Sampling Error and Sampling Distributions

Population

The full collection of all possible individuals, participants, or entities that a researcher is interested in studying (e.g., all human beings, or all UCLA undergraduate psychology students). Because studying an entire population is usually impossible due to finite resources, researchers draw a sample to make inferences about the broader group. (Slide 3; Transcript 00:39:53; Textbook p. 151)

Sample

A subset of individuals selected from a population to represent the larger group. Researchers collect data from a sample and compute sample statistics as approximations or estimates for unknown population parameters. (Slide 3; Transcript 00:06:43, 00:39:53; Textbook p. 151)

Population Parameter

A fixed, true numerical value (such as the population mean mu or population standard deviation sigma) calculated from an entire population. In practice, parameters are almost always unknown because full populations cannot be completely surveyed. (Slide 4; Transcript 00:05:26, 00:40:52; Textbook pp. 151, 158)

Sample Estimate (Sample Statistic)

A numerical value (such as the sample mean x-bar or sample standard deviation s) calculated from observed sample data to approximate an unknown population parameter. Sample statistics vary across different random samples. (Slide 4, 5; Transcript 00:06:51, 00:40:02; Textbook p. 158)

Sampling Error

The random numerical difference between a sample estimate and the true population parameter (calculated as Sample Mean minus Population Mean, x-bar - mu). It represents discrepancy due to chance in selecting sample members and can be positive (sample mean above population mean) or negative, occurring at the sample level rather than the individual score level. (Slide 6, 7; Transcript 00:07:02, 00:10:58, 00:11:10; Textbook p. 151)

Monte Carlo Simulation

A computer-based simulation technique that generates artificial, synthetic sample data drawn repeatedly from a hypothetical population distribution with assumed parameters (e.g., population mean = 3, standard deviation = 1) to empirically demonstrate statistical properties like sampling error and sampling distributions. (Slide 10, 11; Transcript 00:05:32, 00:07:02; Textbook pp. 158-161)

Sampling Distribution of the Mean

A theoretical distribution formed by collecting and plotting the sample means calculated from a very large number (e.g., 10,000) of independent random samples of size N drawn from the same population. It is not a distribution of individual raw scores; it is centered around the true population mean mu and approximates a normal curve as sample size increases. (Slide 13, 14; Transcript 00:13:08, 00:19:29, 00:38:54; Textbook pp. 161-164)

Standard Error of the Mean (SEM)

The standard deviation of the sampling distribution of the mean, representing the average or expected amount of sampling error (the average distance between sample means and the true population mean). Theoretical standard error equals population standard deviation divided by the square root of sample size (sigma / sqrt(N)), and decreases as sample size increases or population variability decreases. (Slide 16, 17, 21; Transcript 00:28:32, 00:31:08, 00:33:19, 00:55:18; Textbook pp. 164-165)

Estimated Standard Error

The practical approximation of standard error calculated when the population standard deviation sigma is unknown, substituting the sample standard deviation s into the formula: s / sqrt(N) (e.g., 1.11 / sqrt(1071) = 0.034). (Slide 22, 23; Transcript 00:57:09, 01:04:37; Textbook pp. 167, 173)

Central Limit Theorem (CLT)

A fundamental statistical theorem establishing three key facts: (1) as sample size N increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the underlying population distribution; (2) the mean of the sampling distribution equals the population mean mu; and (3) the standard deviation of the sampling distribution (standard error) equals sigma / sqrt(N). (Slide 20; Transcript 00:53:03, 00:54:05, 00:55:18; Textbook pp. 164-165)

Law of Large Numbers

A statistical principle stating that as the sample size N drawn from a population increases, the sample estimate (such as the sample mean) becomes increasingly accurate and converges toward the true population parameter, systematically reducing sampling error. (Slide 25; Transcript 01:08:14, 01:09:39; Textbook p. 158)

Simple Random Sample

A sampling procedure in which every individual member of the population has an equal probability of being chosen for the sample. (Slide 3; Transcript 00:06:43; Textbook pp. 153-154)

Sampling With Replacement

A random sampling procedure where an individual unit selected from the population is recorded and then returned to the population before the next unit is drawn, allowing the same population member to be selected multiple times. (Textbook pp. 153-154)

Sampling Without Replacement

A sampling procedure where an individual selected from the population is not returned, ensuring that no individual can be observed or participate more than once in the dataset. (Textbook pp. 153-154)

Stratified Sampling

A sampling scheme where the population is divided into distinct sub-populations or strata, and a separate random sample is collected from each stratum, often used with oversampling to adequately represent rare groups like schizophrenic patients. (Textbook pp. 154-155)

Snowball Sampling

A non-random sampling technique common in social sciences for hard-to-access populations where initial participants provide contact details for additional potential participants, creating an expanding chain. (Textbook pp. 154-155)

Convenience Sampling

A non-random sampling procedure where participants are selected based on convenience or accessibility to the researcher, such as relying on undergraduate psychology students. (Textbook p. 155)

Oversampling

A specific form of stratified sampling that deliberately over-represents rare or small sub-populations in the sample to ensure sufficient statistical power. (Textbook p. 154)

Biased Sampling Scheme

A sampling procedure where selection probabilities are unequal or manipulated in a way that systematically produces unrepresentative samples (e.g., selecting only black chips from a bag). (Textbook pp. 153-154)

Unbiased Estimator

A sample statistic whose expected value across repeated sampling equals the true population parameter, such as the sample mean where the average of sample means across repeated samples equals mu. (Textbook pp. 167-170)

Biased Estimator

A sample statistic whose average value across repeated samples systematically over- or under-estimates the true population parameter, such as the unadjusted sample standard deviation s when dividing by N. (Textbook pp. 167-170)

95% Confidence Interval Formula

An estimated interval of plausible values for the population mean constructed using the sample mean X-bar plus or minus 1.96 times the standard error of the mean (X-bar +/- 1.96 * SEM). (Textbook pp. 173-174)

R Code rnorm Function for Simulation

The explicit R programming command sampledata <- rnorm(N, mean = mu, sd = sigma) used in Monte Carlo simulations to draw a random sample of N artificial scores from a specified normal distribution. (Slide 11, 12; Transcript 00:06:37)

Humor Rating Scale Simulation Example

The specific lecture example using a 1 to 5 humor rating score scale where a hypothetical population with mean = 3 and SD = 1 was sampled to illustrate sample means (2.98 vs 3.04) and standard error reduction from N = 100 to N = 1071. (Slide 11, 12, 24, 25; Transcript 00:06:43, 00:07:02, 01:09:17)

Mathematical Proof of Expected Value of Sample Means

The step-by-step mathematical proof demonstrating that E(x-bar) = E((x1 + x2 + … + xN)/N) = (N*mu)/N = mu, proving that the sample mean is an unbiased estimator of the population mean. (Slide 21; Transcript 01:01:20)

Mathematical Proof of Standard Deviation of Sample Means

The formal algebraic derivation showing that sigma_x-bar = sqrt(Var((x1 + … + xN)/N)) = sqrt((1/N^2) * N * sigma^2) = sigma / sqrt(N), proving the standard error formula. (Slide 21; Transcript 01:01:20)

Comparison Table of Three Distributions

A conceptual reference framework contrasting Sample Distribution (components: individual scores x, center: sample mean x-bar, spread: sample SD s, distance: deviation score x - x-bar), Population Distribution (components: individual scores x, center: population mean mu, spread: population SD sigma, distance: deviation score x - mu), and Sampling Distribution (components: sample means x-bar, center: population mean mu, spread: standard error sigma_x-bar, distance: sampling error x-bar - mu). (Slide 18, 19; Transcript 00:39:53, 00:41:44)

Depression Treatment Multi-Researcher Scenario

A conceptual classroom exercise where 1,000 independent researchers each recruit 100 participants to evaluate a depression treatment, creating 1,000 sample means, 1,000 sampling errors, 100,000 individual scores, and 100,000 deviation scores to reinforce distribution levels. (Slide 18-20; Transcript 00:44:22, 00:47:46, 00:49:29)

Given a single sample mean, what do we know about the true population mean?

The true population mean can be higher or lower than the observed sample mean, because sampling error can be positive or negative depending on which specific individuals were selected. (Slide 8; Transcript 00:07:02)

How does standard error differ from standard deviation?

Standard deviation quantifies the average distance between individual raw scores and the mean within a single distribution, whereas standard error quantifies the average distance between sample means and the true population mean across many random samples. (Slide 17; Transcript 00:28:32, 00:36:45)

What happens to the standard error when sample size N is decreased from 1071 to 100?

The standard error increases (e.g., from approximately 0.03 to 0.10), because smaller samples produce wider sampling distributions with greater variability in sample estimates. (Slide 24, 25; Transcript 01:09:17, 01:09:39)

What happens to the standard error when population standard deviation sigma is reduced from 1.0 to 0.5?

The standard error decreases by half (e.g., from 0.03 to 0.015 for N = 1071), because a less variable population produces sample means that cluster much closer to the population mean. (Slide 26, 27; Transcript 01:08:14)

Do sample means from a study with N = 10 participants and a study with N = 100 participants come from the same sampling distribution?

No, because sampling distributions depend directly on sample size; samples with different sample sizes (or different population parameters) originate from different sampling distributions with distinct standard errors. (Slide 25; Transcript 01:09:57)

If 1,000 researchers each recruit 100 participants, how many sample means, sampling errors, and true standard errors exist?

There are 1,000 sample means, 1,000 sampling errors (one for each sample mean minus mu), and exactly 1 true standard error (since standard error is a fixed single property of the sampling distribution). (Slide 19, 20; Transcript 00:50:01, 00:51:24)

Why is Monte Carlo simulation alone insufficient for real-world researchers estimating sampling error?

Because Monte Carlo simulation requires knowing the true population parameters and drawing thousands of samples, whereas real-world researchers only have access to a single sample and do not know the true population mean. (Slide 20; Transcript 00:52:18, 00:52:58)

Why do researchers prefer larger sample sizes when conducting empirical studies?

Recruiting more participants increases the denominator in the standard error formula (sigma / sqrt(N)), making standard error smaller and sample estimates more accurate and reliable representations of the population. (Slide 23; Transcript 01:08:14, 01:08:44)

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