying causes better grades. Other factors may be involved.
Additional info: Always consider alternative explanations and confounding variables.
2. Sample Data Reported Instead of Measured
Using self-reported data can introduce inaccuracies.
Example: Asking people to report their own weight may lead to incorrect data due to misreporting.
3. Small Samples
Small sample sizes can lead to unreliable results.
Example: Drawing conclusions from a sample of only 5 people is not reliable. Generally, a sample size of 30 or more is considered adequate.
4. Loaded Questions
Questions phrased to elicit a specific response can bias survey results.
Example: Asking, "Netflix is America’s most watched streaming service. What streaming service do you watch the most?" may influence responses.
5. Order of Questions
The order in which questions are asked can affect responses.
Example: Asking about traffic’s contribution to air pollution before or after mentioning other sources can change the perceived importance of traffic.
6. Nonresponse
Nonresponse occurs when people do not participate in a survey, potentially biasing results.
Example: People refusing to respond due to differing views or embarrassment.
7. Percentages
Percentages can be miscalculated or misinterpreted, leading to incorrect conclusions.
How to Work with Percentages
1. Converting Percentages to Numbers
To find how many people correspond to a given percentage in a population:
Move the decimal in the percentage two places to the left and multiply by the total number.
Example: 30% of 3000 people: people.
Example: 10% of 352 people: people.
2. Changing a Decimal to a Percent
Multiply the decimal number by 100.
Example:
Example:
3. Changing a Fraction to a Percent
First, convert the fraction to a decimal, then multiply by 100.
Example:
Example:
4. Changing a Percent to a Decimal
Divide the percent by 100.
Example:
Example:
Summary Table: Common Statistical Terms
Term | Definition | Example |
|---|---|---|
Data | Collection of observations | Student weights |
Population | All individuals or items under study | All students at a college |
Sample | Subset of the population | Students in one class |
Census | Data from every member of the population | Surveying all students |
Voluntary Response | Respondents choose to participate | Online poll |
Statistical Significance | Result unlikely by chance | 95 tails in 100 coin flips |
