Let's start by computing the mean, median, and mode for the given set of pro football player ages:

Data:

$25, 24, 26, 24, 31, 30, 29, 27, 34, 30, 25, 39, 26, 24, 23, 28, 29, 26, 32, 30, 26, 23, 32, 30, 23, 29, 28, 27, 24, 22, 26, 22, 26, 25, 23, 27, 26, 33, 27, 30$

(a) Compute the mean, median, and mode:

1. Mean:

The mean is calculated by adding all the ages and dividing by the total number of values.

Mean = $\frac{\text{Sum of ages}}{\text{Number of players}}$

Let's calculate it:

$\text{Sum of ages} = 25 + 24 + 26 + \ldots + 30 = 1064$

The total number of players is 40.

$\text{Mean} = \frac{1064}{40} = 26.6$

So, the mean is 26.6 years.

2. Median:

The median is the middle value when the data is arranged in ascending order. For an even number of observations, it's the average of the two middle values.

Let's sort the data in ascending order:

$22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27, 27, 28, 28, 29, 29, 29, 29, 30, 30, 30, 30, 31, 32, 32, 33, 34, 39$

The middle values are the 20th and 21st terms:

$26, 27$

So, the median is:

$\frac{26 + 27}{2} = 26.5$

3. Mode:

The mode is the most frequent value in the data set. From the sorted data, we can see that 26 appears the most frequently (7 times).

Thus, the mode is 26.

(b) Compare the averages:

• Mean = 26.6
• Median = 26.5
• Mode = 26

These values are very close to each other, indicating that the data is fairly symmetric.

The most appropriate average is likely the mean, as the data appears mound-shaped symmetric. The mean generally provides the best measure of central tendency in such cases.

Thus, the correct conclusion is:

The averages are very close. The mean seems to represent the ages most accurately since the distribution of the data is mound-shaped symmetric.

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Do you want more details or have any questions about these calculations?

Here are 5 related questions to deepen your understanding:

1. How would the median change if the highest age (39) were removed?
2. What impact does an outlier have on the mean compared to the median?
3. Can a data set have more than one mode? If so, how would that affect analysis?
4. Why is the mean considered a better measure of central tendency for symmetric distributions?
5. How could you visualize the distribution of ages to confirm the shape of the data?

Tip: The median is less affected by extreme values (outliers) than the mean.