Nominal vs. Ordinal Data

Nominal Data (Categorical)

• Nominal data involves labels or names assigned to data points without any intrinsic order or ranking.
• Examples include nationality (e.g., Vietnam, USA, Japan), colors (e.g., Red, Green, Yellow), gender (Male, Female), and blood type (A, B, AB, O).
• Analysis typically focuses on the frequency of occurrence for each category, such as calculating percentages or creating pie charts.

Ordinal Data

• Ordinal data represents a ranking or order between data points, but the intervals between the rankings are not necessarily equal.
• Examples include academic grades (e.g., Excellent, Good, Fair, Average, Poor), ranking in a competition (First, Second, Third), and levels of agreement (e.g., Strongly Agree, Agree, Disagree, Strongly Disagree).
• Analysis involves frequency counts, comparisons of ranks and computation of the median, and bar charts construction.

Interval Data

• Interval data possesses a defined order, and the intervals between values are equal. However, there is no true zero point.
• Temperature in Celsius or Fahrenheit, years (e.g., 2023, 2024, 2025), and IQ scores are examples. Zero does not indicate absence of the property
• Arithmetic operations like addition and subtraction are viable, enabling the calculation of means, standard deviations, and histograms.

Ratio Data

• Ratio data encompasses all the properties of interval data and includes a true zero point, indicating the absence of the measured attribute.
• Weight, height, and the number of products sold serve as examples. A weight of 0 kg signifies that there is no weight.
• All arithmetic operations, including multiplication and division, are permissible, facilitating a wide array of statistical methods.

Important Notes on Data Types

• Quantitative variables can be converted to ordinal, but ordinal variables cannot be converted to quantitative.
• The data matrix serves as a basis for all statistical analyses.
• Frequency tables are used to present results.

Influence of Data Distribution on Statistical Methods

• Data can possess various distributions, which influence the choice of appropriate statistical analysis methods.
• Non-standard data requires non-parametric methods or data transformation.

Normal Distribution

• Characterized by symmetry, bell shape, a mean, median, and mode that coincide.
• Has one peak, with balanced sides, and is common in natural phenomena.
• Examples include adult height, standardized test scores (e.g., SAT, IQ, GMAT), and measurement errors.
• Parametric tests are the appropriate statistical methods such as t-tests, ANOVA, and linear regression.

Right Skewed Distribution

• It has a long tail extending to the right with many small values and infrequent but very large values
• Mean > Median > Mode in this distribution
• Examples include Income distributions and YouTube video views
• Data transformation methods: Log Transformation, Square Root Transformation, Inverse Transformation can be used
• Non-parametric Tests: Mann-Whitney U Test and Kruskal-Wallis Test can be used

Left Skewed Distribution

• It has a long tail extending to the left with many large values and infrequent but very small values
• Mean < Median < Mode in this distribution
• Examples include grades on an easy test, retirement age and time to complete a test
• Data transformation methods: Reverse Transformation can be used
• Non-parametric Tests: Wilcoxon Signed-Rank Test can be used

Bimodal Distribution

• Has two peaks, may represent two distinct groups within the data.
• Cannot be described by a single mean.
• Examples: age of players and coaches in a sports team, the height of sampled males and females, driving time in a city due to rush hour vs. off-peak.
• Appropriate statistical methods include separating the data into two groups or using Gaussian Mixture Models (GMM)

Techniques for Handling Skewed Data

• Data Transformation: Transforms data to approach a normal distribution.
• Includes Log Transformation, Square Root Transformation, and Inverse Transformation, each appropriate under different conditions.
• Non-Parametric Tests: Use when data is too skewed or cannot be transformed.
• Mann-Whitney U Test replaces the t-test, and the Kruskal-Wallis Test replaces ANOVA.

Log Transformation

• Use when: Data is right skewed.
• How: Each data point x is replaced with log(x).
• Improves: Balances data and can normalize distributions
• Only works for positive numbers

Square Root Transformation

• Use when: Data is slightly right skewed.
• How: Each data point x is replaced with √x.
• Still usable if data set has a value of 0
• Less balance than log transformation

Inverse Transformation

• Use when: Data is severely right skewed with extreme outliers.
• How: Each data point x is replaced with 1/x.
• Should not be used if the data set has a value of 0
• Need care of the data relationship reversal

Non-Parametric Tests

• Can be used to evaluate datasets that fail a normality test
• Mann-Whitney U Test: Compares the median of two non-normal groups
• Kruskal-Wallis Test: Compares the median of three or more non-normal groups

T-tests & ANOVA

• T-Tests are used to test the mean between two groups
• ANOVA tests the mean between 3 or more groups

T-test

• Goal: Test for the difference in the mean between two groups
• Formula: t=X1־-X2-S12n1+S22n2t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{S_1^2{{n_1} + \frac{S_2^2}{n_2}}} where X1,X2 are the mean of groups 1 & 2, S1 and S2 are the variants and n1,n2 is the sample size.
• Independent T-tests can be used for testing unrelated groups
• Parired T-tests can be used for paired related groups
• A p value test can be used to determined significance where: p<.05 indicates significance / p>.05 indicates no significance
• Only usable if there are 2 groups and data is normall distributed

ANOVA

• Goal: Measure the difference between the means of 3+ groups
• Formula: F=VarianceBetweenGroupVarianceWithinGroupF = \frac{\text{Variance Between Group}}{\text{Variance Within Group}} large values of F indicate significance
• Multiple types of anova

Summarized Statistics

• Mode: Most frequently occurring value.
• Mean: Average of all values calculated by sum / count of values
• Median: Middle value, after the data has been sorted, not affected by outlier data