Scales of Measurement and Statistics

Scales of Measurement

• Nominal scales categorize data with distinct names, implying no inherent order (e.g., gender, nationality).
• Ordinal scales categorize data with a meaningful order or ranking (e.g., education level).
• Interval scales feature ordered categories with equal intervals but lack a true zero point (e.g., satisfaction ratings on a 1-7 scale); non-comparative rating scales are interval if they have 5+ points.
• Ratio scales represent interval scales possessing a true zero point (e.g., age, weight).

Frequency Distribution

• Frequency distributions summarize the occurrence rate for each value of a variable.
• Frequency distributions do not indicate skewness.

Measures of Central Tendency

• The mean is the arithmetic average.
• The mean is suitable for interval and ratio data types.
• The mean is susceptible to skewed data.
• The median is the middle value in a dataset.
• The median is used for ordinal, interval, and ratio data.
• The median divides a dataset into two equal halves.
• The mode represents the most frequent value in a dataset.
• The mode is applicable to nominal and ordinal variables.
• The mode indicates the distribution's highest peak.
• In a normal distribution, the mean, median, and mode are equal.

Measures of Variability

• The range is the difference between the largest and smallest values in a dataset.
• The range is sensitive to outliers.
• The interquartile range (IQR) is the difference between the 75th and 25th percentiles.
• A small SD indicates values clustered around the mean.
• The coefficient of variation (CV) is the ratio of the standard deviation to the mean.
• The CV is useful for comparing different variables, tracking the same variable over time or across groups, and analyzing variables measured in different units.

Measures of Shape: Skewness & Kurtosis

• Skewness measures the symmetry of a distribution.
• A skewness value less than -2 or greater than 2 indicates high skewness.
• A long right tail indicates positive skewness.
• A long left tail indicates negative skewness.
• Kurtosis measures the peakedness or flatness of a distribution.
• Kurtosis values less than -7 or greater than 7 signify a significant deviation from normal.
• Positive kurtosis indicates a sharply peaked distribution.
• Negative kurtosis indicates a flatter distribution.

Probability

• The total probability of mutually exclusive outcomes equals 1.
• A probability less than 5% is considered an unlikely event.
• The 5% level is a critical threshold in statistical analysis.
• Probability helps assess the likelihood of observing a result if the null hypothesis (H₀) is true.
• Sampling error affects the ability to generalize findings from a sample to a population.
• The standard error of the mean (SE) estimates sampling error based on the sample's standard deviation and size.

Confidence Intervals

• A 95% confidence interval suggests the population mean lies within the range with 95% probability.
• The 95% confidence interval is not a guarantee.

Hypothesis Testing Basics

• Inferential statistics draw conclusions about a population from a sample.
• The null hypothesis (H₀) posits no effect or difference.
• The alternative hypothesis (H₁) suggests an expected effect.
• The p-value indicates the probability of observing a sample result if H₀ is true.
• A p-value less than 0.05 leads to the rejection of H₀.
• A p-value of 0.05 is a common cut-off point for rejecting H₀.
• A p-value higher than 0.05 indicates failure to reject H₀.
• A p-value of 0.001 provides strong evidence against H₀.
• The alpha (α) level is the maximum acceptable probability of a Type I error, commonly set at 0.05.

Type I and Type II Errors

• A Type I error (false positive) involves rejecting H₀ when it is true.
• The risk of a Type I error equals the alpha level (e.g., 5%).
• A Type II error (false negative) involves failing to reject H₀ when it is false.
• Increasing the sample size reduces the risk of a Type II error.

T-Tests

• An independent samples t-test compares means from two different groups (e.g., male vs. female income), assuming equal variances checked by Levene’s Test.
• A paired samples t-test compares two variables within the same group (e.g., satisfaction before vs. after service).
• A one-sample t-test compares a sample mean to a known value (e.g., hotel satisfaction vs. industry average).
• A T-distribution is flatter and wider than a normal distribution, particularly with small samples.
• Levene’s test checks whether two or more groups have equal variances, which is important in tests like independent sample t-tests and ANOVA.

Chi-Square Tests

• Chi-square tests assess the association between categorical variables.
• Chi-square tests cannot be used for continuous variables (e.g., age vs. shopping).
• Expected frequencies represent what we would expect if no association exists.
• Statistical significance is indicated by a p-value less than 0.05.

Experimental Design & Validity

• The goal is to determine causal relationships between variables.
• Causality requires concomitant variation, time-order of variables, and elimination of other causes.
• The independent variable (IV) is manipulated.
• The dependent variable (DV) is measured.
• Threats to internal validity (extraneous variables) include history, maturation, instrumentation, statistical regression, selection bias, mortality, and testing effect.
• Threats to external validity include artificial situations, inappropriate samples, and inappropriate timing.
• Pre-experimental designs such as one-group pretest-posttest lack a control group.
• True experimental designs include a control group and randomization; types include pretest-posttest with control and posttest-only with control.
• Lab experiments offer controlled conditions and high internal validity but may have low external validity.
• Field experiments occur in real-world settings, providing high external validity but potentially lower internal validity.

One-Way ANOVA

• One-way ANOVA compares means across two or more groups.
• The DV is continuous (interval/ratio).
• The IV is categorical (nominal/ordinal).
• The null hypothesis (H₀) states that all group means are equal.
• The F-statistic is the ratio of between-group variance to within-group variance.
• An F-statistic ≈ 1 suggests H₀ is likely true.
• Within-group variance is the variation within each category.
• Between-group variance is the variation between group means.
• A p-value less than 0.05 leads to the rejection of H₀, indicating a significant effect.
• A p-value greater than or equal to 0.05 indicates failure to reject H₀, suggesting no significant effect.

Correlation & Regression

• Pearson’s correlation measures the strength and direction of the linear relationship between two continuous variables.
• r = 1 indicates a perfect positive correlation.
• r = 0 indicates no correlation.
• r = -1 indicates a perfect negative correlation.
• A scatter plot visualizes relationships and helps detect linearity and outliers.
• Simple regression predicts Y (DV) from X (IV) using the equation Y = a + bX, where a is the intercept and b is the slope.
• A positive b indicates a positive relationship.
• A negative b indicates a negative relationship.
• b = 0 indicates no relationship.
• R² (coefficient of determination) represents the proportion of variance in the dependent variable explained by the model.
• Higher R² indicates better predictive power.
• Multiple regression predicts Y from multiple Xs (IVs), including continuous and nominal-binary predictors.
• Multicollinearity in multiple regression can be checked using the variance inflation factor (VIF).
• High VIF indicates a strong correlation between IVs, which can be problematic.