PSYC2002 Lecture 2 – SkoolWerx

The Breakdown

Important

Notations – Variables are typically represented by uppercase letters like X and Y. N denotes the number of scores in a population, while n indicates the number of scores in a sample. The symbol Σ (Sigma) signifies summation, meaning to add up everything that follows it; operations within brackets should always be performed first.
Frequency Distribution Tables – These tables list the actual scores and how frequently they appear in your data, not all the raw data collected. When constructing one, all possible scores between the highest and lowest values must be listed, even if they have a frequency of zero. The sum of all frequencies (f) must equal your sample size (n).
Proportions (p) – Calculated by dividing the frequency (f) of a score by the sample size (n) (p = f/n). The sum of all proportions should always total approximately 1.0 (allowing for minor rounding differences).
Percentages (p100) – Obtained by multiplying the proportion (p) by 100 (p100 = p x 100). The sum of all percentages should always total approximately 100% (allowing for minor rounding differences).
Grouped Frequency Distribution Tables – These are used when there are too many possible scores, making an ordinary frequency table unmanageably large (typically more than 12 rows). The goal is to create 8 to 12 intervals, and all intervals must be of identical size. Interval widths should be simple, such as 2s, 5s, 10s, 20s, or 50s, and not inconvenient numbers like 3s, 4s, 6s, or 7s. When interpreting these tables, if a question refers to a number that falls within an interval, the data for the entire interval must be included.
Histograms – These graphs are exclusively used for data collected on an interval or ratio scale. In a histogram, the bars touch each other, signifying continuity between the data points. The Y-axis is labelled “Frequency,” and the X-axis is labelled with the dependent variable. Consistency is key: if your frequency table is grouped, your histogram should also be grouped.
Bar Graphs – These are used for nominal or ordinal scale data. Unlike histograms, the bars in a bar graph do not touch each other, indicating separate and distinct categories. The height of the bars represents the frequency.
Graph Descriptions – Graphs should be described by their shape (symmetry and skewness), central tendency (modality), and variability (kurtosis).
Graphing Ethics and Distortion – The choice of X and Y-axis scales is critical because it determines what the graph communicates and can easily distort the data. Practices like truncating the scale (eliminating some numbers from an axis) or using misleading imagery can visually misrepresent the impact of the data, making things appear better or worse than they are.
Measures of Central Tendency – These are descriptive statistics that aim to find a single score that best represents the ‘center’ or ‘typical’ score of all the data in a distribution. The three types are mode, median, and mean.
Mode – Identifies the most common score or the highest point in a distribution. Its limitations include being quite affected by the shape of the graph and not including all sample scores in its calculation. It is the only measure of central tendency that can be used with discrete (nominal or ordinal) variables.
Median – Represents the point in the data where 50% of scores fall below it and 50% fall above it, acting as the 50th percentile. It is calculated after arranging scores in numerical order. The median is often the preferred measure when a distribution is skewed, as it is less influenced by extreme scores. However, it does not convey how spread out the scores are.
Mean – The most common and often preferred measure of central tendency because it includes every single score in its calculation, making it the least biased measure for inferring about a population. It is, however, quite affected by extreme scores in skewed distributions, which can pull the average towards the tail. The mean can only be used with interval and ratio scales, not discrete variables.
Normal Distribution – In a perfectly normal distribution (bell curve), the mean, median, and mode are all the same.
Skewed Distribution Effects on Central Tendency – In a positively skewed distribution, the mode is the lowest, followed by the median, with the mean being the highest and closest to the tail. In a negatively skewed distribution, the mean is the lowest and closest to the tail, followed by the median, with the mode being the highest.

Core concepts

Samples and Populations: A population is the entire group that a researcher wishes to generalize their results back to, as defined by the researcher; a sample is a smaller group selected from that defined population.
Parameters vs. Statistics: Parameters are numbers based on the whole population and are usually symbolized by Greek letters (e.g., μ); statistics are numbers based on a sample taken from a population and are usually symbolized by Latin letters (e.g., M, X, SD).
Sampling Error: The inherent difference that exists between a sample and the population from which it was drawn, even with random sampling, because the sample does not include every single person in the population. The goal in research is to minimize this error for valid and reliable results.
Measurement Scales (Variables): Ways to quantify observations for analysis:
- Discrete Observations: Values that can only take on certain numbers, often whole numbers (e.g., number of siblings). Nominal and ordinal variables are always discrete.
- Continuous Observations: Values that can take on all possible numbers within a given range (e.g., reaction time in seconds). Ratio variables are almost always continuous, while interval variables can be either discrete or continuous.
- Nominal Variable: A variable whose values are categories with no inherent order or meaningful numerical difference (e.g., eye color).
- Ordinal Variable: A variable whose values are rank-ordered, but the differences between ranks are not necessarily equal (e.g., finishing place in a race).
- Interval Variable: A variable with equal distances between values, but without a true zero point, meaning zero does not indicate the absence of the quantity (e.g., temperature in Celsius).
- Ratio Variable: A variable with equal distances between values and a true zero point, indicating the complete absence of the quantity (e.g., height, weight).
- Scale Variable: A general term used in statistics to refer to variables that meet the criteria for either an interval or a ratio variable, as they are typically analyzed with the same statistical tests.
Variables and Notations: Variables are typically represented by uppercase letters, most commonly X and Y. N denotes the number of scores in the population, and n denotes the number of scores in a sample. The symbol Σ (Sigma) is used to indicate summation, meaning to add up a list of scores.
Frequency Distribution: A table that describes the pattern of a set of numbers by displaying a count or proportion for each possible value of a variable.
Raw Score: A data point that has not yet been transformed or analyzed.
Frequency Table: A visual depiction of data that shows how often each value occurred, serving as the starting point for organizing data.
Grouped Frequency Table: A type of frequency table that organizes data into intervals (ranges of values) rather than individual data points, useful when the data cover a wide range or have many possible values.
Outlier: An extreme score that is either very high or very low in comparison with the rest of the scores in the sample.
Histogram: A graph that resembles a bar graph but depicts just one variable, typically based on scale data, with the values of the variable on the x-axis and the frequencies on the y-axis; the bars are adjacent to each other.
Bar Graph: A graph typically used for displaying data with a nominal or ordinal independent variable and a scale dependent variable; the bars are separated, indicating distinct categories.
Descriptive Statistics: Methods used to organize, summarize, and communicate a group of numerical observations.
Central Tendency: A descriptive statistic that represents the center of a data set, the particular value that all other data seem to gather around, or the “typical” score.
- Mean (M or μ): The arithmetic average of a set of scores, calculated by summing all scores and dividing by the total number of scores. It is the most commonly used measure of central tendency.
- Median (Mdn): The middle score of all the scores in a sample when arranged in ascending order; if there is an even number of scores, it is the mean of the two middle scores. It is also referred to as the 50th percentile.
- Mode: The most common score of all the scores in a sample.
- Unimodal, Bimodal, Multimodal: Terms to describe distributions based on the number of modes (peaks): unimodal has one mode, bimodal has two, and multimodal has more than two.
Variability: Refers to the extent to which scores in a data set differ from each other or from the mean.
- Range: The difference between the highest and lowest scores in a distribution.
- Interquartile Range (IQR): The range of the middle 50% of the scores, calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
- Deviation from the Mean: The amount that a score in a sample differs from the mean of the sample.
- Sum of Squares (SS): The sum of the squared deviations from the mean.
- Variance (SD², s², MS, or σ²): The average of the squared deviations from the mean.
- Standard Deviation (SD, s, or σ): The square root of the variance, representing the typical amount that a score deviates from the mean.
Shapes of Distributions:
- Normal Distribution: A specific frequency distribution that is bell-shaped, symmetric, and unimodal, crucial for many statistical calculations.
- Skewed Distribution: An asymmetrical distribution where data “leans” to one side. A positive skew indicates the tail extends to the right (higher values), often due to a floor effect. A negative skew indicates the tail extends to the left (lower values), often due to a ceiling effect.
- Symmetry: A distribution is symmetrical if it can be divided into two halves that are mirror images of each other.
- Kurtosis: Describes the “peakedness” or flatness of a distribution:
  - Mesokurtic: A normal, bell-shaped curve.
  - Platykurtic: A flatter curve, meaning scores are more spread out.
  - Leptokurtic: A pointy curve, indicating that many scores are clustered around the middle.
Standardization: A process to convert individual scores from different normal distributions to a shared normal distribution with a known mean, standard deviation, and percentiles, enabling meaningful comparisons.
z-Score: A standardized score that indicates the distance a raw score is from the mean of its distribution in terms of standard deviations. The z-distribution always has a mean of 0 and a standard deviation of 1.
Percentile: The percentage of scores that fall at or below a particular score in a distribution.
Central Limit Theorem: A fundamental concept stating that the distribution of means from multiple samples drawn from the same population will be approximately normal, regardless of the shape of the original population distribution, as long as the sample size is sufficiently large.
Distribution of Means: A distribution composed of many sample means calculated from all possible samples of a given size, all taken from the same population.
Standard Error of the Mean (σM): The standard deviation of the distribution of means, which measures the variability of sample means around the population mean. It generally decreases as the sample size increases.
Hypothesis Testing: A specific, statistics-based process of drawing conclusions about whether a particular relation between variables is supported by the evidence. It involves six steps: identifying populations, stating hypotheses, determining comparison distribution characteristics, setting critical values, calculating the test statistic, and making a decision.
- Assumptions: Characteristics ideally required for the population from which samples are drawn to ensure accurate inferences (e.g., random selection, normally distributed populations, and equal variances among populations – homoscedasticity).
- Null Hypothesis (H0): A statement that posits no difference between groups or no relation between variables in the population.
- Research Hypothesis (H1): A statement that posits a difference or relation between variables in the population, typically what the researcher expects to find.
- Critical Values (Cutoffs): Points on the comparison distribution that delineate the critical region, beyond which a test statistic must fall for the null hypothesis to be rejected.
- Test Statistic: A calculated value (e.g., z-statistic, t-statistic) that quantifies how far a sample result deviates from the null hypothesis, used to make a decision about the hypothesis.
- Statistically Significant: A finding is statistically significant if the data differ from what would be expected by chance if the null hypothesis were true; often corresponds to a p-value less than 0.05.
- One-tailed vs. Two-tailed Test: A one-tailed test is used with a directional hypothesis (predicting an increase or decrease), while a two-tailed test is used with a nondirectional hypothesis (predicting simply a difference). One-tailed tests are rare in research.
Errors in Hypothesis Testing:
- Type I Error: Occurs when the null hypothesis is rejected, but it is actually true (a false positive).
- Type II Error: Occurs when the null hypothesis is failed to be rejected, but it is actually false (a false negative).
Replication/Reproducibility: The process of duplicating scientific results, ideally in a different context or with a sample that has different characteristics, to confirm the reliability of findings. It is considered a crucial “feature” of scientific discovery that leads to a better understanding of contexts.
Confidence Interval: A type of interval estimate that provides a range of plausible values for a population parameter (e.g., population mean) based on a sample, typically centred around the sample mean. It offers more information than a simple hypothesis test by giving a sense of the uncertainty behind the statistics.
Effect Size (Cohen’s d): A standardized measure of the magnitude or strength of an observed effect, independent of sample size. It helps understand the practical importance of a finding, beyond just statistical significance. Cohen’s conventions for d are: small (0.2), medium (0.5), and large (0.8).
Meta-analysis: A research study that involves the calculation of a mean effect size from the individual effect sizes of more than one study that addressed the same research question. It helps to combine findings across multiple studies.
Statistical Power: The probability that a researcher will correctly reject the null hypothesis when it is false (i.e., detecting a real effect if one exists). Ideally, a study should aim for at least 80% statistical power. It is primarily affected by sample size.
p-hacking: An unethical research practice where researchers manipulate data or analyses (e.g., stopping data collection once p < 0.05) until they find statistically significant results.
HARKing (Hypothesizing After the Results are Known): An unethical practice where researchers change or formulate their hypotheses to match findings they have already observed. Ethical reporting requires distinguishing original hypotheses from those developed after seeing results and reporting all variables and analyses.

Theories and Frameworks

VIPERR Principles for Teaching Statistics: A set of research-backed techniques guiding the teaching and writing of the textbook, designed to enhance learning and retention. These principles include Vivid examples, Integration of new knowledge with existing knowledge, Practice and participation, Examination of misconceptions, Real-time feedback, and Repetition.
Data Ethics/Open Science: A set of principles and practices for conducting research that emphasizes transparency and ethical considerations at all stages, from design to reporting. It encourages sharing methodology, data, and analyses to allow others to question and recreate findings.
Severe Testing: A concept, originally coined by Karl Popper and further developed by Deborah Mayo, that involves subjecting a hypothesis to rigorous statistical scrutiny aimed at uncovering any potential flaws. An ethical researcher aggressively attempts to uncover weaknesses in their data analyses and claims.
The New Statistics: A movement in the behavioural sciences advocating for the increased use of statistical measures like effect sizes, confidence intervals, and meta-analysis, which focus on providing estimates of the magnitude of effects and their precision, rather than solely relying on traditional hypothesis testing (p < 0.05).

Notable Individuals

No notable individuals, other than the professor (who is explicitly excluded by the prompt), were identified as being mentioned in the lecture transcripts or slides.