Statistics: Samples, Distributions, and Central Tendency

• Disseminating information is integral to science.
• Statistics help reduce cognitive load when disseminating information.

Ethical Science Approval

• The two main regulatory boards that monitor and approve ethical science are called Institutional Review Boards (IRB).

Statistics Definitions

• Statistics involve numerical values, graphical figures, or output analysis that represents a larger data subset.
• Statistics serve as a language used to interpret data.
• A good statistic is succinct and representative.
• Each included statistic is equally weighted against other observations.
• A single value explains the behavior of a larger group.

Variables

• Data collection involves gathering the number of variables in a dataset.
• Independent Variable: Something that is changed in an experiment.
• Dependent Variable: The variable expected to change due to the independent variable.
• The change observed in the dependent variable is dependent on the change made to the independent variable.

Types of Variables

• Qualitative Variables: Descriptive and non-numerical.
• Quantitative Variables: Numerical and measurable.
• Discrete Variables: Levels are 'whole' with no intermediates (e.g., hair color, eye color, movie, gender).
• Continuous Variables: Infinite levels between each level (e.g., time to respond to a question like 1.642371 seconds).
• Levels of Independent Variable: Different presentations of the IV, such as control vs. experiment or placebo vs. no placebo.
• Extraneous Variables: Variables not accounted for that may impact results if uncontrolled.
  • Environmental variables are easy to control and should be addressed before starting.
  • Personal variables are difficult to control and often not considered unless important.
• Confounding Variables: Extraneous variables that change systematically along with the variables of interest and can confound results.
• A control group helps maintain consistency between experimental sessions.
• An experimental group with known outcome is compared against a null/negative control group.

Scales of Measurement

• Least to most complex: Nominal, Ordinal, Interval, Ratio (NOIR).
• Nominal Scale: Names or categories with no specific order (e.g., eye color, restaurant style).
• Ordinal Scale: Unique levels with an inherent order, allowing comparisons (e.g., income brackets, satisfaction scales).
• Interval Scale: Ordered levels with a set distance or magnitude between each level (e.g., temperature in Celsius or Fahrenheit, Likert scales).
• Ratio Scale: Name categories for each object where numbers serve as labels, same difference at two places on the scale (interval) AND 0 is real and meaningful (e.g., height, weight, salary).

Additional Measurement Notes

• Categories involve different numbers or names expressing different things.
• Rank order follows an inherent logic of sequentiality.
• Equal spacing means the magnitude between values is consistent.
• True zero means a measurement of 0 represents a true lack of observation.
• Likert scales are ordinal but often treated as interval.

Likert Scales

• Rating scales used to measure attitudes, opinions, or perceptions.
• Participants rank agreement or disagreement with a series of arguments.
• Psychologists are unsure if Likert scales provide true numerical differences or just ranked categories.
• Scales with 7-11+ points tend to behave more like intervals.

Populations and Samples

• Samples are collected to make inferences about a population.
• A population is as wide or narrow as defined, with a shared trait.
• A sample is a subset of a population.
• Sampling biases mean conclusions apply only to the sample.

Types of Sampling

• Simple Random Sample: Every population member has an equal chance of selection.
• Small Sample: Not representative.
• Large Sample: More representative.
• Stratified Sampling: Random sampling along specific guidelines to ensure equal group representation.
• Convenience Sampling: Using an easily accessible sample.

Research Design

• Includes quasi-experimental and non-experimental designs.

Distributions

• Distributions can take various forms, identifiable in bar charts/histograms using measures of central tendency.
• Central Tendency: Average where the center of a distribution tends to fall.
• Average: Mode, median, mean.

Measurement of Central Tendency

• Mode: The most commonly observed level, the highest peak or bar in a distribution.
  • Unimodal: one peak
  • Bimodal: two peaks
  • Multimodal: many peaks
• Median: The point where 50% of scores are above and 50% are below, sensitive to data set observations, and used for ordinal data.
• Mean: The average of values in a data set, representing the balancing point.
  • Best for continuous data.
  • Sensitive to the number and values of observations.
  • The sum of distance between the mean and scores is always 0.

Distributions

• Uniform Distributions: Do not change as levels change.
• Normal Distributions: Have a peak in the middle and taper off equally.
• Skewed Distributions: Have a longer tail on one side.
  • Positive Skew: Long tail points towards the positive (right) side.
  • Negative Skew: Long tail points towards the negative (left) side.

Identifying Skew

• Outliers cause skewed distributions.
  • If Mean = Median = Mode: Normal distribution.
  • If Mean < Mode: Negative skew.
  • If Mean > Mode: Positive skew.

When to Use Measures of Central Tendency

• Mode: For nominal data only.
• Median: For ordinal data only.
• Mean: For continuous data only.

Averages

• Central tendency is considered good, expressing where most people score, giving a central point.
• Central tendency pairs with variability measures to indicate distribution spread.

Variance

• Variance is important in psychology because humans are diverse.
• Humans are too variable to use as a model organism when identifying systematic differences.
• Variability indicates how spread out a dataset is.
• It measures how much dataset values differ from each other and from the central tendency.
• Systematic differences follow a clear pattern.

Measures of Variability

• Range: Distance between the highest and lowest scores.
• Interquartile Range: The 50 percentile points centered around the mean.
  • For discrete variables, count the number of levels.
  • For continuous variables, calculate Xmax - Xmin.
• Variance: The mean of squared deviance scores.
  • The average squared distance between an observation and the mean.
• Standard Deviation: The square root of variance.
  • A corrected/adjusted average distance from the mean.

Identifying Outliers

• Interquartile range measures the range between the 25th and 75th percentiles.
• Most commonly used in continuous data and when describing ordinal data variability.
• Divide into 4 equally portioned sections:
  • Organize ordinally.
  • Count the number of variables and divide into 4 equally portioned sections.
  • Find the range of the inner 2 (highest - lowest).

Variance Calculation Steps

• Squaring residuals helps accentuate extreme scores and remove negative values.
• The mean is a fulcrum and will always equal 0.
• Variance measures the spread of values as the average squared distance from the mean.
• Taking the square root of the variance gives the standard deviation and brings it back closer to original score.
• High variability reduces predictability and reliability, leading to uncertainty.
• Low variability increases predictability and reliability, aiding forecasting.
• Reliability: The ability to get the same results in similar tests.
• Validity: The ability of a test to measure what it intends to.

Measurements

• Measurements of reliability and validity ensure accurate construct measurement through operational definitions.
• Constructs are intangible or hard-to-measure.
• Operational definitions define how a measurement supports or refutes an internal construct.
• Using N instead of n-1 produces a biased statistic.
• Degrees of freedom indicate the number of pieces of information to assess or estimate statistics.
• DF = N-1 for measures of central tendency.
• DF = N for variance with degrees of freedom reduced because the mean is estimated.

Causality

• Causality is the relationship where a change in one variable directly causes a change in another.
  • Temporal sequencing means the independent variable happens before the dependent.
  • To find a non-spurious relationship, IV is not related to DV by chance.
• Eliminate alternative causes by randomly assigning people to treatment/control groups to eliminate bias, then record results.

Construct Measurement

• Reliability and validity ensure accurate measurement of a construct using operational definitions.
• Test-Retest Reliability examines consistency over time in the same people, assessed by correlation.
• Internal Consistency is consistency between questions addressing the same construct and is assessed through split-half correlation.
• Inter-rater reliability is consistency between researchers/scorers.
• Validity measure how something correct construct.
• Face validity shows how something measures the correct construct.
• Content validity identifies if construct measurements match construct aspects.
• Criterion validity measures how well construct measurements match other measurements for construct.

Graphs

• Good graphs:
  • Include graph titles, legends, and trend lines, with clear data.
  • Use histograms for continuous data, with connected bars representing different variable levels.
  • Use bar charts for discrete data, with disconnected bars representing variables .

Frequency Tables

• They summarizes data, reduces cognitive load, and uses ‘f’ to denote frequency.
• Ungrouped tables: Every level of the variable/count is shown, including even levels with f=0. Often present continuous, discrete
• Grouped tables: Equal Magnitude, start at the lowest distribution. Helps with nuance. If more than 10 continuums, should group data. Reduces cognitive load, but oversimplification would be an issue.

Distributions

• Normal Distributions- symmetric, with mean/median/mode the same. Also defined by mean and SD. Area will equal 1. Occurs naturally but use lots of distributions to figure them out.
• Central Limit theorem and distribution sampling is important for hypothesizing.

Z-Scores

• Z-Score for each variable for a known population. Helpful in outliers, measure significant.
• Use in standardized variables comparisons, see how things are spread, direct comparison

Explore sample size impact, stratified sampling, and sampling bias. Understand unimodal vs. bimodal distributions, and mode's use in data types. Learn about mean sensitivity and comparing mean vs. median with outliers. Determine when the median is a better central tendency measure.