Let's solve this problem step-by-step.

Step 1: Calculate the Median and Mean of the Initial 7 Distances

Given distances are: $74, \ 88, \ 94, \ 80, \ 95, \ 102, \ 91 \ \text{yards}$

1.1 Calculating the Median

To find the median, we first need to sort the distances in ascending order:

$74, \ 80, \ 88, \ 91, \ 94, \ 95, \ 102$

Since there are 7 numbers, the median is the middle number, which is the 4th value in the sorted list.

$\text{Median} = 91 \ \text{yards}$

1.2 Calculating the Mean

The mean (average) is calculated by summing all the values and dividing by the number of values.

$\text{Mean} = \frac{74 + 80 + 88 + 91 + 94 + 95 + 102}{7}$

Calculating the sum:

$74 + 80 + 88 + 91 + 94 + 95 + 102 = 624$

Now, divide by 7:

$\text{Mean} = \frac{624}{7} \approx 89 \ \text{yards}$

So, the median is 91 yards, and the mean is 89 yards.

Step 2: Effect of Adding Two More Distances of 91 Yards

The new distances are:

$74, \ 80, \ 88, \ 91, \ 94, \ 95, \ 102, \ 91, \ 91$

2.1 Recalculating the Median

First, sort the new list:

$74, \ 80, \ 88, \ 91, \ 91, \ 91, \ 94, \ 95, \ 102$

With 9 values, the median is the 5th value:

$\text{Median} = 91 \ \text{yards}$

The median remains 91 yards.

2.2 Recalculating the Mean

Now, we recalculate the mean with the two additional throws:

$\text{New Mean} = \frac{74 + 80 + 88 + 91 + 91 + 91 + 94 + 95 + 102}{9}$

Calculating the new sum:

$74 + 80 + 88 + 91 + 91 + 91 + 94 + 95 + 102 = 806$

Now, divide by 9:

$\text{New Mean} = \frac{806}{9} \approx 90 \ \text{yards}$

The mean increases to 90 yards.

Conclusion

• Median: Remains the same at 91 yards.
• Mean: Increases from 89 yards to 90 yards.

Summary

Adding two distances that are equal to the original median did not change the median, but it slightly increased the mean because the added values were closer to the upper half of the original data set.

Would you like more details on any part of this solution, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the mode of the updated list of distances?
2. How would the mean change if the additional throws were 85 yards instead?
3. What is the range of the original list and the updated list?
4. How does adding more values close to the mean affect the standard deviation?
5. If another throw was added at 100 yards, how would the mean and median change?

Tip: When analyzing data, it's essential to understand how additional data points can affect measures like the mean and median to interpret results correctly.