To find the 25th percentile (P25) of the given radiation levels, we first need to sort the data in ascending order. Then, we calculate the position of the 25th percentile using the formula: \( P = \frac{n+1}{100} \times k \), where \( n \) is the number of data points and \( k \) is the desired percentile (25 in this case). If the position is a whole number, the value at that position is the percentile. If it's not, we interpolate between the two closest values.
The given radiation levels are sorted in ascending order:
\[
\text{radiation\_levels} = [0.21, 0.21, 0.25, 0.32, 0.5, 0.56, 0.62, 0.65, 0.67, 0.75, 0.86, 0.88, 0.88, 0.89, 0.92, 0.94, 0.96, 1.01, 1.01, 1.08, 1.1, 1.1, 1.12, 1.14, 1.14, 1.16, 1.18, 1.19, 1.21, 1.21, 1.23, 1.24, 1.26, 1.26, 1.27, 1.27, 1.28, 1.31, 1.34, 1.34, 1.36, 1.4, 1.42, 1.43, 1.44, 1.46, 1.46, 1.52, 1.52, 1.59]
\]
Using the formula for the position of the 25th percentile:
\[
P = \frac{n + 1}{100} \times k
\]
where \( n = 50 \) and \( k = 25 \):
\[
P = \frac{50 + 1}{100} \times 25 = 12.75
\]
Since the position \( P = 12.75 \) is not an integer, we find the values at the indices \( 12 \) and \( 13 \):
- Lower value: \( \text{radiation\_levels}[12 - 1] = 0.88 \)
- Upper value: \( \text{radiation\_levels}[13 - 1] = 0.89 \)
Now, we interpolate to find \( P_{25} \):
\[
P_{25} = \text{lower value} + (P - \text{lower index}) \times (\text{upper value} - \text{lower value})
\]
Substituting the values:
\[
P_{25} = 0.88 + (12.75 - 12) \times (0.89 - 0.88) = 0.88 + 0.75 \times 0.01 = 0.88 + 0.0075 = 0.8875
\]
The 25th percentile \( P_{25} \) is:
\[
\boxed{P_{25} = 0.8875}
\]
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