Five Number Summary and Measures of Spread

Five Number Summary

• Definition: A list of five key statistics that summarize a set of data. These values divide the data into four equal sections called quartiles.
• Components:
  • $X_{min}$: Minimum value
  • $Q_1$: Lower Quartile
  • $Q_2$: Median
  • $Q_3$: Upper Quartile
  • $X_{max}$: Maximum value

Example: Finding the Five Number Summary

• Data set: 3, 2, 8, 6, 1, 5, 3, 7, 6
  1. List the data in ascending order: 1, 2, 3, 3, 5, 6, 6, 7, 8
  2. Find the median (Q2): Median = 5
  3. Find Q1 (the median of the lower section):
     • Lower section: 1, 2, 3, 3
     • $Q_1 = (2 + 3) / 2 = 2.5$
  4. Find Q3 (the median of the upper section):
     • Upper section: 6, 6, 7, 8
     • $Q_3 = (6 + 7) / 2 = 6.5$

Measures of Spread

• Definition: Measures of spread describe how far data values are spread out from the center or from each other.

Minimum and Maximum Values

• Minimum Value: $X_{min} = 1$
• Maximum Value: $X_{max} = 8$

Range

• Definition: The range is a measure of how spread out a data set is.
• Formula: $Range = X<em>{max} - X</em>{min}$

Interquartile Range (IQR)

• Definition: The IQR measures the spread of the middle 50% of the data.
• Advantage: It is not affected by outliers, making it a more reliable measure of spread.
• Formula: $IQR = Q<em>3 - Q</em>1$

Example: Finding the Range and IQR

• Data set: 2.1, 3.5, 3.9, 4.0, 4.7, 4.8, 5.2
• Five Number Summary:
  • $X_{min} = 2.1$
  • $Q_1 = 3.5$
  • $Median = 4.0$
  • $Q_3 = 4.8$
  • $X_{max} = 5.2$
• Range Calculation:
  • $Range = X<em>{max} - X</em>{min} = 5.2 - 2.1 = 3.1$
• IQR Calculation:
  • $IQR = Q<em>3 - Q</em>1 = 4.8 - 3.5 = 1.3$

Stem and Leaf Plot Example

• Key: 2 | 3 = 23
• Plot:
  • 0 | 9
  • 1 | 0 1 1 2 4 5 8 9
  • 2 | 1 3 3 3 7 9
  • 3 | 0 0 1 2 2