Let's analyze the image.

The data provided is:

32, 33, 35, 36, 37, 43, 54, 55, 59, 62, 66, 69, 69, 75, 78, 89

1. What is the percentile of the data value 55?

To calculate the percentile of a given data value $X$, use the formula:

$P = \left( \frac{L}{N} \right) \times 100$

Where:

• $P$ is the percentile,
• $L$ is the number of values less than $X$,
• $N$ is the total number of values.

For $X = 55$:

• There are 8 values less than 55: 32, 33, 35, 36, 37, 43, 54.
• $N = 16$ (since there are 16 data points in total).

Now calculate:

$P = \left( \frac{8}{16} \right) \times 100 = 50\%$

So, the percentile for the data value 55 is 50 (rounded to the nearest whole number).

2. Which data value represents the 20th percentile?

To find the data value corresponding to a specific percentile, use the formula:

$L = \frac{P}{100} \times (N + 1)$

Where:

• $L$ is the rank position,
• $P$ is the percentile,
• $N$ is the total number of data points.

For the 20th percentile: $L = \frac{20}{100} \times (16 + 1) = 0.2 \times 17 = 3.4$

Since the rank position is 3.4, we look between the 3rd and 4th data points (35 and 36). To interpolate, the value at the 20th percentile will be slightly closer to 35.

So, the data value representing the 20th percentile is approximately 35.4.

Would you like further details on the calculations?

Here are five related questions to explore:

1. What is the 90th percentile for the given data?
2. How would you calculate the median for the given data set?
3. Can you calculate the interquartile range for this data set?
4. What is the range of the data?
5. How would the percentile change if another data value is added to the set?

Tip: Percentile ranks show how a particular value compares to the rest of the data and are useful in understanding the relative position within the dataset.