Quartiles, Percentiles, and Median

The Big Idea: Dividing Data

All three concepts are about locating positions within a dataset — they help you understand how values are distributed and where any particular value stands relative to the rest. The median is a midpoint of the distribution.

I. Median

The median is the middle value of an ordered dataset. It splits data into two equal halves.

How to find it:

Sort your data from smallest to largest
If n is odd → the median is the middle value
If n is even → the median is the average of the two middle values

Example:

Dataset: 3, 7, 8, 12, 15 → Median = 8 (middle value)
Dataset: 3, 7, 8, 12 → Median = (7 + 8) / 2 = 7.5

The median is also known as the 50th percentile, because 50% of values fall below it.

II. Quartiles

Quartiles divide ordered data into four equal parts (quarters).

Quartile	Symbol	Also Called	What it means
First Quartile	Q1	Lower Quartile	25% of data falls below this
Second Quartile	Q2	Median	50% of data falls below this
Third Quartile	Q3	Upper Quartile	75% of data falls below this

Example — Dataset: 2, 4, 6, 8, 10, 12, 14, 16

Q2 (Median) = (8 + 10) / 2 = 9
Q1 = median of the lower half {2, 4, 6, 8} = (4 + 6) / 2 = 5
Q3 = median of the upper half {10, 12, 14, 16} = (12 + 14) / 2 = 13

*Note: We are using the most widely taught method of finding quartiles which is to exclude the median from the halves (exclusive method). There are other conventions where the median is included in both halves (inclusive methods).

How to find quartiles (Even number of data values)
1. Sort your data from smallest to largest
2. Find the median (second quartile) as the average of the two middle data values.

3. Because the median is excluded when finding the quartiles, you will have an even number of data values in both halves of the data set.

4. Find the median of the lower half (first quartile)

5. Find the median of the upper half (third quartile)

Example data (10 values):
2, 3, 5, 6, 7, 9, 11, 13, 14, 18

1. The data is already sorted from smallest to largest
2. Median Q2: There are 10 values. The middle two are the 7 and 9. Q2 = (7 + 9)/2 =8.

3. Excluding the median, you have an even number of data values in both halves of the data set.

2, 3, 5, 6, 7 | 9, 11, 13, 14, 18

4. The 5 is the middle value of the lower half so Q1 = 5

5. The 13 is the middle value of the upper half so Q3 = 13

Example data (9 values):

2, 3, 5, 6, 7, 9, 11, 13, 14

III. Percentiles

Percentiles divide data into 100 equal parts. The pth percentile is the value below which p% of the data falls.

Formula (finding the position):

L = \frac{p}{100} \times n

Where p = percentile and n = number of data points. If L is a whole number, average the Lth and (L+1)th values. If L is a decimal, round up.

Example — Find the 30th percentile of: 5, 10, 15, 20, 25, 30, 35, 40 (n = 8)

L = \frac{30}{100} \times 8 = 2.4 → \text{round up to position 3}

The value at position 3 is 15, so the 30th percentile = 15.

How They All Connect

0%          25%         50%         75%        100%
|___________|___________|___________|___________|
Min         Q1        Median        Q3         Max
           (25th      (50th        (75th
         percentile) percentile)  percentile)