How to Find Q1 and Q3 (by Data Type)

A) Ungrouped (Raw) Data

1. Arrange the data in ascending order.
2. Let n be the number of observations.

3. Locate quartile positions using the common inclusive method:

   • Position of Q₁ = (n + 1) / 4^th item
   • Position of Q₃ = 3(n + 1) / 4^th item

   Note: Some textbooks use n/4 and 3n/4 (exclusive method). State your convention and be consistent.

4. If the position is not an integer, interpolate between neighboring values.

B) Discrete Frequency

1. List distinct values x with their frequencies f.
2. Compute cumulative frequencies (cf) and total N = Σf.
3. Find the item positions:
   • Q₁ position = (N + 1) / 4
   • Q₃ position = 3(N + 1) / 4
4. Identify the values whose cumulative frequencies first reach or exceed these positions.

C) Continuous Grouped Data

Use linear interpolation within the quartile class:

Quartile class is the class whose cumulative frequency just exceeds the target: 
For Q₁: target = N/4 
For Q₃: target = 3N/4

Interpolation formula (applies to both Q₁ and Q₃):

Q_k = L + [(T − cf_prev) / f] × h

• L = lower class boundary of the quartile class
• T = target (N/4 for Q₁, 3N/4 for Q₃)
• cf_prev = cumulative frequency before the quartile class
• f = frequency of the quartile class
• h = class width