Percentiles and Quantiles

Reading

Section 1.6.5

Practice Problems

1.9.6 (Page 65)

1.46, 1.48, 1.49a, 1.50, 1.51, 1.53, 1.54

Notes

• Percentiles measure how many of the values are below a given value.
• So the 90th percentile is the x value such that 90% of the values are below it.
• Lower percentiles correspond to lower values.
• Important percentiles:
  • 25th percentile = first quartile = Q1
  • 50th percentile = median = M
  • 75th percentile = third quartile = Q3
• These three percentiles, together with the min and max, split the data into 4 quarters:
  • from min to Q1
  • from Q1 to M
  • from M to Q3
  • from Q3 to max
• Each of these quarters has 25% of the data (the same number of values!)
• These numbers are called the five number summary.

• Interquartile Range is the distance between Q1 and Q3:

  \[IQR = Q3 - Q1\]

• Boxplots are a graphical representation of the 5 number summary.
  • A “box” from Q1 to Q3
  • A thick line marking the median
  • “whiskers” extend to min/max
  • These boxplots can be misleading in the presence of outliers.
• Outlier test. Values that are more than \(1.5\times IQR\) from the quartiles are suspected outliers.
• Modified boxplot:
  • Whiskers extend to last values before the suspected outliers
  • suspected outliers marked with a symbol

Practice

The following table contains the populations for each state, in millions. Determine the 5 number summary and draw the boxplot.