Five Number Summary and Box and Whisker Plots

Five Number Summary

• Definition: A concise summary of a dataset using five key points.
  • Minimum: The smallest data point.
  • Q1: The first quartile, separating the lowest 25% of the data.
  • Median (Q2): The middle value of the dataset.
  • Q3: The third quartile, marking 75% of the data.
  • Maximum: The largest data point.

Box and Whisker Plot

• Description: A graphical representation of the five number summary.
  • Box: Extends from Q1 to Q3, with a line indicating the median (Q2).
  • Whiskers: Lines extend from the box to the minimum and maximum values.
  • Scale: Always includes a horizontal scale showing data values.

Example 4

• Task: Draw a box and whisker plot using a dataset.
• Process:
  • Determine the horizontal scale from minimum (11) to maximum (35).
  • Align Q1 (23), median (25), and Q3 (30) on the scale.
  • Complete the box and whiskers representation.

Percentiles and Fractiles

Definitions

• Percentiles: Divide data into 100 equal parts.
• Deciles: Divide data into 10 equal parts.
• Quartiles: Divide data into 4 equal parts.

Usage

• Educational/Health Applications: Indicate relative standing (e.g., a child in the 95th percentile is taller than 95% of peers).
• Quartiles and Percentiles:
  • Q1 = 25th percentile
  • Q2 (Median) = 50th percentile
  • Q3 = 75th percentile

Example 5

• Task: Interpret percentiles using an ogive (cumulative frequency graph).
• Example: The 80th percentile on a SAT score graph is approximately 1250.

Example 6

• Task: Calculate the percentile corresponding to a data point ($34,000 tuition).
• Process:
  • Count data entries less than $34,000 (8 entries of 25 total).
  • Calculate percentile: ((8/25) \times 100 = 32%
  • Interpretation: $34,000 tuition is higher than 32% of the dataset.

Standard Scores (Z-Scores)

Definition

• Z-Score: Measures how many standard deviations a data point is from the mean.
  • Positive if above mean, negative if below.
  • Formula: (z = \frac{x - \text{mean}}{\text{standard deviation}})

Typical Ranges

• Normal scores: Within ±2 standard deviations from the mean.
• Unusual scores: More than ±2 standard deviations.
• Very unusual: More than ±3 standard deviations.

Example 7

• Dataset: Vehicle speeds with mean = 56 mph, standard deviation = 4 mph.
• Speeds: Car 1 = 62 mph, Car 2 = 47 mph, Car 3 = 56 mph.
• Calculations:
  • Z-score for Car 1 (62 mph): 1.5 (normal)
  • Z-score for Car 2 (47 mph): -2.25 (unusual)
  • Z-score for Car 3 (56 mph): 0 (exactly the mean)