Unit 5: Sampling Distributions

Population

The entire group of individuals you want to learn about.

Sample

The individuals actually observed from the population.

Repeated sampling

The idea of taking many random samples of the same size from a fixed population to see how a statistic varies.

Sampling variability

Natural variation in a statistic from sample to sample due to random sampling.

Inference

Using sample data to estimate or test claims about a population while accounting for sampling variability (e.g., confidence intervals and significance tests).

Parameter

A numerical value describing a population; fixed but usually unknown (e.g., p, μ, σ).

Statistic

A numerical value computed from a sample; varies from sample to sample (e.g., p̂, x̄).

Population proportion (p)

The true fraction of the population with a characteristic.

Population mean (μ)

The true average of the population.

Population standard deviation (σ)

The true spread (SD) of the population.

Sample proportion (p̂)

The fraction of the sample with the characteristic; p̂ = X/n.

Sample mean (x̄)

The average of the sample; x̄ = (x1 + x2 + ··· + xn)/n.

Sampling distribution

The distribution of a statistic’s values across all possible random samples of a fixed size n from the same population.

Population distribution

The distribution of values for all individuals in the population.

Sample distribution

The distribution of the observed data values in one particular sample.

Standard error

The standard deviation of a statistic’s sampling distribution (how much the statistic typically varies from sample to sample).

Bias

Systematic error where the sampling distribution is centered away from the true parameter.

Variability (of an estimator)

How spread out the sampling distribution is; how much the statistic jumps from sample to sample.

Unbiased estimator

A statistic whose sampling distribution mean equals the true parameter (centered at the parameter).

Sample maximum (as an estimator)

Typically a biased estimator of the population maximum because sample maxima tend to be below the population maximum for a fixed n.

Center–Spread–Shape framework

Organizing a sampling distribution by its mean (center), standard deviation/standard error (spread), and approximate form (shape).

Standardization

Converting a value to a z-score by subtracting the mean and dividing by the standard deviation.

z-score

The number of standard deviations a value is from the mean; z = (value − mean)/SD.

Standard Normal distribution

The Normal distribution with mean 0 and standard deviation 1 (distribution of Z).

normalcdf (TI-84)

Calculator function that returns the probability (area) between two bounds under a Normal curve (can use z-scores or raw scores with mean and SD).

invNorm (TI-84)

Calculator function that returns the z-score (or raw score, if mean and SD are provided) with a given area to the left.

Raw score

A value in the original measurement units (not standardized).

Tail probability

A probability in the extreme left or right tail of a distribution (e.g., P(Z > 1.6)).

Large Counts condition

For using a Normal model with p̂: require np ≥ 10 and n(1 − p) ≥ 10.

10% condition

When sampling without replacement, observations are approximately independent if n ≤ 0.10N.

Independence (in sampling)

The assumption that one observation does not affect another; supported by sampling with replacement or by the 10% condition when sampling without replacement.

Sampling with replacement

A sampling method that supports independence because selections don’t change the population composition.

Sampling without replacement

A sampling method where independence is only approximate; typically justified with the 10% condition.

Mean of p̂ (μ_p̂)

The center of the sampling distribution of p̂; μ_p̂ = p.

SD of p̂ (σ_p̂)

The standard error of p̂; σ_p̂ = √(p(1 − p)/n).

Normal model for p̂

If conditions hold, p̂ ≈ N(p, √(p(1 − p)/n)).

Mean of x̄ (μ_x̄)

The center of the sampling distribution of x̄; μ_x̄ = μ.

SD of x̄ (σ_x̄)

The standard error of x̄; σ_x̄ = σ/√n.

Central Limit Theorem (CLT)

For sufficiently large n, the sampling distribution of x̄ becomes approximately Normal, with mean μ and SD σ/√n, regardless of population shape (given randomness/independence).

Noise cancellation by averaging

Intuition for the CLT: random deviations above and below μ tend to cancel when averaged, making x̄ less variable.

Sum (S) under CLT

For large n, S = x1 + ··· + xn is approximately Normal with μS = nμ and σS = σ√n.

Difference in sample proportions (p̂1 − p̂2)

A statistic comparing two groups by subtracting their sample proportions.

Mean of (p̂1 − p̂2)

μ_(p̂1−p̂2) = p1 − p2.

SD of (p̂1 − p̂2)

σ_(p̂1−p̂2) = √( p1(1−p1)/n1 + p2(1−p2)/n2 ) for independent samples.

Difference in sample means (x̄1 − x̄2)

A statistic comparing two groups by subtracting their sample means.

Mean of (x̄1 − x̄2)

μ_(x̄1−x̄2) = μ1 − μ2.

SD of (x̄1 − x̄2)

σ_(x̄1−x̄2) = √(σ1^2/n1 + σ2^2/n2) for independent samples.

Variances add for independent differences

For independent samples, the variance of a difference equals the sum of the variances (add variances, not SDs).

Conservative planning value (p = 0.5)

When p is unknown for planning a sample size for proportions, use p = 0.5 because p(1 − p) is largest (0.25), giving the largest required n.

Sample size planning for x̄ standard error

To target σ_x̄ ≤ k, choose n ≥ (σ/k)^2 (using σ/√n ≤ k).