Box plots are powerful tools for visualizing data distribution. They show the median, quartiles, and outliers, giving a quick snapshot of how test scores are spread out. This makes it easy to compare different datasets side-by-side.

By looking at box plots, you can quickly see if test scores are skewed, symmetric, or have outliers. This helps identify patterns and differences between groups, like comparing scores across different classes or subjects.

Box Plots

Construction of box plots

• Five-number summary consists of:
  • Minimum value represents the smallest value in the dataset (lowest test score)
  • First quartile (Q1) represents the median of the lower half of the dataset (25th percentile)
  • Median (Q2) represents the middle value of the dataset (50th percentile)
  • Third quartile (Q3) represents the median of the upper half of the dataset (75th percentile)
  • Maximum value represents the largest value in the dataset (highest test score)
• Constructing a box plot involves:
  • Drawing a horizontal line representing the range of the data from minimum to maximum value
  • Drawing a box from Q1 to Q3, with a vertical line inside the box at the median (Q2)
  • Drawing whiskers extending from the box to the minimum and maximum values
  • Identifying outliers, which are data points that fall more than 1.5 times the interquartile range (IQR) below Q1 or above Q3, where $IQR = Q3 - Q1$ (extremely low or high test scores)

Interpretation of box plot data

• Box in the plot represents the middle 50% of the data, or the interquartile range (IQR) (middle 50% of test scores)
• Median line inside the box indicates the center of the dataset
  • Median closer to Q1 indicates the data is skewed right (more low test scores)
  • Median closer to Q3 indicates the data is skewed left (more high test scores)
  • Median roughly in the middle of the box indicates the data is approximately symmetric (evenly distributed test scores)
• Whiskers show the range of the data, excluding outliers
  • Longer whiskers indicate a wider spread of data (greater variability in test scores)
  • Shorter whiskers indicate a narrower spread of data (less variability in test scores)
• Outliers, represented by individual points beyond the whiskers, are unusual or extreme values in the dataset (exceptionally low or high test scores)

Data Distribution and Variability

• Box plots provide insights into the data distribution and variability of a dataset
• Quartiles divide the data into four equal parts, helping to visualize the spread and central tendency
• The box and whiskers together show the overall variability of the data
• Box plots are a form of descriptive statistics, summarizing key features of the data distribution visually

Comparison using side-by-side box plots

• Side-by-side box plots allow for visual comparison of the distribution and spread of multiple datasets (comparing test scores between classes)
• Comparing datasets involves:
  1. Observing the relative positions of the boxes to compare the central tendencies (medians) of the datasets (which class has higher or lower median test scores)
  2. Comparing the lengths of the boxes (IQRs) to assess the spread of the middle 50% of each dataset (which class has more or less variability in the middle 50% of test scores)
  3. Examining the lengths of the whiskers to compare the overall ranges of the datasets, excluding outliers (which class has a wider or narrower range of test scores)
  4. Identifying and comparing any outliers present in each dataset (which class has more or fewer exceptionally low or high test scores)
• Differences in box plot characteristics can help identify similarities and differences between the datasets, such as:
  • Which dataset has a higher or lower median (which class has higher or lower median test scores)
  • Which dataset has a larger or smaller spread (which class has more or less variability in test scores)
  • Which dataset has more or fewer outliers (which class has more or fewer exceptionally low or high test scores)