| Describe in detail what you need | Given 72, 85, 91, 77, 65, 88, find the 25th, 50th and 75th percentile. |
| How many pages | 1 |
In statistics, percentiles are used to understand the distribution of a dataset by dividing it into 100 equal parts. In this analysis, we will calculate the 25th (Q1), 50th (Q2 or median), and 75th (Q3) percentiles for the dataset consisting of the following numbers: 72, 85, 91, 77, 65, 88.
To calculate the percentiles, we first need to sort the data in ascending order.
The formula to calculate the position of the 25th percentile is given by:
[ P = \frac{n + 1}{4} ]
Where:
In our case, ( n = 6 ):
[ P_{25} = \frac{6 + 1}{4} = \frac{7}{4} = 1.75 ]
Since 1.75 is not an integer, we take the value at the 1st and 2nd position in our ordered data:
The 25th percentile is calculated as:
[ Q1 = \text{Value at 1st position} + 0.75 \times (\text{Value at 2nd position} - \text{Value at 1st position}) ]
[ Q1 = 65 + 0.75 \times (72 - 65) ] [ Q1 = 65 + 0.75 \times 7 = 65 + 5.25 = 70.25 ]
The 50th percentile, or median, can be found using the formula:
[ P_{50} = \frac{n + 1}{2} ]
For our dataset:
[ P_{50} = \frac{6 + 1}{2} = \frac{7}{2} = 3.5 ]
This means the median is the average of the 3rd and 4th values:
Thus, the calculation is:
[ Q2 = \frac{\text{Value at 3rd position} + \text{Value at 4th position}}{2} ]
[ Q2 = \frac{77 + 85}{2} = \frac{162}{2} = 81 ]
The formula for finding the 75th percentile is:
[ P_{75} = \frac{3(n + 1)}{4} ]
For our dataset:
[ P_{75} = \frac{3(6 + 1)}{4} = \frac{21}{4} = 5.25 ]
Here, we take the 5th position and interpolate it with the 6th position:
The calculation for Q3 is:
[ Q3 = \text{Value at 5th position} + 0.25 \times (\text{Value at 6th position} - \text{Value at 5th position}) ]
[ Q3 = 88 + 0.25 \times (91 - 88) ] [ Q3 = 88 + 0.25 \times 3 = 88 + 0.75 = 88.75 ]
After performing the necessary calculations, we find that:
These percentiles provide valuable insights into the distribution of our dataset.