Measures of the Location of the Data

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.

I. Median

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

How to find it:

1. Sort your data from smallest to largest
2. If n is odd → the median is the middle value
3. 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

Interquartile Range (IQR) = Q3 − Q1 = 13 − 5 = 8 The IQR measures the spread of the middle 50% of data, and is useful for detecting outliers.

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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.

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How They All Connect

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

• Median = Q2 = 50th percentile
• Q1 = 25th percentile
• Q3 = 75th percentile
• Every quartile is a percentile, but not every percentile is a quartile

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