🎓 Statistics & Data • Descriptive Statistics
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Quartile Calculator (Q₁, Q₂, Q₃)
Divides datasets into four equal parts, calculating Q1 (25th percentile), Q2 (median), and Q3 (75th percentile).
Descriptive Statistics Engine
Dataset Analysis & Five-Number Summary
Input comma or space separated numbers. Computes central tendency, spread, shape moments, Tukey fences, and interactive plots.
Raw Data Values10 observations parsed
Presets:
Mean (x̄)24.7Sum / N
Median (Q₂)23.550th percentile
Sample Std Dev (s)8.7819s² = 77.1222
IQR (Q₃ - Q₁)12Middle 50%
Interactive Data Visualization
IQR Box (Q₁ to Q₃) Median (23.5)✕ Mean (24.7) Outlier (Outside 1.5×IQR)
Comprehensive Metrics Table
Count (N)10
Sum (Σx)247
Mode22
Range28 (12 - 40)
Sample Variance (s²)77.1222
Pop Variance (σ²)69.41
Standard Error (SE)2.7771
Coef of Variation (CV)35.55%
10% Trimmed Mean24.375
First Quartile (Q₁)18
Third Quartile (Q₃)30
Tukey Fences [LF, UF][0, 48]
Skewness0.319 (Symmetric)
Excess Kurtosis-0.525 (Platykurtic)
Detected OutliersNone detected
Single Value Evaluator (Z-Score & Percentile)Standardized deviation from sample mean: z = (x - x̄) / s
Standardized Z-Score+0.034 σ0.034 standard deviations above mean
Percentile Rank55th PercentileBetter than or equal to 55% of sample
Step-by-Step Statistical Derivations
1. Sorted Ascending Dataset:
12, 15, 18, 22, 22, 25, 28, 30, 35, 40
2. Central Tendency:
- Mean: x̄ = Σx / n = 247 / 10 = 24.7
- Median: Ordered middle position average of items 5 & 6 = 23.5
- Mode: 22 (Frequency = 2)
3. Sum of Squared Deviations & Variance:
- SS = Σ(x_i - x̄)² = 694.1
- Sample Variance: s² = SS / (n - 1) = 694.1 / 9 = 77.1222
- Sample Standard Deviation: s = √s² = √77.1222 = 8.7819
4. Five-Number Summary & Outlier Fences:
- Min = 12, Q₁ = 18, Median = 23.5, Q₃ = 30, Max = 40
- IQR = Q₃ - Q₁ = 30 - 18 = 12
- Lower Fence = Q₁ - 1.5×IQR = 18 - 18 = 0
- Upper Fence = Q₃ + 1.5×IQR = 30 + 18 = 48
What are Quartiles?
Quartiles are values that divide sorted data into four quarters: Q1 separates the lowest 25%, Q2 is the median (50%), and Q3 separates the highest 25%.
Formula & Step-by-Step Calculation
Q₁ = Median(Lower Half), Q₂ = Median(Total), Q₃ = Median(Upper Half)
Five-number summary boundary landmarks.
Worked Step-by-Step Examples
Example 1
Find quartiles of: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49
Solution: Q₁ = 15, Q₂ = 40, Q₃ = 43
• Sorted 11 items: Median Q2 = 40; Lower half median Q1 = 15; Upper half median Q3 = 43
Common Real-World & Academic Use Cases
- ✓ Constructing Box-and-Whisker plots
- ✓ Income inequality quartile breakdowns
- ✓ Academic performance band grading
How to Use the Quartile Calculator (Q₁, Q₂, Q₃)
1
Paste Values
Enter dataset.
2
Read Quartiles
Inspect Q1, Q2, Q3, and 5-number summary.
Frequently Asked Questions
Q: What is the relationship between quartiles and box plots?
The box in a box plot spans from Q1 to Q3, with a line at Q2 (the median).