| Describe in detail what you need | 72, 85, 91, 77, 65, 88. what is the 25th percentile base on median? |
| How many pages | 1 |
To calculate the 25th percentile of a given dataset, we first need to understand what a percentile is. A percentile indicates the value below which a given percentage of observations fall. The 25th percentile, also known as the first quartile (Q1), is the value below which 25% of the data points in the dataset lie.
We have the following dataset of numbers:
72, 85, 91, 77, 65, 88The first step in calculating the 25th percentile is to sort the data in ascending order:
65, 72, 77, 85, 88, 91Next, we determine the number of data points in our dataset. In this case, we have a total of 6 points:
The formula to find the position ( P ) of the k-th percentile within a sorted list is given by:
[ P = \frac{k}{100} \times (N + 1) ]
Where:
Using the formula, we can calculate the position for the 25th percentile:
[ P = \frac{25}{100} \times (6 + 1) = 0.25 \times 7 = 1.75 ]
Since ( P = 1.75 ), we need to find the value at this position. This value falls between the 1st and 2nd entries of our sorted list. In this case:
Given that the calculated position is not a whole number, we proceed to interpolate between these two values:
To find the value at the 25th percentile, we can use the formula for interpolation:
[ Q1 = \text{Value at position 1} + (0.75 \times \text{Difference between adjacent values}) ] [ Q1 = 65 + (0.75 \times (72 - 65)) ] [ Q1 = 65 + (0.75 \times 7) ] [ Q1 = 65 + 5.25 = 70.25 ]
Thus, the 25th percentile (Q1) based on the median of the dataset is approximately 70.25.
The process of calculating the 25th percentile involves sorting the dataset, determining the position using the appropriate percentile formula, and then interpolating if necessary. In this case, the 25th percentile for the dataset ( 72, 85, 91, 77, 65, 88 ) is calculated to be 70.25.