DESCRIBING NUMERICAL DATA

Chapter 4, Describing Numerical Data, focuses on how to categorise, organise, and summarise quantitative information using mathematical tools and graphical representations.

1. Types of Numerical Variables

Numerical data is split into two main types based on how the values are generated:

2. Organising Numerical Data

Data must be organised to be understandable.

• Discrete Data: If there are few distinct values, they are listed in a frequency table where each value is treated as its own category.
• Continuous Data: This is grouped into classes (bins).
  • Terminology: The Lower/Upper class limits are the smallest/largest values in a class. The Class width is the difference between consecutive lower limits, and the Class mark is the average of the two limits.

3. Measures of Central Tendency

These measures identify the “typical” value or the centre of the dataset .

4. Measures of Dispersion (Variation)

These quantify the “spread” or variability of the data .

1. Range: The difference between the largest and smallest values ($Max - Min$). It is highly sensitive to outliers.
2. Variance ($S^2$): Measures how much data values deviate from the mean.
   • Formula (Sample): $S^2 = \frac{\sum (X_i - \bar{X})^2}{n - 1}$.
   • Property: Adding a constant does not change the variance. Multiplying by $c$ changes the variance by $c^2$.
3. Standard Deviation: The square root of the variance. It is expressed in the same units as the original data (e.g., kg or years).

5. Relative Standing and Summaries

• Percentiles: The $100p$ percentile is the value below which $100p$ percent of the data falls.
• Quartiles: These divide the data into four equal parts.
  • Q1 (First Quartile): 25th percentile.
  • Q2 (Median): 50th percentile.
  • Q3 (Third Quartile): 75th percentile.
• Interquartile Range (IQR): The difference between Q3 and Q1 ($Q3 - Q1$).
• Five Number Summary: A set consisting of: Minimum, Q1, Median, Q3, and Maximum.

Practice Session