I see the data set and the percentiles you are working with. Let's go through the steps of how the 16th and 89th percentiles were calculated based on the given data set.

Data Set:

The numbers in the data set are: $54, 44, 68, 74, 59, 71, 59, 40, 32, 74, 44, 34, 63, 32, 36, 44, 46, 55$

Step 1: Arrange the Data in Ascending Order

First, we sort the data from smallest to largest:

$32, 32, 34, 36, 40, 44, 44, 44, 46, 54, 55, 59, 59, 63, 68, 71, 74, 74$

Step 2: Percentile Calculation

To find a percentile, you use the formula: $P = \left( \frac{n}{100} \times N \right)$ Where:

• $P$ is the position of the percentile,
• $n$ is the desired percentile (e.g., 16th, 89th),
• $N$ is the total number of data points.

Here, $N = 18$ since there are 18 values.

16th Percentile Calculation:

For the 16th percentile: $P = \left( \frac{16}{100} \times 18 \right) = 2.88$

This means we take the 3rd value in the sorted data set, which is 36.

So, the 16th percentile is 36.

89th Percentile Calculation:

For the 89th percentile: $P = \left( \frac{89}{100} \times 18 \right) = 16.02$

This means we take the 16th value in the sorted data set, which is 74.

So, the 89th percentile is 74.

Would you like any further details or clarifications on these calculations?

Follow-up questions:

1. How is the 50th percentile (median) calculated in a dataset?
2. What is the difference between percentiles and quartiles?
3. Can you explain how to handle fractional positions in percentile calculations?
4. How would outliers affect percentile values in a dataset?
5. What would be the interquartile range (IQR) for this dataset?

Tip:

When finding percentiles, if the position is not a whole number, you can interpolate between the two nearest data points to get a more precise result.