First Quartile Range Calculator

The First Quartile Range Calculator helps you measure the spread of the lowest 25% of a dataset. It specifically calculates the distance between the smallest value and the first quartile (Q1) in a sorted list of numbers. This range gives insights into the consistency, concentration, or spread of data at the lower end.

This tool belongs to the Statistical Data Analysis calculator category. It simplifies complex manual computations and supports quick decision-making in areas like academic grading, quality control, research, and business data analysis.

Formula

Q1 = Value at position (n + 1) × 1/4
First Quartile Range = Q1 − Minimum Value

Where:

• Q1 (First Quartile) is the value below which 25% of the data fall
• Minimum Value is the smallest number in the dataset
• n is the number of data points

If (n + 1) × 1/4 is not a whole number, you must interpolate between the closest ranks.

Step-by-step:

1. Sort the dataset in ascending order
2. Calculate position P = (n + 1) × 1/4
3. If P is a whole number: use value at position P
4. If P is decimal:
   Q1 = Value at floor(P) + decimal × (Value at ceiling(P) − Value at floor(P))
5. Subtract the minimum value from Q1 to get the range

Reference Table

Number of Data Points	Q1 Position	Example Min Value	Example Q1 Value	Q1 Range
5	1.5	3	5.5	2.5
7	2	4	6	2
10	2.75	2	5.75	3.75
12	3.25	1	6.25	5.25
15	4	3	7	4

This table offers quick estimates to help users visualize what the range looks like for datasets of varying sizes.

Example

Dataset: 2, 4, 5, 7, 9, 11, 12, 13

Step 1: n = 8
Step 2: Calculate position for Q1: (8 + 1) × 1/4 = 2.25

Step 3: Value at position 2 = 4, position 3 = 5
Q1 = 4 + 0.25 × (5 − 4) = 4.25

Minimum Value = 2
First Quartile Range = 4.25 − 2 = 2.25

So, the first quartile range is 2.25

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