Lecture 2

• A sample is a model of the population from which it is drawn
• If the sample is representative, it provides insight into how a population works abstractly
• Abstraction occurs because certainty about how well the model compares to the real thing is impossible
• This leads to the necessity of making generalizations from the sample

Models

• Form is a component of models
• Form considers what the topic of interest looks like, how it is distributed, and its shape

Function

• Function considers how a topic of interest works or interacts, involving other topics, characteristics, processes, and environments
• Characteristics allow discernment of underlying traits that shape form/function

Abstraction

• Understanding statistics involves considering both the workings of the sample (concrete elements) and how it models population functions (abstract elements)
• Local/concrete/physical elements involve computations, statistics, relationships in the sample, and visualizations
• Global/abstract elements encompass estimation, simulations, relationships, understanding, knowledge creation, and hypothesis testing

Distribution Basics

• Distributions describe the "sample space," representing the range of possible data points for a given variable and the "density" of scores at different values
• They are important because inferences depend on the distribution type, and tell the likelihood of getting any specific value in a random sample
• Distributions come in different "shapes" with scores/values arranged in order and plotted according to frequency
• Several "parameters" describe these shapes

Distribution of a Sample

• Distribution of a sample is a histogram of value of data points in a single sample, while each point is a single observation
• Interpreting a sample requires knowing how likely each data point is to occur

Population Distribution

• The current population of Canada is 38,583,352, with an average age of 41.9 (median = 41.6)

Population Distribution Detail

• Each point is a single observation from one participant
• Every possible participant is represented
• These distributions are theoretical

Population Distributions

• Population distributions in psychology are mainly theoretical, due to the huge populations and the need to sample most populations
• Distributions are assumed based on sampled data

Probability Density Function (PDF)

• PDF is a distribution plot for a given variable that illustrates the probability of observing a given data value within a distribution

PDF Importance

• Ideally random sampling is used to take samples from populations
• The samples allow generalizations/inferences about that population
• Understanding the likelihood of a given value enables characterizing a sample and the population more accurately
• PDF tells the relationship between the value and the population mean

Distribution Types

• Common distributions include:

Uniform

• Uniform is symmetric, outcomes are bounded (lowest/highest values known), and all outcomes are equally likely
• Discrete uniform distribution has finite number of outcomes
• Continuous uniform distribution infinite number of outcomes

Binomial

• Binomial is the probability of a 'win' or 'lose' outcome
• 'bi' means two possible outcomes
• Number of observations are fixed, and has trials that are all independent
• Probability of 'win' is identical from one trial to the next

Normal Distribution

• Normal Distribution is also known as "Gaussian Distribution," which is symmetric and described by the mean and standard deviation

Other Parameters

Two Other Parameters

• Skew measures the symmetry of a distribution's tails
• Positive skew
• Negative skew
• Kurtosis measures the "tailedness" of a distribution
• Leptokurtosis
• Platykurtosis

Estimating Populations

• Sample statistics are used to make an estimate about population parameters from which it came
• Now we are going to quantify uncertainty in the distribution, and to do this, we will build a type of distribution

Sampling Distribution

• The sampling distribution statistics from samples
• It consists of the mean, instead of individual scores
• It will plot the means of the people randomly selected within in the sample size

What Does the Distribution of Sample Means Look Like?

• Population distribution has characterization by a mean (µ) and standard deviation (σ)
• The mean of the DOSM converges on the population mean
• The standard deviation of this distribution is much smaller
• The standard deviation of distribution of sample means is the "Standard Error"

Sampling Distribution Utility

• A sampling distribution is efficient, cost effective, and tells about the characteristics within a sample
• These allow estimates about the full population and the probability of a particular outcome occurring

Population Distribution Shape

• With enough samples of large enough sizes the DOSM converges to normal
• This is the central limit theorem in action

Theoretical Populations

• Populations are assumed with distribution shapes
• Basic parameters are set, like the mean and standard deviation, and can either be arbitrary or based on previous research
• Produce a sampling distribution by taking random samples from this population

Monte Carlo Sampling

• Repeated random sampling is used to generate draws from a probability distribution
• Repeated
• Independent
• The "expected value" of a variable is estimated empirically because samples converge on the true value

Bootstrap Resampling

• Bootstrap Resampling is used if you only have one sample of data
• PDFs are created from data by "re-sampling" them randomly and independently
• Need to sample with replacement to allow independent samples of same size as original
• Sample statistics are estimated by treating data as a population

Method

• Random and independent draws of scores are taken from the sample, while being the same size and replacement
• Sample statistic is calculated and a plot is created as a frequency histogram, repeated multiple times

Bootstrap Use

• Normal distribution is not needed for data and wanted comparisons across groups
• More precise estimates of parameters are wanted
• The underlying population distribution is unknown
• When there is a small sample size

Bootstrap Caveats

• If sample is too small, non-representative of the population, etc., boostrapping

Explore the concepts of samples, models, and abstraction in statistical analysis. Understand how representative samples provide insights into population dynamics. Grasp the interplay between form and function and the role of abstraction in statistical modeling.