Describe in detail what you need	72, 85, 91, 77, 65, 88. Find the 25th percentile, 50th percemntile and 75th percentile.
How many pages	1

Percentile Calculation for the Given Data Set

To find the 25th, 50th (median), and 75th percentiles for the dataset provided, we will follow a systematic approach. The dataset provided is as follows:

72, 85, 91, 77, 65, 88

Step 1: Order the Data

We first need to sort the dataset in ascending order:

65, 72, 77, 85, 88, 91

Step 2: Calculate the Percentiles

Percentiles are values in a dataset that divide it into segments based on percentages.

• The 25th percentile (P25) is the value below which 25% of the data falls.
• The 50th percentile (P50), also known as the median, is the value below which 50% of the data falls.
• The 75th percentile (P75) is the value below which 75% of the data falls.

25th Percentile (P25)

To calculate the 25th percentile, we use the formula:

[ P_k = \left( \frac{k}{100} \times (N + 1) \right) ]

Where:

• ( k ) is the desired percentile (25 in this case)
• ( N ) is the number of data points

In our case:

• ( N = 6 )
• Thus, ( P_{25} = \frac{25}{100} \times (6 + 1) = 0.25 \times 7 = 1.75 )

Since 1.75 is not an integer, we need to interpolate between the 1st and 2nd data points.

1st data point (D1) = 65
2nd data point (D2) = 72

To find the value: [ P_{25} = D1 + 0.75 \times (D2 - D1) ]

Calculating: [ P_{25} = 65 + 0.75 \times (72 - 65) = 65 + 0.75 \times 7 = 65 + 5.25 = 70.25 ]

50th Percentile (P50)

For the 50th percentile, we repeat the same steps:

Calculating: [ P_{50} = \frac{50}{100} \times (6 + 1) = 0.5 \times 7 = 3.5 ]

Again, since 3.5 is not an integer, we interpolate between the 3rd and 4th data points.

3rd data point (D3) = 77
4th data point (D4) = 85

Now, we calculate: [ P_{50} = D3 + 0.5 \times (D4 - D3) ]

Calculating: [ P_{50} = 77 + 0.5 \times (85 - 77) = 77 + 0.5 \times 8 = 77 + 4 = 81 ]

75th Percentile (P75)

Now, for the 75th percentile:

Calculating: [ P_{75} = \frac{75}{100} \times (6 + 1) = 0.75 \times 7 = 5.25 ]

Again, since 5.25 is not an integer, we interpolate between the 5th and 6th data points.

5th data point (D5) = 88
6th data point (D6) = 91

Calculating: [ P_{75} = D5 + 0.25 \times (D6 - D5) ]

Calculating: [ P_{75} = 88 + 0.25 \times (91 - 88) = 88 + 0.75 = 88.75 ]

Summary of Percentiles

• 25th Percentile (P25): 70.25
• 50th Percentile (P50): 81
• 75th Percentile (P75): 88.75

These calculations provide a clear understanding of how the data is distributed and can be useful for further statistical analysis.