5.5 	5.7 	5.8 	5.9 	6.1 	6.1 	6.4 	6.4 	6.5 	6.6
6.7 	6.7 	6.7 	6.9 	7 	7 	7 	7.1 	7.2 	7.2
7.4 	7.5 	7.7 	7.7 	7.8 	8 	8.1 	8.1 	8.3 	8.7, find the 82nd percentile

To find the 82nd percentile of a data set, we first need to determine its position within the ordered data. Here's the step-by-step process:

### Step 1: Organize the data
The given data is already ordered from smallest to largest.

**Data Set**:
\[ 5.5, 5.7, 5.8, 5.9, 6.1, 6.1, 6.4, 6.4, 6.5, 6.6, 6.7, 6.7, 6.7, 6.9, 7, 7, 7, 7.1, 7.2, 7.2, 7.4, 7.5, 7.7, 7.7, 7.8, 8, 8.1, 8.1, 8.3, 8.7 \]

There are **30 data points**.

### Step 2: Calculate the rank (position) for the 82nd percentile
The formula to calculate the rank for a given percentile is:

\[
P = \frac{n+1}{100} \times k
\]

Where:
- \( P \) = percentile rank position
- \( n \) = number of data points (30 in this case)
- \( k \) = the desired percentile (82)

Substitute the values:

\[
P = \frac{30+1}{100} \times 82 = \frac{31}{100} \times 82 = 25.42
\]

### Step 3: Interpret the rank
The rank position is **25.42**, which is between the 25th and 26th values in the ordered data set.

### Step 4: Find the values at the 25th and 26th positions
- The 25th value is **7.8**.
- The 26th value is **8**.

### Step 5: Interpolate between these values
Since 25.42 is slightly above 25, we can interpolate between the 25th and 26th values. To do this:

\[
\text{Interpolated Value} = 7.8 + (0.42 \times (8 - 7.8)) = 7.8 + (0.42 \times 0.2) = 7.8 + 0.084 = 7.884
\]

### Final Answer:
The 82nd percentile of the data set is approximately **7.88**.

Would you like further clarification on any steps?

Here are 5 related questions for deeper exploration:
1. How would the result change if we were looking for the 90th percentile?
2. What is the 50th percentile (median) for this data set?
3. How can you find the interquartile range (IQR) from the given data set?
4. How would you calculate the percentile rank of the value 7.5 in the data set?
5. What is the difference between percentile and percentile rank?

**Tip**: Percentiles are useful for understanding how a particular value compares to the rest of the data set.

The query submitted by the user: 5.5 5.7 5.8 5.9 6.1 6.1 6.4 6.4 6.5 6.6 6.7 6.7 6.7 6.9 7 7 7 7.1 7.2 7.2 7.4 7.5 7.7 7.7 7.8 8 8.1 8.1 8.3 8.7, find the 82nd percentile.

Learn how to find the 82nd percentile from a data set with 30 values. This step-by-step guide walks you through calculating the rank position using the percentile rank formula and finding the interpolated value. Suitable for students in Grades 9-12.

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