Computing Quartiles, Deciles, and Percentiles for Ungrouped Data

Key Concepts

• Quartiles divide a distribution into four equal parts.

  • First Quartile (Q1): 25%
  • Second Quartile (Q2): 50%
  • Third Quartile (Q3): 75%

• Deciles divide a distribution into ten equal parts.

  • First Decile (D1), Second Decile (D2), ..., Tenth Decile (D10)

• Percentiles divide a distribution into 100 equal parts.

  • First Percentile (P1), ..., Hundredth Percentile (P100)

Formulas

• Quartiles:

  • Formula: ( Q_k = k \times \frac{n + 1}{4} )

• Deciles:

  • Formula: ( D_k = k \times \frac{n + 1}{10} )

• Percentiles:

  • Formula: ( P_k = k \times \frac{n + 1}{100} )

Steps for Calculation

1. Arrange the data in ascending order.
2. For quartiles, deciles, or percentiles, use the respective formula.
3. Determine whether to round up or down based on the position.
4. Use the rounded position to extract the value from the dataset.

Examples

Quartiles Example

• Data Set: 8, 12, 15, 2, 7, 25, 9, 18, 16, 6, 10
• Step 1: Arrange data: 2, 6, 7, 8, 9, 10, 12, 15, 16, 18, 25
• First Quartile (Q1): ( 0.25 \times (13) = 3.5 ) -> Round up to 4th data point = 6
• Third Quartile (Q3): ( 0.75 \times (13) = 10.5 ) -> Round down to 10th data point = 18

Deciles Example

• Data Set: 4, 8, 9, 12, 7, 15, 14, 6, 10, 11
• Step 1: Arrange data: 4, 6, 7, 8, 9, 10, 11, 12, 14, 15
• Third Decile (D3): ( 3 \times \frac{11}{10} = 3.3 ) -> Round up to 4th data point = 8
• Eighth Decile (D8): ( 8 \times \frac{11}{10} = 8.8 ) -> Round down to 8th data point = 12

Percentiles Example

• Data Set: 20, 18, 16, 10, 12, 15, 13, 9, 11, 16, 15, 16, 18, 20
• Step 1: Arrange data: 9, 10, 11, 12, 13, 15, 15, 16, 16, 16, 18, 18, 20, 20
• 15th Percentile (P15): ( 15 \times \frac{15}{100} = 2.25 ) -> Round up to 3rd data point = 11
• 80th Percentile (P80): ( 80 \times \frac{15}{100} = 12 ) -> 12th data point = 18

Important Notes

• Always remember to arrange data in ascending order before calculations.
• The choice of rounding up or down depends on the position calculated.
• Quartiles, deciles, and percentiles give a way to understand data distribution.
• Rounding rules:
  • Round up for quartiles and deciles if not a whole number.
  • Round down for percentiles unless otherwise stated.