Variables, Constructs, Populations and Samples

Variables vs Constructs

• Variables include items like scrambled words solved in 1 minute, mood pleasantness rating, response time and treatment condition
• Constructs include items like verbal ability, agreeableness, efficiency, and treatment received
• A score on an unscrambling task is a "raw score" (X)

Standard Deviation

• Standard deviation is needed to see if a score is good or bad

Samples vs Populations

• A set of scores on a variable comes from a sample
• A sample represents a population

Populations

• A population contains all members of a specified group

Descriptive Parameters

• 𝛍 (“mu”) is the population mean
• 𝛔 (“sigma”) is the standard deviation (SD)
• Parameters are often unavailable

Descriptive Statistics

• A sample is a population subset
• 𝑿 ̅ (“x-bar”) is the mean
• 𝒔 is the standard deviation

Sample Estimate Accuracy

• 𝑿 ̅ is slightly different from 𝝁
• Statistics from a sample (e.g. 𝑿 ̅) are the best estimates of population parameters (e.g. 𝝁)
• Estimates might be inaccurate due to sampling error or bias

Sampling Error

• Random variation exists between any sample and the population

Sampling Bias

• Specific study methods may cause samples to differ consistently from the population
• Some groups might be over or under-represented
• Sampling bias is avoidable through random sampling
• Every member of the target population should have an equal chance of selection
• It is not always a problem depending on the research question and the nature of the bias

Distribution

• Distribution is a graph associating a frequency/probability with each variable value
• A normal distribution is bell-shaped, unimodal, and symmetrical
• Mode=mean=median in a normal distribution
• Tails extend indefinitely, area under the curve = 100%
• The dependent variable is usually assumed to be normally distributed for parametric tests
• Probability of falling at different places along the distribution is known
• This allows inferences from a sample about population parameters

Standard/Z Scores

• When measuring something, scales are arbitrary, for example, weight
• A z-transformation converts a normal distribution into a standard normal distribution, putting everything on the same scale
• z = (score - mean) / standard deviation
• A Z score indicates how far a data point is from the mean in standard deviation units

Z Transformation

• Converting raw scores into z-scores and plotting them on a frequency distribution is a standard normal distribution
• Standard normal distribution mean = 0 and SD = 1
• A standard normal distribution allows comparison of performance across different tests
• Population mean (u) and standard deviation (σ) calculates probability of values within a specified interval
• Any normal distribution transforms into standard normal distribution (z-transformation)
• Only one set of tables is needed

Inferences

• Making inferences about the probability that different scores are obtained is possible

Group Scores

• Research looks at groups of scores (samples) rather than individual scores

Caffeine example

• To see if caffeine improves performance, one must compare a mean (X) for the sample with the population mean (µ)
• If a sample of 50 students taking PSYC2010 have a mean caffeine consumption of 105mg, and the general UQ population mean is 115mg, caffeine consumption might be lower amongst the PSYC2010 group
• Differences between the sample mean and the population could be due to other factors

Inferential Statistics

• Sample data creates inferences about population parameters
• Sample statistics estimate population parameters
• Samples might not give good population estimates due to sampling error and sampling bias

Sampling Error Explained

• No two samples from the same population are identical, being that there is natural variation in scores

Sampling Bias Explained

• Faulty sampling methods may cause over or under-representation of subgroups
• This is systematic variation
• In a 1948 telephone poll for US elections, Thomas Dewey (Republican) was predicted to win by a large margin but Harry S Truman (Democrat) won easily because the sample was biased
• Sampling bias is avoidable through random sampling, therefore every member of a target population should be equally likely to be selected

Statistical Inference

• Statistical inferences use sample data to create inferences about population parameters
• Statistical inference is the foundation of hypothesis testing
• It can determine the probability that a sample is from one population and not another
• It tests the "truth" of a hypothesis as if a whole population was available instead of just a small representative sample