DESCRIPTIVEDescriptive StatisticsStatistics Calculator
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Quartiles โ€” Q1, Q2, Q3, IQR

Q1=25th, Q2=median, Q3=75th percentile. Four methods: inclusive (Tukey), exclusive, interpolation (Excel), nearest rank. IQR, midhinge, Bowley skewness.

Concept Fundamentals
Lower quartile
Q1 (25th %ile)
First quartile
Median
Q2 (50th %ile)
Middle value
Upper quartile
Q3 (75th %ile)
Third quartile
Q3 โˆ’ Q1
IQR
Interquartile range
Compute QuartilesQ0, Q1, Q2, Q3, Q4

Why This Statistical Analysis Matters

Why: Quartiles summarize distribution without assuming normality. IQR and fences detect outliers. Midhinge is robust center.

How: Enter data. Choose method. Get Q0โ€“Q4, IQR, semi-IQR, midhinge, quartile deviation coefficient, Bowley skewness.

  • โ—IQR = Q3 โˆ’ Q1
  • โ—Midhinge = (Q1+Q3)/2
  • โ—1.5ร—IQR fence rule
Sources:NIST Engineering Statistics HandbookWikipedia Quartile
Q
STATISTICSDescriptive Statistics

Quartile Calculator โ€” Q1, Q2, Q3 with 4 Methods

Compute quartiles with Inclusive (Tukey), Exclusive, Interpolation (Excel), Nearest Rank. IQR, Midhinge, Bowley skewness.

Five Number Summary โ†’IQR Calculator โ†’

Real-World Scenarios โ€” Click to Load

Data Input

quartile_results.sh
CALCULATED
$ quartile --n=15 --method="inclusive"
Q0 (Min)
55.0000
Q1
73.5000
Q2 (Median)
82.0000
Q3
91.0000
Q4 (Max)
98.0000
IQR
17.5000
Semi-IQR
8.7500
Midhinge
82.2500
Bowley Skewness
0.0286
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Quartile Result
Q1, Q2, Q3 โ€” IQR, Midhinge, Bowley
IQR = 17.500
Q1: 73.50Median: 82.00Q3: 91.00Midhinge: 82.25Skew: 0.029
numbervibe.com/calculators/statistics/quartile-calculator

Box Plot with Quartile Lines

Q0Q1Q2Q3Q4

Orange = Q0/Q4; Dashed = Q1/Q3; Black = Q2 (median). Box spans Q1 to Q3.

Quartile Values Comparison

Data Distribution with Quartile Zones

Red: below Q1; Orange: Q1โ€“Q2; Green: Q2โ€“Q3; Blue: above Q3.

Calculation Breakdown

QUARTILES
Q0 (Min)
55.0000
ext{First} ext{value} ext{after} ext{sorting}
Q1 (25th percentile)
73.5000
Method: inclusive
Q2 (Median)
82.0000
50th ext{percentile}
Q3 (75th percentile)
91.0000
Method: inclusive
DERIVED STATISTICS
Q4 (Max)
98.0000
ext{Last} ext{value} ext{after} ext{sorting}
IQR
17.5000
Q3 โˆ’ Q1 = 91.00 โˆ’ 73.50
Semi-IQR
8.7500
IQR/2 = 17.50/2
Midhinge
82.2500
(Q1+Q3)/2 = (73.50+91.00)/2
Bowley Skewness
0.0286
(Q3+Q1โˆ’2ร—Q2)/(Q3โˆ’Q1)
Q.D. Coefficient
0.1064
IQR/(Q3+Q1)

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿ“ˆ Statistical Insights

Q1

25th percentile

โ€” Definition

IQR

Q3 โˆ’ Q1, middle 50%

โ€” Spread

Midhinge

(Q1+Q3)/2 robust center

โ€” Tukey

Key Takeaways

  • โ€ข Q0 = Min, Q1 = 25th percentile, Q2 = Median (50th), Q3 = 75th percentile, Q4 = Max
  • โ€ข IQR = Q3 โˆ’ Q1 measures spread of the middle 50%
  • โ€ข Semi-IQR (Quartile Deviation) = IQR/2
  • โ€ข Quartile Deviation Coefficient = IQR/(Q3+Q1) โ€” relative spread measure
  • โ€ข Midhinge = (Q1+Q3)/2 โ€” midpoint of the quartile range, robust center
  • โ€ข Bowley Skewness = (Q3+Q1โˆ’2ร—Q2)/(Q3โˆ’Q1) โ€” skewness from quartiles only

Quartile Methods Explained

1. Inclusive (Tukey)

Split data at median, find median of each half. Include median in both halves when n is odd.

2. Exclusive

Split at median, exclude median from halves. Uses rank = (k/4)ร—n.

3. Interpolation (Excel QUARTILE.INC)

Position = 1 + (k/4)(nโˆ’1). Linear interpolation between adjacent values.

4. Nearest Rank

Q_k at position โŒˆ(k/4)ร—nโŒ‰. Simple integer indexing.

Did You Know?

๐Ÿ“ŠDifferent software uses different quartile formulas โ€” Excel, R, and Python can give slightly different Q1/Q3 values.Source: Wikipedia
๐Ÿ“Bowley's quartile skewness is robust: it uses only quartiles, not the mean, so it resists outliers.Source: Wolfram MathWorld
๐Ÿ”The midhinge is the average of Q1 and Q3 โ€” a robust measure of center.Source: NIST Handbook
๐Ÿ“ˆIQR is the basis of the box in a box plot โ€” the box spans from Q1 to Q3.Source: Tukey EDA
๐ŸงชFor very small samples (n < 4), quartile methods can produce identical results.Source: Khan Academy
๐Ÿ“‰Quartile Deviation Coefficient is dimensionless โ€” useful for comparing spread across different scales.Source: NIST Handbook

Expert Tips

Method Choice

Inclusive matches many textbooks; Interpolation matches Excel QUARTILE.INC. State your method in reports.

Midhinge

Robust alternative to mean when data has outliers. Midhinge = (Q1+Q3)/2.

Semi-IQR

Quartile Deviation = IQR/2. Used in coefficient of quartile variation and robust statistics.

Bowley Skewness

Quartile skewness โˆˆ [โˆ’1, 1]. Positive = right tail; negative = left tail.

Method Comparison Table

MethodQ1 FormulaSoftware Match
Inclusive (Tukey)Split at median, median of lower halfMany textbooks
Exclusiverank = (1/4)ร—nR type 7
Interpolationposition = 1 + (1/4)(nโˆ’1)Excel QUARTILE.INC
Nearest Rankposition = โŒˆ(1/4)ร—nโŒ‰Simple indexing

Frequently Asked Questions

Why do different methods give different results?

Quartiles are not uniquely defined for finite samples. Different methods use different interpolation rules between data points.

What is the midhinge?

Midhinge = (Q1+Q3)/2. It's a robust measure of center, resistant to outliers.

What is Bowley's quartile skewness?

Skewness = (Q3+Q1โˆ’2ร—Q2)/(Q3โˆ’Q1). Ranges from โˆ’1 to 1. Positive = right skew; negative = left skew.

When to use IQR vs standard deviation?

Use IQR for skewed data or when you want robustness to outliers. SD is better for symmetric, roughly normal data.

Which method matches Excel?

Interpolation (Excel QUARTILE.INC) matches Excel's QUARTILE.INC function.

What is the Quartile Deviation Coefficient?

IQR/(Q3+Q1). A dimensionless relative spread measure, useful for comparing across different scales.

How do I interpret Bowley skewness?

Positive = right-skewed (long right tail); Negative = left-skewed; Near 0 = symmetric.

What is Semi-IQR?

Semi-IQR (Quartile Deviation) = IQR/2. Half the spread of the middle 50% of data.

Quartiles by the Numbers

5
Quartile Points
25%
Data in Each Q
50%
Data in IQR
4
Methods

Formulas Reference

IQR = Q3 โˆ’ Q1

Semi-IQR (Quartile Deviation) = IQR / 2

Quartile Deviation Coefficient = IQR / (Q3 + Q1)

Relative spread measure (dimensionless)

Midhinge = (Q1 + Q3) / 2

Robust measure of center

Bowley Skewness = (Q3 + Q1 โˆ’ 2ร—Q2) / (Q3 โˆ’ Q1)

Skewness from quartiles only

Disclaimer: Quartile methods can yield different Q1/Q3 values. For publication, state which method you used. Results may differ from Excel, R, or Python depending on the method.

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