Variance & Standard Deviation
Quartiles divide a sorted dataset into four equal parts. Q1 (25th percentile) is the median of the lower half, Q2 is the overall median, Q3 (75th percentile) is the median of the upper half. The IQR (Q3−Q1) measures the middle 50% spread.
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Tip: For odd numbers of data points: exclude the median when calculating Q1 and Q3. For even: include all data in each half. Different methods exist — statistical software may calculate quartiles slightly differently.
- 1Sort data in ascending order
- 2Q2 = median of all data
- 3Q1 = median of lower half (below Q2)
- 4Q3 = median of upper half (above Q2)
- 5IQR = Q3 − Q1; Outlier if < Q1 − 1.5×IQR or > Q3 + 1.5×IQR
1, 2, 3, 4, 5, 6, 7, 8=Q1=2.5, Q2=4.5, Q3=6.5, IQR=4
10, 20, 30, 40, 50, 100=Q1=20, Q2=35, Q3=50, IQR=30100 is an outlier: > Q3+1.5×30 = 95
| Context | Use of Quartiles |
|---|---|
| Box plots | Visual display of Q1, Q2, Q3, whiskers |
| Outlier detection | Values beyond 1.5×IQR from Q1/Q3 |
| Test scoring | 75th percentile = top 25% of scores |
| Salary data | Salary range for middle 50% of earners |
| Finance | Portfolio return distribution analysis |
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Fun Fact
Box-and-whisker plots (box plots), invented by John Tukey in 1970, show quartiles visually. They are particularly useful for comparing distributions across groups and quickly spotting outliers and skew.
References
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