Calculate Percentiles
Online calculator for percentiles, quartiles, and related statistical measures: minimum, maximum, mean, count, and interquartile range
Percentile Calculator
Percentile Calculation
Calculate the p-th percentile and related statistical measures: minimum, maximum, mean, count, and interquartile range. Enter the percentile value as 25, 50, 75, or 0.25, 0.50, 0.75.
Percentile Distribution
Visualization of percentile distribution
Percentiles divide data into equal groups
● Min (0%) ● Q1 (25%) ● Median (50%) ● Q3 (75%) ● Max (100%)
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Understanding Percentiles - Fundamentals
A percentile is a statistical measure indicating the value below which a given percentage of observations in a group fall.
- Percentile Definition: The p-th percentile is the value below which p% of the data lies
- Quartiles: Special percentiles that divide data into four equal parts (Q1=25%, Q2=50%, Q3=75%)
- Deciles: Percentiles at 10%, 20%, ..., 90% dividing data into ten equal parts
- Interpolation: When the exact position falls between two values, interpolation (Type-7/R-method) is used
- Position (h): The position h = (n-1)·p + 1 indicates where the percentile falls in sorted data
Calculating Percentiles
Step-by-step explanation for calculating percentiles using Type-7 (R) interpolation:
Step 1: Sort Data
- Arrange all data values in ascending order
- Example: [7, 3, 9, 1, 5] → [1, 3, 5, 7, 9]
- Index starts at 1 (1-based indexing)
Step 2: Calculate Position
- Compute h = (n - 1) · p + 1
- n = number of values, p = percentile (0-1)
- Example: For 50th percentile (p=0.5) and n=10: h = 9 · 0.5 + 1 = 5.5
Step 3: Interpolate
- j = floor(h), g = h - j (fractional part)
- Result = (1-g) · x[j-1] + g · x[j]
- Linear interpolation between two values
Step 4: Related Statistics
- Calculate Q1 (25th percentile)
- Calculate Q3 (75th percentile)
- Compute IQR = Q3 - Q1
Applications and Significance
Percentiles are widely used in practical applications:
Medical & Health
- Growth charts: height/weight percentiles
- Test score evaluation
- Benchmarking performance
- Outlier detection in patient data
Finance & Business
- Risk assessment (Value at Risk)
- Income distribution analysis
- Market data analysis
- Performance evaluation
Education
- Standardized test scores
- Grade distribution
- Class ranking
- Performance comparison
Manufacturing & Quality
- Quality control limits
- Process capability analysis
- Tolerance specifications
- Six Sigma methodologies
Formulas and Definitions
Percentile (Type-7/R)
h = position, p = percentile (0-1), n = data count
Quartiles
Quartiles divide data into four equal parts
Interquartile Range
Contains the middle 50% of all data
Arithmetic Mean
Average of all values
Symbol Explanations
| \(p\) | Percentile (0-1) |
| \(n\) | Number of data points |
| \(h\) | Position in sorted data |
| \(g\) | Fractional part (0-1) |
| \(x_i\) | i-th data point |
| \(IQR\) | Interquartile range |
Example Calculations
Example 1: Test Scores (10 Students)
Calculate: 25th, 50th, 75th percentiles and IQR
1. Data Preparation
Data already sorted in ascending order
2. Calculate Q1 (25th)
Between x[2]=58 and x[3]=63
3. Calculate Q2 (50th/Median)
Between x[4]=71 and x[5]=75
4. Calculate Q3 & IQR
Interquartile range contains middle 50%
Example 2: Employee Salaries (in $1000s)
Calculate: 90th percentile to find top earners
Results for Common Percentiles
10th Percentile
39.1
25th Percentile (Q1)
48.25
75th Percentile (Q3)
82.75
90th Percentile
108.1
Example 3: Interpretation
If a test score is at the 75th percentile: This means 75% of all test takers scored equal to or below this score, and only 25% scored higher.
If height is at the 50th percentile (median): Half of the population is shorter and half is taller than this height.
Mathematical Foundations
Percentiles are fundamental concepts in statistics for analyzing and comparing data distributions. They provide context-independent measures for understanding relative position within a dataset.
Types of Percentile Methods
There are 9 different methods to calculate percentiles, with Type-7 (R/default) being the most common:
- Type-1 to Type-3: Inverse of empirical distribution (discontinuous)
- Type-4 to Type-9: Linear interpolation methods (continuous)
- Type-7 (R/default): Standard in R, SAS, and most statistical software
- Type-5 (Maple): Alternative linear interpolation
Key Properties of Percentiles
- Monotonicity: If p₁ < p₂, then P_{p₁} ≤ P_{p₂}
- Bounds: min(x) = P₀ and max(x) = P₁
- Median: P_{0.5} is the 50th percentile (median)
- Robustness: Percentiles are more robust to outliers than mean
- Scalability: Percentiles scale with data (multiply by k, percentiles multiply by k)
Relationship to Other Statistics
Box-and-Whisker Plot
Uses five-number summary: Min, Q1 (25%), Q2 (50%), Q3 (75%), Max. Visually represents percentile-based distribution.
Outlier Detection
Values < Q1 - 1.5·IQR or > Q3 + 1.5·IQR are considered outliers. Percentile-based detection is robust and distribution-free.
Common Percentile Applications
Deciles: 10%, 20%, ..., 90% (10 equal groups)
Percentiles: 1%, 2%, ..., 99% (100 equal groups)
Finance (VaR): 1%, 5% percentiles for risk
Testing: Rank percentiles for scoring
Summary
Percentiles provide a powerful, distribution-free way to understand and compare data. They are especially useful for non-normal distributions and have intuitive interpretations. Combined with other measures, percentiles form the foundation of modern exploratory data analysis.
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