To find the first quartile (Q1) and the third quartile (Q3) of a data set, follow these steps:

1. Order the data from smallest to largest.
2. Find the median (Q2) of the entire data set. This median divides the data into two halves.
3. Find Q1 (the first quartile):
   • Take the lower half of the data (all values below the median).
   • Find the median of this lower half. This median is Q1, the value below which 25% of the data lies.
4. Find Q3 (the third quartile):
   • Take the upper half of the data (all values above the median).
   • Find the median of this upper half. This median is Q3, the value below which 75% of the data lies.

Note:

• If the total number of data points is odd, exclude the median when splitting the data into halves for finding Q1 and Q3.
• If even, split the data evenly.

Alternatively, you can use the position formula to find the quartiles:

• Q1=(n+1)4Q1=\frac{(n+1)}{4}Q1=4(n+1)​th term
• Q3=3(n+1)4Q3=\frac{3(n+1)}{4}Q3=43(n+1)​th term
  where nnn is the number of data points. This gives the position of Q1 and Q3 in the ordered data

Example

Given data: 15, 16, 17, 17, 17, 18, 19, 19, 19, 20, 20, 20, 21, 23, 25

• Median (Q2) = 19 (middle value)
• Lower half: 15, 16, 17, 17, 17, 18, 19 → median is Q1 = 17
• Upper half: 19, 20, 20, 20, 21, 23, 25 → median is Q3 = 20

Thus, Q1 = 17 and Q3 = 20

. This method helps you understand the spread and distribution of data by dividing it into four equal parts.