Glossary
Accuracy of Estimates
How well a sample statistic approximates the true, unknown population parameter.
Example:
A larger sample size generally leads to a better accuracy of estimates for the population mean.
Basis for Inference
The foundation provided by sampling distributions for making conclusions or predictions about a population based on sample data, crucial for hypothesis testing and confidence intervals.
Example:
Understanding the shape and spread of a sampling distribution provides the basis for inference when deciding if a new drug is effective.
Central Limit Theorem (CLT)
A fundamental theorem stating that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the original population's distribution.
Example:
Even if individual student test scores are skewed, the Central Limit Theorem tells us that the distribution of sample mean test scores from many large samples will be approximately normal.
Continuous Random Variables
Random variables that can take on any value within a given range, often representing measurements.
Example:
The exact time it takes for a student to complete a test is a continuous random variable, as it can be any value within a range (e.g., 45.32 minutes, 58.75 minutes).
Discrete Random Variables
Random variables that can only take on a finite number of values or a countable number of values, typically whole numbers.
Example:
The number of defective items in a batch of 100 is a discrete random variable, as you can only have whole numbers of defective items (0, 1, 2, ..., 100).
Population Mean (μ)
The true average value of a variable for an entire population.
Example:
If we could measure the income of every single household in the US, that average would be the population mean (μ) income.
Population Parameters
Measures that describe a characteristic of an entire population; these are fixed, true values, though often unknown.
Example:
The true average height of all adult giraffes in the world is a population parameter.
Population Proportion (ρ)
The true proportion or percentage of individuals in an entire population that possess a certain characteristic.
Example:
The actual percentage of all registered voters in a state who support a particular ballot measure is the population proportion (ρ).
Population Standard Deviation (σ)
The true measure of the spread or variability of data for an entire population.
Example:
The actual variability in the lifespan of all lightbulbs produced by a factory is the population standard deviation (σ).
Random Variables
Variables whose values are numerical outcomes of a random phenomenon.
Example:
The number of cars passing a certain point on a highway in an hour is a random variable because its value is determined by chance.
Sample Mean (x̄)
The average value of a variable calculated from a sample, used to estimate the population mean.
Example:
After surveying 100 students, the calculated average GPA of those 100 students is the sample mean (x̄).
Sample Proportion (p̂)
The proportion or percentage of individuals in a sample that possess a certain characteristic, used to estimate the population proportion.
Example:
If 60 out of 100 surveyed customers prefer a new product, then 0.60 is the sample proportion (p̂).
Sample Standard Deviation (s)
The measure of the spread or variability of data within a sample, used to estimate the population standard deviation.
Example:
The variability in the heights of 30 randomly chosen basketball players is represented by the sample standard deviation (s).
Sample Statistics
Measures that describe a characteristic of a sample; these are estimates of the population parameters.
Example:
The average height of 50 randomly selected giraffes from a zoo is a sample statistic.
Sampling Distribution
A distribution of a statistic (like a mean or proportion) calculated from all possible samples of a given size from a population.
Example:
Imagine taking 1000 samples of 50 students' GPAs from a university and plotting all 1000 sample means; this plot would show the sampling distribution of the sample mean GPA.
Sampling Variability
The natural variation that occurs in sample statistics from sample to sample, even when samples are drawn from the same population.
Example:
If you repeatedly survey 100 people about their favorite color, the percentage for 'blue' will likely differ slightly in each survey due to sampling variability.
Standard Error
The standard deviation of a sampling distribution, which measures the typical variability or spread of a sample statistic.
Example:
A small standard error for the sample mean indicates that sample means from different samples tend to be very close to each other.
Variances Add
A rule stating that when combining two independent random variables, their variances are always added to find the variance of their sum or difference.
Example:
If you're looking at the difference in heights between male and female students, you would use the 'variances add' rule to find the variance of that difference, even though you're subtracting means.