Calculate Upper Quartile (Q3)
Online calculator to compute the upper quartile (75th percentile) of a data series
Upper Quartile Calculator
The Upper Quartile (Q3)
The upper quartile Q3 is the 75th percentile of a data series. It separates the upper 25% from the lower 75% of sorted data.
Quartile Visualization
Quartiles divide data into four equal parts.
Q3 lies between median and maximum.
● Upper Quartile (Q3) ● Median (Q2) ● Lower Quartile (Q1)
What is the Upper Quartile (Q3)?
The upper quartile Q3 is an important measure of location in descriptive statistics:
- Definition: Value that separates 75% of data from the upper 25%
- Designation: Also called 75th percentile or third quartile
- Position: Lies between median and maximum
- Property: Robust against outliers in upper range
- Application: Box plots, interquartile range, dispersion measures
- Interpretation: 75% of values lie below Q3
Overview of All Four Quartiles
Quartiles divide a sorted data series into four equal-sized parts:
Q1
Lower Quartile
25th Percentile
25% below
Q2
Median
50th Percentile
50% below
Q3
Upper Quartile
75th Percentile
75% below
IQR
Interquartile Range
IQR = Q3 - Q1
Middle 50%
Applications of Upper Quartile
The upper quartile Q3 is used in many fields:
Data Analysis
- Box plot representation (upper box boundary)
- Interquartile range (IQR = Q3 - Q1)
- Outlier detection (values > Q3 + 1.5·IQR)
- Five-number summary (Min, Q1, Q2, Q3, Max)
Practical Applications
- Income distribution (upper quarter)
- Performance evaluation (top 25%)
- Quality control (upper tolerance limit)
- Benchmarking (above-average performance)
Calculation of Upper Quartile
Upper Quartile (Q3)
Q3 corresponds to the 75th percentile
Calculate Position
Position of Q3 in sorted list (n = number of values)
Interquartile Range
Span of the middle 50% of data
Outlier Threshold (Upper)
Values above this are considered outliers
Symbol Explanations
| \(Q_3\) | Upper Quartile |
| \(Q_1\) | Lower Quartile |
| \(P_{75}\) | 75th Percentile |
| \(n\) | Number of Values |
| \(IQR\) | Interquartile Range |
| \(k\) | Position in Sorted List |
Example Calculations for Upper Quartile
Example 1: Odd Number of Values
Calculate: Upper Quartile Q3
1. Sort Data
2, 5, 4, 8, 3, 7, 9, 3, 1, 6
Sorted:
1, 2, 3, 3, 4, 5, 6, 7, 8, 9
2. Calculate Position
Position between 8th and 9th value
3. Determine Q3
8th value = 7
9th value = 8
\[Q_3 = 7 + 0.25(8-7)\] \[= \color{blue}{7.25}\]Example 2: Five-Number Summary and Box Plot
Determine all five statistics
Five-Number Summary
| Minimum: | 12 |
| Q1 (25%): | 18 |
| Q2 (50%, Median): | 28 |
| Q3 (75%): | 40 |
| Maximum: | 50 |
Additional Statistics
| IQR: | Q3 - Q1 = 40 - 18 = 22 |
| Lower Threshold: | Q1 - 1.5·IQR = -15 |
| Upper Threshold: | Q3 + 1.5·IQR = 73 |
| All values are within thresholds → No outliers | |
Interpretation
Q3 = 40: 75% of values lie at or below 40.
IQR = 22: The middle 50% of data has a range of 22.
Q3 - Q2 = 12: The upper half of data is slightly more dispersed than the lower half (Q2 - Q1 = 10).
This indicates a slightly right-skewed distribution.
Example 3: Outlier Detection with Q3
Is 95 an outlier?
Calculate Quartiles
| Q1: | 15 |
| Q2 (Median): | 21 |
| Q3: | 28 |
| IQR: | 28 - 15 = 13 |
Outlier Test
Upper Threshold:
Q3 + 1.5 · IQR
= 28 + 1.5 · 13
= 28 + 19.5
= 47.5
95 > 47.5
→ 95 is an outlier!
Quantile Calculation Methods
There are nine different methods for calculating quartiles and percentiles. These methods differ in how they interpolate between data points.
Standard (Type 6)
Linear interpolation of expected values for order statistics.
Used by: Excel, SAS-4, SciPy-(0,0), Maple-5
R (Type 7)
Linear interpolation of modes for order statistics.
Used by: R, Excel, SciPy-(1,1), Maple-6
Maple (Type 8)
Linear interpolation of approximate medians.
Used by: Maple-7, SciPy-(1/3,1/3)
All Nine Methods in Detail:
| Method | Description | Equivalent to |
|---|---|---|
| Type 1 | Inverse of the empirical distribution function | R-1, SAS-3, Maple-1 |
| Type 2 | Like Type 1, but with averaging at discontinuities | R-2, SAS-5, Maple-2 |
| Type 3 | Nearest data count to Np | R-3, SAS-2 |
| Type 4 | Linear interpolation of empirical distribution function | R-4, SAS-1, SciPy-(0,1), Maple-3 |
| Type 5 | Piecewise linear function with nodes at step midpoints | R-5, SciPy-(.5,.5), Maple-4 |
| Type 6 | Linear interpolation of expected values for order statistics (Standard) | R-6, Excel, SAS-4, SciPy-(0,0), Maple-5 |
| Type 7 | Linear interpolation of modes for order statistics (R-standard) | R-7, Excel, SciPy-(1,1), Maple-6 |
| Type 8 | Linear interpolation of approximate medians (Maple-standard) | R-8, SciPy-(1/3,1/3), Maple-7 |
| Type 9 | Approximately unbiased for expected order statistics (normal distribution) | R-9, SciPy-(3/8,3/8), Maple-8 |
Recommendation
The standard method (Type 6) is suitable for most applications. For compatibility with R use Type 7. The choice of method typically only has significant impact on small datasets. With large datasets, all methods converge to similar results.
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