Quartiles Calculator
Use this unified calculator to find quartiles (Q1, Q2, Q3) for both ungrouped (raw) data and grouped (frequency distribution) data.
Quick Start
Choose your data type, enter your values, and click Calculate:
| Quartiles Calculator | |
|---|---|
| Data Type | Ungrouped (Raw Data) Grouped (Frequency Distribution) |
| Enter the X Values (Separated by comma,) | |
| Type of Frequency Distribution | DiscreteContinuous |
| Enter the Classes for X (Separated by comma,) | |
| Enter the frequencies (f) (Separated by comma,) | |
| Results | |
| Number of Observations (N): | |
| Ascending order of X values : | |
| First Quartile : ($Q_1$) | |
| Second Quartile : ($Q_2$) | |
| Third Quartile : ($Q_3$) | |
How to Use This Calculator
For Ungrouped (Raw) Data
Step 1: Select “Ungrouped (Raw Data)” as your data type
Step 2: Enter your data values separated by commas (e.g., 2, 4, 6, 8, 10, 12, 14)
Step 3: Click “Calculate”
Results will show:
- Number of observations (N)
- Sorted values in ascending order
- First Quartile (Q₁) - 25th percentile
- Second Quartile (Q₂) - 50th percentile (median)
- Third Quartile (Q₃) - 75th percentile
For Grouped (Frequency Distribution) Data
Step 1: Select “Grouped (Frequency Distribution)” as your data type
Step 2: Choose frequency distribution type:
- Discrete: For individual values (e.g., 2, 3, 4, 5, 6)
- Continuous: For class intervals (e.g., 10-20, 20-30, 30-40)
Step 3: Enter class values or intervals separated by commas
Step 4: Enter the corresponding frequencies separated by commas
Step 5: Click “Calculate”
Results will show:
- Number of observations (N)
- First Quartile (Q₁)
- Second Quartile (Q₂)
- Third Quartile (Q₃)
Understanding Quartiles
Quartiles are values that divide data into four equal parts. Each part contains 25% of the data.
| Quartile | Name | Position | Meaning |
|---|---|---|---|
| Q₁ | First Quartile | 25th percentile | 25% of data below this value |
| Q₂ | Second Quartile | 50th percentile | 50% of data below (Median) |
| Q₃ | Third Quartile | 75th percentile | 75% of data below |
Formulas & Theory
For Ungrouped Data
The i-th quartile for ungrouped data is the value at position:
$$Q_i = \text{Value of } \left(\frac{i(N+1)}{4}\right)^{\text{th}} \text{ observation}, \quad i=1,2,3$$
Where:
- $N$ = total number of observations
- Data must be arranged in ascending order
Example: For N=7 observations:
- Q₁ position = (1×8)/4 = 2nd observation
- Q₂ position = (2×8)/4 = 4th observation
- Q₃ position = (3×8)/4 = 6th observation
For Grouped Data - Discrete
For discrete frequency distribution, the i-th quartile is:
$$Q_i = \left(\frac{i(N)}{4}\right)^{\text{th}} \text{ value}, \quad i=1,2,3$$
Where:
- $N$ = total number of observations
- Find the cumulative frequency ≥ $\frac{iN}{4}$
- The corresponding value is the quartile
For Grouped Data - Continuous
For continuous frequency distribution, the i-th quartile is:
$$Q_i = l + \left(\frac{\frac{iN}{4} - F_<}{f}\right) \times h, \quad i=1,2,3$$
Where:
- $l$ = lower boundary of the quartile class
- $N$ = total number of observations
- $F_<$ = cumulative frequency before the quartile class
- $f$ = frequency of the quartile class
- $h$ = class width (upper limit - lower limit)
Related Measures: IQR and Quartile Deviation
Interquartile Range (IQR)
The IQR measures the spread of the middle 50% of data:
$$\text{IQR} = Q_3 - Q_1$$
Interpretation:
- Small IQR = data clustered around median
- Large IQR = data spread out
- Used to identify outliers: values beyond Q₁ - 1.5×IQR or Q₃ + 1.5×IQR
Quartile Deviation (QD)
Quartile Deviation is half the IQR:
$$\text{QD} = \frac{Q_3 - Q_1}{2}$$
Used for:
- Comparing spread between datasets
- Non-parametric measure (robust to outliers)
- Symmetric distributions
Worked Examples
Example 1: Ungrouped Data - Test Scores
Data: 8 students’ test scores: 45, 52, 68, 75, 82, 88, 91, 95
Solution:
Step 1: Data already sorted in ascending order 45, 52, 68, 75, 82, 88, 91, 95
Step 2: Find Q₁ position $$Q_1 = \left(\frac{1(8+1)}{4}\right)^{\text{th}} = (2.25)^{\text{th}} \text{ observation}$$
Position between 2nd (52) and 3rd (68): $$Q_1 = 52 + 0.25(68-52) = 52 + 4 = 56$$
Step 3: Find Q₂ position $$Q_2 = \left(\frac{2(8+1)}{4}\right)^{\text{th}} = (4.5)^{\text{th}} \text{ observation}$$
Position between 4th (75) and 5th (82): $$Q_2 = 75 + 0.5(82-75) = 75 + 3.5 = 78.5$$
Step 4: Find Q₃ position $$Q_3 = \left(\frac{3(8+1)}{4}\right)^{\text{th}} = (6.75)^{\text{th}} \text{ observation}$$
Position between 6th (88) and 7th (91): $$Q_3 = 88 + 0.75(91-88) = 88 + 2.25 = 90.25$$
Results:
- Q₁ = 56 (25% of students scored below 56)
- Q₂ = 78.5 (50% of students scored below 78.5 - Median)
- Q₃ = 90.25 (75% of students scored below 90.25)
- IQR = 90.25 - 56 = 34.25
Interpretation: The middle 50% of scores range from 56 to 90.25, a spread of 34.25 points.
Example 2: Grouped Data (Discrete) - Student Absences
Data: Absences of 35 students
| Days Absent ($x$) | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| Students ($f$) | 1 | 15 | 10 | 5 | 4 |
Solution:
Step 1: Create cumulative frequency table
| $x_i$ | $f_i$ | Cumulative Freq |
|---|---|---|
| 2 | 1 | 1 |
| 3 | 15 | 16 |
| 4 | 10 | 26 |
| 5 | 5 | 31 |
| 6 | 4 | 35 |
Step 2: Find Q₁ $$Q_1 = \left(\frac{1(35)}{4}\right)^{\text{th}} = (8.75)^{\text{th}} \text{ value}$$
Cumulative frequency ≥ 8.75 is 16 (class value = 3) $$Q_1 = 3 \text{ days}$$
Step 3: Find Q₂ $$Q_2 = \left(\frac{2(35)}{4}\right)^{\text{th}} = (17.5)^{\text{th}} \text{ value}$$
Cumulative frequency ≥ 17.5 is 26 (class value = 4) $$Q_2 = 4 \text{ days}$$
Step 4: Find Q₃ $$Q_3 = \left(\frac{3(35)}{4}\right)^{\text{th}} = (26.25)^{\text{th}} \text{ value}$$
Cumulative frequency ≥ 26.25 is 31 (class value = 5) $$Q_3 = 5 \text{ days}$$
Results:
- Q₁ = 3 days (25% had ≤ 3 days absent)
- Q₂ = 4 days (50% had ≤ 4 days absent)
- Q₃ = 5 days (75% had ≤ 5 days absent)
- IQR = 5 - 3 = 2 days
Example 3: Grouped Data (Continuous) - Test Scores
Data: Score distribution for 60 students
| Score Range | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |
|---|---|---|---|---|---|
| Students | 8 | 12 | 20 | 15 | 5 |
Solution:
Step 1: Create cumulative frequency table
| Class | Midpoint | $f_i$ | Cumulative Freq |
|---|---|---|---|
| 50-60 | 55 | 8 | 8 |
| 60-70 | 65 | 12 | 20 |
| 70-80 | 75 | 20 | 40 |
| 80-90 | 85 | 15 | 55 |
| 90-100 | 95 | 5 | 60 |
Step 2: Find Q₁ $$Q_1 = \left(\frac{1(60)}{4}\right)^{\text{th}} = (15)^{\text{th}} \text{ value}$$
Q₁ is in class 60-70 (CF: 20 ≥ 15) $$Q_1 = 60 + \left(\frac{15-8}{12}\right) \times 10 = 60 + 5.83 = 65.83$$
Step 3: Find Q₂ $$Q_2 = \left(\frac{2(60)}{4}\right)^{\text{th}} = (30)^{\text{th}} \text{ value}$$
Q₂ is in class 70-80 (CF: 40 ≥ 30) $$Q_2 = 70 + \left(\frac{30-20}{20}\right) \times 10 = 70 + 5 = 75$$
Step 4: Find Q₃ $$Q_3 = \left(\frac{3(60)}{4}\right)^{\text{th}} = (45)^{\text{th}} \text{ value}$$
Q₃ is in class 80-90 (CF: 55 ≥ 45) $$Q_3 = 80 + \left(\frac{45-40}{15}\right) \times 10 = 80 + 3.33 = 83.33$$
Results:
- Q₁ = 65.83 (25% scored below ~66)
- Q₂ = 75 (50% scored below 75 - Median)
- Q₃ = 83.33 (75% scored below ~83)
- IQR = 83.33 - 65.83 = 17.5
Interpretation: The middle 50% of students scored between 65.83 and 83.33, with a median of 75.
Key Differences: Ungrouped vs. Grouped
| Aspect | Ungrouped Data | Grouped Data |
|---|---|---|
| Data Format | Individual raw values | Classes with frequencies |
| Calculation | Direct positioning method | Cumulative frequency method |
| Accuracy | Exact quartile positions | Approximate (uses class intervals) |
| When to Use | Small datasets | Large datasets, already grouped |
| Formula | Position-based | Cumulative frequency-based |
| Discrete vs Continuous | Not applicable | Both available |
Interpreting Quartile Results
What Q₁ Tells You
- Lower quartile - represents 25th percentile
- 25% of data values fall below Q₁
- Useful for identifying lower outliers
What Q₂ Tells You
- Median - represents 50th percentile
- 50% of data below, 50% above
- Central measure of location
What Q₃ Tells You
- Upper quartile - represents 75th percentile
- 75% of data values fall below Q₃
- Useful for identifying upper outliers
Outlier Detection
Using quartiles to identify outliers:
$$\text{Lower Bound} = Q_1 - 1.5 \times \text{IQR}$$ $$\text{Upper Bound} = Q_3 + 1.5 \times \text{IQR}$$
Values outside these bounds are considered outliers.
When to Use Ungrouped vs. Grouped
Use Ungrouped Data Calculator When:
- You have individual data points
- Dataset is relatively small (< 100 values)
- Need exact quartile positions
- All original values available
- Examples: Individual test scores, small survey responses
Use Grouped Data Calculator When:
- Data organized into classes/intervals
- Large dataset already grouped
- Only frequency distribution available
- Want summary statistics
- Examples: Income distribution, test score ranges across years
Related Calculators & Concepts
Related Dispersion Measures:
- Percentiles - Extended quartiles (100 divisions)
- Deciles - Quartiles divided further (10 divisions)
- Quartile Deviation - Half the IQR
- Five-Number Summary - Min, Q₁, Q₂, Q₃, Max
Related Concepts:
- Box Plot Visualization - Visual representation of quartiles
- Interquartile Range - Spread measurement
Learn More: