Glossary
Central Limit Theorem (CLT)
A fundamental theorem stating that the sampling distribution of the sample mean of a large number of independent, identically distributed random variables will be approximately normal, regardless of the original population distribution.
Example:
Even if individual incomes in a city are heavily skewed, the Central Limit Theorem ensures that the distribution of average incomes from many large random samples will be approximately normal.
Confidence Interval
A range of plausible values for an unknown population parameter, calculated from sample data, along with a level of confidence that the interval contains the true parameter.
Example:
A 95% confidence interval for the proportion of students who prefer online textbooks might be (0.45, 0.55), suggesting that between 45% and 55% of all students prefer them.
Hypothesis Test
A formal statistical procedure used to evaluate a claim or hypothesis about a population parameter using sample data, comparing observed data to what would be expected under a null hypothesis.
Example:
A pharmaceutical company might conduct a hypothesis test to determine if a new drug significantly reduces symptoms compared to a placebo.
Inference for proportions
The statistical process of using sample data to draw conclusions or make predictions about unknown population proportions, typically through confidence intervals or hypothesis tests.
Example:
When a political pollster uses a sample of voters to estimate the true percentage of the population supporting a candidate, they are performing inference for proportions.
Large Counts Condition
A condition for inference about proportions, requiring at least 10 successes and 10 failures in each sample to ensure the sampling distribution is approximately normal.
Example:
Before performing a z-test for the difference in proportions of students who pass an exam using two different study methods, you must verify the Large Counts Condition for both groups.
Nonresponse bias
A type of bias that occurs when individuals selected for a sample do not participate or respond, leading to a sample that is not representative of the population.
Example:
If a survey about online learning is only completed by students who are highly engaged or highly disengaged, this could introduce nonresponse bias, skewing the results.
Point Estimate
A single value calculated from sample data that serves as the best guess or approximation for an unknown population parameter.
Example:
If 75 out of 100 surveyed students prefer a new cafeteria menu item, then 0.75 is the point estimate for the true proportion of all students who prefer it.
Pythagorean Theorem of Statistics
An informal name for the principle that variances add when combining independent random variables, analogous to how the squares of sides add in the Pythagorean theorem.
Example:
To find the standard deviation of the difference between two independent sample proportions, you apply the Pythagorean Theorem of Statistics by adding their variances first, then taking the square root.
Sampling Distribution of the Difference in Proportions
The distribution of all possible differences between sample proportions that could be obtained from repeated random sampling from two independent populations.
Example:
If you repeatedly survey two different groups of consumers about their preference for a new product, the collection of all possible differences in their sample proportions forms the sampling distribution of the difference in proportions.
Standard Deviation (of difference)
A measure of the typical variability or spread of the sampling distribution of the difference between two sample statistics, such as proportions or means.
Example:
If you're comparing the average heights of students from two different schools, the standard deviation of the difference tells you how much that observed difference might vary if you took many pairs of samples.
Standard Error
An estimate of the standard deviation of a sampling distribution, calculated using sample statistics when the true population parameters are unknown.
Example:
When constructing a confidence interval for a population proportion, you use the standard error of the sample proportion because the true population proportion is unknown.
Variances ALWAYS add
A fundamental rule stating that when combining independent random variables, their variances are added to find the variance of their sum or difference.
Example:
When calculating the total variability of two independent sources of error in a measurement, you must remember that variances ALWAYS add to get the combined spread.