Design, Data and Decisions Lecture 1 Notes

Module 1 – Descriptive Statistics

Lecture Overview

• Focus on key concepts:

  • Graphical summaries of a single variable (Categorical and Continuous)

  • Numerical summaries of a single variable

    • Location of the data

    • Spread of the data

Structure of Data

• Data Forms:

  • Transactional: Records interactions between systems

  • Images: Digital files like MRIs

  • Ongoing Data: Continuous data generation

• Data Structure:

  • Organized into attributes

  • Example:

    Obs	Attribute 1	Attribute 2	Attribute 3	...	Attribute p
    1	22.3	Aus	4	...	Low
    2	41.7	Overseas	7	...	High
    ...	...	...	...	...	...

Types of Variables

Categorical (Qualitative)

• Nominal: No order (e.g., nationality, gender)

• Ordinal: Natural ordering (e.g., age group, level of education)

Quantitative (Numerical)

• Discrete: Whole number values (e.g., number of birds)

• Continuous: Any real number (e.g., height, temperature)

Categorical Variables

Frequency Reporting

• Count observations per category and report frequencies.

• Use relative frequencies as percentages:

  • Example:[ f_a = \frac{f_a}{f_t} \times 100 ]

  • Self-reported activity level example:

    Activity Level	Frequency	%
    Slight	10	10.9%
    Moderate	61	66.3%
    High	21	22.8%
    Total	92	100.0%

Graphical Summaries for Categorical Variables

Bar Chart and Pie Chart

• Bar Chart:

  • Uses bars to show frequency/relative frequency

• Pie Chart:

  • Slices represent relative frequencies

  • Angle of slice proportional to frequency

Choosing Between Charts

• Bar Charts are suitable for:

  • Ordinal variables

  • Nominal variables

  • Comparing distributions of two categorical variables

• Pie Charts should be used cautiously:

  • Only for single nominal variables with few categories

  • Avoid 3D as it distorts size perception

Quantitative Variables – Graphical Summaries

Stem-and-Leaf Plot Example

• Example data set height in cm:| Stem & Leaf | Frequency ||-------------|-----------|| 15 | 4 || 16 | 677779 || 17 | 00000002222|| 18 | 0000001222 || 19 | 000 |(Other plotting options can include histograms)

Quantitative Variables – Center and Spread

Center Measurement Stats

• Mode: Most common value, applicable for discrete/ordinal variables

• Median: Middle value when data is ordered; essential for understanding skewed distributions

• Mean: Average when the total of observations is evenly distributed

Spread Measurement Stats

• Range: Difference between the largest and smallest values (e.g., 52 - 17 = 35 years)

• Inter-Quartile Range (IQR): Middle 50% of ordered data

• Standard Deviation: Measure of how spread out observations are from the mean

  • Example Calculation of Median:

    1. Order the dataset

    2. Find middle value (specific rules for odd/even numbers of observations)

Inter-Quartile Range Calculation Steps

1. Order the data

2. Identify the first quartile ( Q1 ) and the third quartile ( Q3 ) positions in the data

3. Calculate IQR as ( Q3 - Q1 )

Box and Whisker Plot Example

• Comprised of:

  • Lower Quartile

  • Median

  • Upper Quartile

Standard Deviation Calculation

• Evaluate distance each observation is from the sample mean

• Calculate variance by summing squared deviations and dividing accordingly

  • Formula for Sample Standard Deviation:[ s^2 = \frac{1}{n-1} \sum (x_i - \overline{x})^2 ]

Z-scores

• Useful for comparing values across different variables

  • Example: Comparing scores across two subjects

• Z-score Calculation:[ z = \frac{o - \mu}{\sigma} ]

  • General rule of thumb: 95% of observations within +/- 2 standard deviations of the mean.