Range and Quartiles

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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