This lesson covers Range and Quartiles.

Sources:

• https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch12/5214890-eng.htm

Range

• Difference between largest and smallest observed values
• Actual spread of the data, including outliers
• Can be expressed as lower limit - upper limit e.g. $1-10$ or as range upper limit - lower limit e.g. $9$
• Doesn’t measure spread of the majority of the data spread since it only measures spread between lowest and highest value

Quartiles

• Median divides the dataset in two equal sets.

• Lower Quartile ($Q1$) - first quartile

  • middle of first set
  • $25\%$ of values are smaller than $Q1$ and $75\%$ values are higher than $Q1$

• Upper Quartile ($Q3$) - third quartile

  • middle of second set
  • $75\%$ of values are smaller than $Q3$ and $25\%$ values are higher than $Q3$

• Second Quartile ($Q2$)

  • Median of the data set

• Example

  • Data: 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36

  • Ordered data: 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49

  • Median: 41

    • 1st part: 6, 7, 15, 36, 39
    • 2nd part: 41, 43, 43, 47, 49

    middle value not included since total odd number else it is included in both parts

  • Upper quartile: 43

  • Lower quartile: 15

  • Second quartile: 41

Interquartile range

• Difference between upper quartile and lower quartile
• Measures spread of the data
• Spans $50\%$ of the data and eliminate influence of outliers by removing highest and lowest quarters
• $IQR = Q3 - Q1$

Example

• Question: A year ago, Angela began working at a computer store. Her supervisor asked her to keep a record of the number of sales she made each month. The following data set is a list of her sales for the last $12$ months: $34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37$. Use Angela’s sales records to find:
  • the median
  • the range
  • the upper and lower quartiles
  • the interquartile range
• Answer:
  • $Median = 26$
  • $Range = 56$
  • $Q1 = 17$
  • $Q3 = 42$
  • $IQR = 25$

Semi-quartile Range

• One half of the difference between $Q3$ and $Q1$ i.e. $\frac{Q3-Q1}{2}$
• If the Distribution is Symmetric
  • $50\%$ of the data is covered by $Median \pm SQR$
  • Not true if it skewed
• $SQR$ Is hardly affected by higher values. Good measure of spread for skewed distribution
• Rarely used for datasets that have normal distribution
  • Standard Deviation is used

Box and Whisker Plot

Source: mathsisfun.com

• Question: $4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11$
• Answer:
  • In order: $3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18$
  • Cut in Quarters:
    • $3, 4, 4, 4, 7, 10,~~~~ 11, 12, 14, 16, 17, 18$
    • $Q2=10.5;~ Q1=4;~ Q3=15$
    • $3, 4, 4,~~~ 4, 7, 10,~~ 11, 12, 14,~~~ 16, 17, 18$
  • $Min=3;~ Max=18$
  • $IQR = 11$

Five -Number Summary

• $Q2=10.5;~ Q1=4;~ Q3=15$

• $Min=3;~ Max=18$

• 

MS Excel:

• Select Data and Insert -> Statistical -> Box and Whisker

More Questions at: https://www.mathsisfun.com/data/quartiles.html