Statistical Methods for Ungrouped Data

Overview

• Focus on calculating quartiles, deciles, percentiles for ungrouped data.
• These methods help in understanding the distribution of data and identifying key positions within a dataset.

Key Concepts and Formulas

Quartiles

• Divide data set into four equal parts.
• Formula: [ Q_i = \left( \frac{i(n+1)}{4} \right) \text{th value of the observation} ]
  • Where ( i = 1, 2, 3 )

Deciles

• Divide data set into ten equal parts.
• Formula: [ D_i = \left( \frac{i(n+1)}{10} \right) \text{th value of the observation} ]
  • Where ( i = 1, 2, 3, , ..., , 9 )

Percentiles

• Divide data set into hundred equal parts.
• Formula: [ P_i = \left( \frac{i(n+1)}{100} \right) \text{th value of the observation} ]
  • Where ( i = 1, 2, 3, , ..., , 99 )

Examples

Example 1

• Data: 3, 13, 11, 11, 5, 4, 2
• Sorted Data: 2, 3, 4, 5, 11, 11, 13
• n: 7

1. Quartile-1 (Q1):

   • [ Q_1 = \left( \frac{8}{4} \right) \text{th value} = 2\text{nd value} = 3 ]

2. Decile-3 (D3):

   • [ D_3 = \left( \frac{3 \times 8}{10} \right) \text{th value} = 2.4\text{th value} ]
   • Calculation: 2nd observation + 0.4 [3rd - 2nd] = 3 + 0.4(1) = 3.4

3. Percentile-20 (P20):

   • [ P_{20} = \left( \frac{20 \times 8}{100} \right) \text{th value} = 1.6\text{th value} ]
   • Calculation: 1st observation + 0.6 [2nd - 1st] = 2 + 0.6(1) = 2.6_

Example 2

• Data: 85, 96, 76, 108, 85, 80, 100, 85, 70, 95
• Sorted Data: 70, 76, 80, 85, 85, 85, 95, 96, 100, 108
• n: 10

1. Quartile-2 (Q2):

   • [ Q_2 = \left( \frac{2 \times 11}{4} \right) \text{th value} = 5.5\text{th value} ]
   • Calculation: 5th observation + 0.5 [6th - 5th] = 85 + 0 = 85

2. Decile-6 (D6):

   • [ D_6 = \left( \frac{6 \times 11}{10} \right) \text{th value} = 6.6\text{th value} ]
   • Calculation: 6th observation + 0.6 [7th - 6th] = 85 + 6 = 91

3. Percentile-45 (P45):

   • [ P_{45} = \left( \frac{45 \times 11}{100} \right) \text{th value} = 4.95\text{th value} ]
   • Calculation: 4th observation + 0.95 [5th - 4th] = 85 + 0 = 85_

Additional Information

• Use these calculations in statistical analysis to summarize or describe a set of data.
• Ensure data is sorted in ascending order before performing calculations.

Resources

• For further details, refer to the textbook or resources provided by your instructor.