Calculate Lower Quartile (Q1)
Online calculator to calculate the lower quartile (25th percentile) of a data series
Lower Quartile Calculator
Lower Quartile (Q1)
The lower quartile Q1 is the 25th percentile of a data series. It divides the lower 25% from the upper 75% of sorted data.
Quartile Visualization
Quartiles divide data into four equal parts.
Q1 lies between minimum and median.
● Lower Quartile (Q1) ● Median (Q2) ● Upper Quartile (Q3)
What is the Lower Quartile (Q1)?
The lower quartile Q1 is an important measure of position in descriptive statistics:
- Definition: Value that separates 25% of data from the upper 75%
- Also Called: 25th percentile or first quartile
- Position: Lies between minimum and median
- Property: Robust against outliers in lower range
- Application: Box-plots, interquartile range, spread measures
- Interpretation: 25% of values lie below Q1
Overview of Four Quartiles
Quartiles divide sorted data into four equally-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 Lower Quartile
The lower quartile Q1 is used in many fields:
Data Analysis
- Box-plot representation (lower box boundary)
- Interquartile range (IQR = Q3 - Q1)
- Outlier detection (values < Q1 - 1.5·IQR)
- Five-number summary (Min, Q1, Q2, Q3, Max)
Practical Applications
- Income distribution (lower quarter)
- Performance rating (lower 25%)
- Quality control (lower tolerance limit)
- Benchmarking (below-average performance)
Calculation of Lower Quartile
Lower Quartile (Q1)
Q1 corresponds to the 25th percentile
Calculate Position
Position of Q1 in sorted list (n = count)
Interquartile Range
Span of middle 50% of data
Outlier Boundary (Lower)
Values below are considered outliers
Symbol Explanations
| \(Q_1\) | Lower quartile |
| \(Q_3\) | Upper quartile |
| \(P_{25}\) | 25th percentile |
| \(n\) | Number of values |
| \(IQR\) | Interquartile range |
| \(k\) | Position in sorted list |
Example Calculations for Lower Quartile
Example 1: Odd Number of Values
Calculate: Lower Quartile Q1
1. Sort
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 2nd and 3rd value
3. Determine Q1
2nd value = 2
3rd value = 3
\[Q_1 = 2 + 0.75(3-2)\] \[= \color{blue}{2.75}\]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 Boundary: | Q1 - 1.5·IQR = -15 |
| Upper Boundary: | Q3 + 1.5·IQR = 73 |
| All values lie within boundaries → No outliers | |
Interpretation
Q1 = 18: 25% of values lie at or below 18.
IQR = 22: The middle 50% of data have a range of 22.
Q2 - Q1 = 10: The lower half of data is slightly less dispersed than upper half (Q3 - Q2 = 12).
This indicates a slightly right-skewed distribution.
Example 3: Outlier Detection with Q1
Is 5 an outlier?
Calculate Quartiles
| Q1: | 12 |
| Q2 (Median): | 19 |
| Q3: | 26.5 |
| IQR: | 26.5 - 12 = 14.5 |
Outlier Test
Lower Boundary:
Q1 - 1.5 · IQR
= 12 - 1.5 · 14.5
= 12 - 21.75
= -9.75
5 > -9.75
→ 5 is NOT 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 expectations 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 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 point 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 knots at step midpoints | R-5, SciPy-(.5,.5), Maple-4 |
| Type 6 | Linear interpolation of expectations for order statistics (Standard) | R-6, Excel, SAS-4, SciPy-(0,0), Maple-5 |
| Type 7 | Linear interpolation of modes for order statistics (R default) | R-7, Excel, SciPy-(1,1), Maple-6 |
| Type 8 | Linear interpolation of approximate medians (Maple default) | 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. Method choice typically has significant impact only with small datasets. With large datasets, all methods converge to similar results.
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