D3.2.15 Box-and-whisker plots to represent data for a continuous variable such as student height Notes

Using Box-and-Whisker Plots to Visualize Continuous Data

1. It uses five key statistics, along with the identification of outliers:
   1. Minimum: The smallest data value, excluding outliers.
   2. First Quartile (Q1): The median of the lower half of the data.
   3. Median (Q2): The middle value of the dataset.
   4. Third Quartile (Q3): The median of the upper half of the data.
   5. Maximum: The largest data value, excluding outliers.
   6. Outliers: Data points that fall significantly outside the typical range, calculated as more than 1.5 × IQR above Q3 or below Q1.

Why Are Box-and-Whisker Plots Useful?

1. Box-and-whisker plots provide a clear, concise way to:
   1. Summarize Data: Visualize the range, spread, and central tendency of the dataset.
   2. Identify Outliers: Spot unusual data points that lie far from the majority.
   3. Compare Datasets: Use multiple box plots side by side to compare distributions across groups.

How to Create a Box-and-Whisker Plot

Step 1: Organize the Data

Arrange your data values in ascending order.

Step 2: Calculate Key Statistics

1. Find the Median (Q2): Divide the dataset into two equal halves.
2. Find Q1 and Q3: Identify the medians of the lower and upper halves.
3. Calculate the Interquartile Range (IQR): Subtract Q1 from Q3: IQR=Q3−Q1IQR = Q3 - Q1IQR=Q3−Q1

Step 3: Identify Outliers

1. Multiply the IQR by 1.5: 1.5×IQR1.5 \times IQR1.5×IQR
2. Add this value to Q3 to find the upper bound for outliers.
3. Subtract this value from Q1 to find the lower bound for outliers.

Step 4: Draw the Plot

1. Box: Draw a rectangle from Q1 to Q3.
2. Median: Add a line inside the box at Q2.
3. Whiskers: Extend lines from the box to the minimum and maximum values within the non-outlier range.
4. Outliers: Plot outliers as individual points beyond the whiskers.

Example question

Example: Heights of Students in a Class

Consider this dataset of student heights (in cm):

193, 183, 176, 163, 193, 152, 160, 175, 184, 180, 173, 186, 153, 172, 180, 195, 176, 201, 177.

Solution

1. Organize Data: Arrange in ascending order:
   1. 152, 153, 160, 163, 172, 173, 175, 176, 176, 177, 180, 180, 183, 184, 186, 193, 193, 195, 201.
2. Calculate Q1, Q2 (Median), and Q3:
   1. Q1 = 172
   2. Median (Q2) = 176
   3. Q3 = 184
3. Find IQR: IQR=Q3−Q1=184−172=12IQR = Q3 - Q1 = 184 - 172 = 12 IQR=Q3−Q1=184−172=12
4. Identify Outliers:
   1. Upper bound = Q3+1.5×IQR=184+18=202Q3 + 1.5 \times IQR = 184 + 18 = 202Q3+1.5×IQR=184+18=202
   2. Lower bound = Q1−1.5×IQR=172−18=154Q1 - 1.5 \times IQR = 172 - 18 = 154Q1−1.5×IQR=172−18=154
   3. Any value above 202 or below 154 is an outlier.
5. Draw the Plot:
   1. Box: From Q1 (172) to Q3 (184).
   2. Median: Line at 176.
   3. Whiskers: Extend to 153 (minimum) and 201 (maximum).
   4. Outliers: None in this dataset.

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Tips for Analyzing Box-and-Whisker Plots

1. Check the Spread: The length of the whiskers and box indicates variability.
   1. A longer box suggests greater spread within the quartiles.
2. Look for Symmetry: If the median is centered within the box, the data is evenly distributed. Otherwise, it’s skewed.
3. Compare Multiple Plots: When analyzing datasets from different groups, compare the boxes for medians, spread, and outliers.

Step 1: Organize the Data

1. Arrange the data in ascending order.

Step 4: Draw the Plot

1. Box: Draw a box from $$Q1$$ to $$Q3$$.
2. Median: Draw a line inside the box at the median ($$Q2$$).
3. Whiskers: Extend lines from the box to the minimum and maximum values (excluding outliers).
4. Outliers: Plot any outliers as individual points.

Why Use Box-and-Whisker Plots?

1. Summarize Data: Quickly visualize the spread and central tendency of the data.
2. Identify Outliers: Easily spot unusual data points.
3. Compare Datasets: Multiple box plots can be used side by side to compare different groups.

Practical Applications

1. Biology: Analyzing traits like plant height or enzyme activity.
2. Medicine: Comparing patient data, such as blood pressure or cholesterol levels.
3. Education: Evaluating test scores to identify trends or outliers.

Box-and-whisker plots are a powerful tool for summarizing and interpreting data. By highlighting key statistics and outliers, they provide valuable insights into the variability and distribution of a dataset.