Five-Number Summary and Box Plot Construction
Key Concepts
- Five-Number Summary: Consists of five key data points: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
- Box Plot: A graphical representation of the five-number summary.
Steps to Find the Five-Number Summary
- Order the Data: Arrange the dataset from least to greatest.
- Identify the Minimum and Maximum:
- Minimum is the smallest number.
- Maximum is the largest number.
- Find the Median:
- For an even number of data points, average the two middle numbers.
- For an odd number, it is the middle number.
- Determine Quartiles:
- Split data into two halves.
- Q1: Median of the lower half.
- Q3: Median of the upper half.
Example Dataset
- Ordered Data: Assume data is [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
- Minimum: 0
- Maximum: 10
- Median: 3.5 (average of 3 and 4)
- Q1: 2
- Q3: 5
Constructing a Box Plot
- Draw a Number Line: Span from minimum (0) to maximum (10).
- Vertical Lines: At each of the five numbers in the summary (minimum, Q1, median, Q3, maximum).
- Box and Whiskers:
- Box from Q1 to Q3 with a line at the median.
- Whiskers extend from the box to the minimum and maximum.
- Interquartile Range (IQR): Width of the box (Q3 - Q1).
Modified Box Plot and Outlier Detection
- Outliers:
- Calculate fences using IQR: A value is an outlier if
- Less than Q1 - 1.5 * IQR
- Greater than Q3 + 1.5 * IQR
- Example Calculation:
- IQR = Q3 - Q1 = 3
- Lower Fence: Q1 - 1.5 * IQR = -2.5 (no lower outliers)
- Upper Fence: Q3 + 1.5 * IQR = 9.5
- Maximum 10 is an outlier (greater than 9.5)
- Representation:
- Maximum whisker to the largest non-outlier value
- Outliers (e.g., 10) represented as dots
Conclusion
- Box plots visually represent data spread and identify outliers.
- Modified box plots provide a clear view of data distribution when outliers are present.
Let me know if you have any questions or need further clarification. Check related resources for additional practice.