Box Plot and Outliers

Box Plot and Outliers

• Quartiles

  • Divide dataset into four equal parts

  • Q1 (25th percentile), Q2 (50th percentile or median), Q3 (75th percentile)

  • Each quartile represents 25% of the data.

• Interquartile Range (IQR)

  • Represents the range of the middle 50% of the data.

  • Calculated as:
    IQR = Q3 - Q1

  • Example: If Q1 = 28 and Q3 = 38, then
    IQR = 38 - 28 = 10

  • The IQR indicates the spread in the middle 50% of the data points.

• Five Number Summary

  • Comprises: Minimum, Q1, Q2, Q3, Maximum.

  • Provides a quick summary of the data's spread and center.

• Box Plot Construction

  • Draw a box from Q1 to Q3, with a line at Q2 (median).

  • Extend lines (whiskers) to the minimum and maximum values (without outliers).

  • If outliers exist, whiskers only extend to the adjacent values within fences.

• Outlier Identification

  • Use fences to identify outliers:

  • Lower Fence:
    Q1 - 1.5 * IQR

  • Upper Fence:
    Q3 + 1.5 * IQR

  • Data points outside these fences are considered outliers.

• Example Application

  • For car speeds, if Q1 = 28 and Q3 = 38, IQR = 10:

  • Lower Fence = 28 - 15 = 13

  • Upper Fence = 38 + 15 = 53

  • Hence, any speed < 13 or > 53 is an outlier.

• Impact of Outliers on Box Plots

  • If no outliers: whiskers extend to min & max values.

  • If outliers exist: whiskers stop at the closest non-outlier value (adjacent value).

• Histograms and Box Plots

  • Box plots provide compact representation of data (similar to histograms).

  • Useful to visualize data skewness and compare distributions.

  • Skewed distributions can be identified using box plots:

  • Right skew (long tail on the right)

  • Left skew (long tail on the left)

• Comparing Two Datasets

  • Box plots are preferable for side-by-side comparisons of two groups.

  • Ensure the same numerical axis to compare effectively.