6.4 Quartiles, outliers, and distribution shape

Ordered summaries show how data are distributed and provide measures that remain useful when observations are skewed or include extremes.

6.4.1 Five-number summaries and boxplots

Interquartile range. The spread of the middle half of the data, \(IQR=Q_3-Q_1\).

The five-number summary is minimum, \(Q_1\), median, \(Q_3\), maximum. A boxplot displays these values. Potential outliers lie below \(Q_1-1.5IQR\) or above \(Q_3+1.5IQR\). A flagged point should be investigated, not automatically deleted.

Worked Example: Finding outlier fences

If \(Q_1=12\) and \(Q_3=20\), then \(IQR=8\). The fences are \(12-12=0\) and \(20+12=32\). Values below 0 or above 32 are potential outliers.

Try It. Find fences when \(Q_1=4\) and \(Q_3=10\).
Check Your Work \(IQR=6\); fences are \(-5\) and 19.

Worked Example: Constructing a five-number summary

For \(2,4,5,7,9,12,15\), the median is 7, \(Q_1=4\), and \(Q_3=12\). The five-number summary is \((2,4,7,12,15)\).

Try It. Find the five-number summary of \(1,3,4,6,8\).
Check Your Work Using medians of the halves excluding the overall median: \((1,2,4,7,8)\).

6.4.2 Skewness and choosing summaries

Skewed distribution. A distribution with a longer tail on one side than the other.

Right-skewed distributions usually have mean greater than median; left-skewed distributions usually have mean less than median. For roughly symmetric data without strong outliers, use mean and standard deviation. For skewed data or data with outliers, use median and IQR.

Worked Example: Choosing resistant summaries

Most repair times are under 4 h, but one major failure requires 38 h. The distribution is right-skewed, so median and IQR better describe a typical repair time and spread.

Try It. Which summaries suit a symmetric set with no outliers?
Check Your Work Mean and standard deviation.

Worked Example: Predicting the effect of an extreme value

Increasing the largest observation from 64,401 to 150,000 raises the mean and standard deviation substantially. The median and IQR may remain unchanged because the ordered middle positions do not move.

Try It. What happens to the mean if an unusually large value is added?
Check Your Work It generally increases, often noticeably.
Applied Problem: Summarizing filter run times. Run times in hours are \(18,19,20,20,21,22,23,24,25,52\). Describe the likely shape and choose appropriate measures of centre and spread. Explain the effect of 52.
One Possible Answer The value 52 creates a long right tail, so the distribution is right-skewed. Median and IQR are preferable. The value 52 raises the mean and standard deviation much more than it affects the median and IQR and should be investigated as a possible unusual operating event.

6.4.3 Practice Problems

  1. Name the five values in a five-number summary.
    Check Your Work Minimum, \(Q_1\), median, \(Q_3\), maximum.
  2. Find the IQR if \(Q_1=14\) and \(Q_3=29\).
    Check Your Work 15.
  3. Find the upper fence for \(Q_1=10\), \(Q_3=18\).
    Check Your Work \(IQR=8\), so upper fence \(=30\).
  4. Which is more resistant, mean or median?
    Check Your Work Median.
  5. For right skew, which is usually larger, mean or median?
    Check Your Work Mean.
  6. Pressure data contain one sensor error of 900 kPa among values near 400 kPa. Which summaries should be reported initially?
    Check Your Work Median and IQR, with the extreme value investigated.
  7. Flow quartiles are 12 and 28 L/s. Find IQR and outlier fences.
    Check Your Work \(IQR=16\); the fences are \(-12\) L/s and 52 L/s.
  8. Repair times have a long right tail. Choose centre and spread.
    Check Your Work Median and IQR.