Statistics helps us turn raw data into useful information. Descriptive statistics focuses on organizing and summarizing the data you actually have, such as a class set of test scores or a list of daily temperatures. Inferential statistics uses data from a sample to make reasonable conclusions about a larger population.
Knowing the difference matters because summaries describe what was observed, while inferences estimate what may be true beyond the observed data.
Descriptive statistics includes measures like mean, median, range, and standard deviation, often shown with tables, graphs, and charts. Inferential statistics depends on sampling, probability, confidence intervals, and hypothesis tests to handle uncertainty. A good inference requires a sample that represents the population well, not just a large sample.
In science, business, medicine, and public policy, inferential statistics helps people make decisions when measuring every individual is impossible.
Key Facts
- Descriptive statistics summarize the data collected, such as mean, median, mode, range, and standard deviation.
- Inferential statistics use a sample to estimate or test claims about a larger population.
- Mean = sum of all data values / number of data values.
- Range = maximum value - minimum value.
- Sample proportion: p-hat = x / n, where x is the number of successes and n is the sample size.
- A confidence interval has the form estimate ± margin of error, showing a plausible range for a population value.
Vocabulary
- Population
- The entire group of individuals or items that a study wants to understand.
- Sample
- A smaller group selected from a population and used to collect data.
- Descriptive statistics
- Methods used to organize, display, and summarize the data that were actually measured.
- Inferential statistics
- Methods used to make estimates, predictions, or decisions about a population based on sample data.
- Confidence interval
- A range of values calculated from sample data that is likely to contain a true population value.
Common Mistakes to Avoid
- Calling every graph inferential statistics, which is wrong because many graphs only describe the data that were collected.
- Using a sample mean as if it must equal the population mean, which is wrong because samples vary and usually include sampling error.
- Ignoring how the sample was chosen, which is wrong because a biased sample can lead to a misleading conclusion about the population.
- Thinking a larger sample always fixes bias, which is wrong because a big biased sample can still represent the wrong group.
Practice Questions
- 1 A teacher records quiz scores for 8 students: 6, 7, 7, 8, 8, 9, 10, 10. Find the mean, median, and range of the data set.
- 2 In a random sample of 200 voters, 118 say they support a new school policy. Find the sample proportion p-hat and express it as a decimal and a percent.
- 3 A website surveys only its most active users and concludes that 92 percent of all users love a new design. Explain whether this is descriptive or inferential, and identify one reason the conclusion may be unreliable.