Question

you have been provided with the following data set:
2, 4, 6, 8, 10, 12, 14, 16, and 18
for the given data set, is the 25th percentile, is the 50th percentile,
and is the 75th percentile.
i dont know 2 attempts

Explanation:

Step1: Calculate the position of percentiles

For a data set of size \(n = 9\).
The formula for the position of the \(p\)th percentile is \(L=\frac{p}{100}\times n\).
For the 25th percentile (\(p = 25\)), \(L=\frac{25}{100}\times9 = 2.25\).
For the 50th percentile (\(p = 50\)), \(L=\frac{50}{100}\times9=4.5\).
For the 75th percentile (\(p = 75\)), \(L=\frac{75}{100}\times9 = 6.75\).

Step2: Interpolate to find the values

• For \(L = 2.25\) (25th percentile):

The first value \(x_1\) at position \(2\) is \(4\), and the value \(x_2\) at position \(3\) is \(6\).
Using linear interpolation \(x=x_1+(x_2 - x_1)\times0.25\), so \(x = 4+(6 - 4)\times0.25=4 + 0.5=5\).

• For \(L = 4.5\) (50th percentile):

The value \(x_1\) at position \(4\) is \(8\), and the value \(x_2\) at position \(5\) is \(10\).
Using linear interpolation \(x=x_1+(x_2 - x_1)\times0.5\), so \(x=8+(10 - 8)\times0.5=8 + 1=9\).

• For \(L = 6.75\) (75th percentile):

The value \(x_1\) at position \(6\) is \(12\), and the value \(x_2\) at position \(7\) is \(14\).
Using linear interpolation \(x=x_1+(x_2 - x_1)\times0.75\), so \(x=12+(14 - 12)\times0.75=12+1.5 = 13.5\).

Answer:

The 25th percentile is \(5\), the 50th percentile is \(9\), and the 75th percentile is \(13.5\).