When & Why to Use
Notched Box Plots?
Notched box plots add approximate confidence intervals around the median, helping you compare group medians visually before using a formal statistical test.
Direct Answer
A notched box plot shows an approximate confidence interval around the median. If two notches do not overlap, their medians may differ; if they overlap, the plot alone is not strong evidence of a difference.
notch = median Β± 1.57 Γ IQR / sqrt(n)1. What Are Notched Box Plots?
A notched box plot is a standard box plot with an additional feature: a notch (or confidence interval) around the median. This notch visualizes uncertainty in the median estimate, helping you screen whether group medians may differ.
The notch appears as a narrowing of the box around the median line. If you're comparing multiple groups in a grouped box plot, the notches help you quickly spot groups that may have different medians.
Key Insight
Rule of thumb: If the notches of two groups don't overlap, their medians may differ. If the notches overlap, the plot alone is not strong evidence of a median difference.
2. How Notches Work: The Math Behind It
The notch represents an approximate 95% confidence interval for the median. It's calculated using the interquartile range (IQR) and sample size:
Formula
Standard Error = 1.57 Γ (IQR / βn)
where n is the sample size
The notch extends from:
- Lower bound: median - standard error
- Upper bound: median + standard error
The constant 1.57 is chosen to approximate a 95% confidence interval for the median under normal distribution assumptions.
Example
If median = 80, IQR = 15, n = 20:
Standard Error = 1.57 Γ (15 / β20) = 1.57 Γ 3.35 β 5.26
Notch extends from 80 - 5.26 = 74.74 to 80 + 5.26 = 85.26
3. Interpreting Notches: Reading Median Differences
The key to interpreting notched box plots is understanding what the notch overlap (or lack thereof) means:
Notches Don't Overlap = Possible Median Difference
If the notches of two groups don't overlap, their medians may be different by this visual rule.
Example: If Group A's notch extends from 75-85 and Group B's notch extends from 90-100, there's no overlap. This suggests Group B's median may be higher than Group A's.
Notches Overlap = Difference Is Not Clear Visually
If the notches of two groups overlap, the plot does not give strong visual evidence that their medians differ. Formal statistical testing is still recommended.
Example: If Group A's notch extends from 75-85 and Group B's notch extends from 80-90, they overlap (80-85 range). This suggests the difference is not clear visually, so you should verify with a formal test before making a conclusion.
Multiple Groups Comparison
When comparing multiple groups in a grouped box plot, look for groups whose notches don't overlap with others. These are candidates for follow-up testing.
Example: In a comparison of 5 groups, if Groups A, B, and C have overlapping notches, but Group D's notch doesn't overlap with any, Group D is a candidate for follow-up testing.
4. When to Use Notched Box Plots
Useful For:
- Comparing multiple groups visually
- Quick visual screening of median differences
- Exploratory data analysis
- Presentations where visual clarity matters
- When sample sizes are moderate to large (n β₯ 10)
- When you want to avoid formal hypothesis testing initially
Consider Alternatives For:
- Very small sample sizes (n < 10)
- When you need exact p-values
- Formal hypothesis testing requirements
- When notches are too wide (low precision)
- When comparing only two groups (t-test may be clearer)
Best Practice
Use notched box plots as a visual screening tool to identify possible median differences. For formal conclusions, follow up with appropriate statistical tests (e.g., ANOVA, Kruskal-Wallis, or t-tests).
5. Practical Examples
Example 1: Classroom Test Scores
Scenario: Compare test scores across three classes to see which medians may differ.
Data:
Class A: 85, 87, 88, 90, 92, 93, 95 Class B: 75, 78, 80, 82, 85, 87, 90 Class C: 70, 72, 74, 76, 78, 80, 82
Result with Notches: Class A's notch likely doesn't overlap with Class B or C, suggesting Class A may have a higher median. Class B and C may have overlapping notches, so their difference is not clear from the plot alone.
β Try this example in PlotNerd (enable notches) βExample 2: A/B Testing Results
Scenario: Compare conversion rates across three website variants.
Data:
Control: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6 Variant A: 3.0, 3.1, 3.2, 3.3, 3.4, 3.5 Variant B: 2.5, 2.6, 2.7, 2.8, 2.9, 3.0
Result with Notches: Variant A's notch likely doesn't overlap with Control, suggesting a possible median improvement. Variant B may have overlapping notches with both, suggesting it is visually intermediate.
β Try this example in PlotNerd (compare with notches) β6. Limitations and Considerations
Sample Size Matters
Notches are most reliable with moderate to large sample sizes (n β₯ 10). For very small samples, notches can be very wide, making them less informative. For large samples, notches become narrow, making differences easier to detect.
Approximation, Not Exact Test
Notched box plots provide a visual approximation of median uncertainty, not an exact hypothesis test. They are based on assumptions (e.g., normal distribution) that may not hold for your data. Always verify with formal statistical tests when making conclusions.
Distribution Assumptions
The notch calculation assumes approximately normal distributions. For highly skewed or non-normal data, notches may be less reliable. Consider using MAD outlier detection or transformation for skewed data.
7. FAQ
Q: What confidence level do notches represent?
A: Notches represent an approximate 95% confidence interval for the median. The constant 1.57 in the formula is chosen to approximate this confidence level under normal distribution assumptions.
Q: Can I use notched box plots with grouped comparisons?
A: Yes! Notched box plots work excellently with grouped box plots. In PlotNerd, you can enable notches for both single and grouped box plots. Simply toggle the "Show Notches" option in the visualization panel.
Q: What if notches are too wide or too narrow?
A: Wide notches (small sample sizes) indicate high uncertaintyβdifferences may be harder to detect. Narrow notches (large sample sizes) indicate high precision, so even small visual gaps may merit follow-up testing. If notches extend beyond the box (Q1-Q3 range), they are automatically clipped to the box boundaries for visual clarity.
Q: Should I always use notched box plots?
A: Not necessarily. Use notched box plots when you want to screen median differences visually between groups. For simple descriptions or when comparing only two groups, standard box plots or formal tests (t-tests) may be clearer. Notched box plots are most valuable for exploratory analysis with multiple groups.
Q: Can notched box plots replace formal statistical tests?
A: No. Notched box plots are a visual screening tool, not a replacement for formal hypothesis testing. They help you identify potential median differences, but you should follow up with appropriate statistical tests (e.g., ANOVA, Kruskal-Wallis) for formal conclusions, especially in research or publication contexts.
8. Conclusion
Notched box plots are useful tools for visually screening median differences between groups. By displaying approximate intervals around medians, they help you decide which group comparisons deserve formal follow-up tests.
Key takeaways:
- Non-overlapping notches suggest possible median differences
- Overlapping notches mean the difference is not clear from the plot alone
- Use notches as a visual screening tool, not a replacement for formal tests
- Works best with moderate to large sample sizes (n β₯ 10)
With PlotNerd, you can easily enable notches in your box plots, whether you're comparing single groups or multiple groups in a grouped visualization. Combine notches with robust outlier detection methods for comprehensive statistical analysis.
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