Let’s solve the given problem step-by-step:

Given Data Set:

$\{9, -8, 7, 5, 3, 1, 6, -9, -5\}$

(a) Find the average $A$ of the data set:

The average $A$ is given by: $A = \frac{\text{Sum of all data points}}{\text{Number of data points}}$

Step 1: Sum of all data points:

$9 + (-8) + 7 + 5 + 3 + 1 + 6 + (-9) + (-5) = 9 - 8 + 7 + 5 + 3 + 1 + 6 - 9 - 5 = 9$

Step 2: Number of data points:

There are 9 numbers in the set.

Step 3: Calculate $A$:

$A = \frac{9}{9} = 1$

So, the average $A = 1$.

(b) Find the median $M$ of the data set:

Step 1: Arrange the data in ascending order:

$\{-9, -8, -5, 1, 3, 5, 6, 7, 9\}$

Step 2: Identify the median:

The median is the middle number in the ordered set. Since there are 9 data points (an odd number), the median is the 5th number.

$M = 3$

So, the median $M = 3$.

(c) Consider the data set $\{9, -8, 7, 5, 3, 1, 6, -9, -5, 1\}$, obtained by adding one more data point ($1$) to the original data set. Find the average and the median of this data set.

Step 1: New data set:

$\{9, -8, 7, 5, 3, 1, 6, -9, -5, 1\}$

Average $A$:

1. Sum of all data points: $9 + (-8) + 7 + 5 + 3 + 1 + 6 + (-9) + (-5) + 1 = 9 + 1 = 10$

2. Number of data points: There are 10 numbers in the new set.

3. Calculate $A$: $A = \frac{10}{10} = 1$

So, the average $A = 1$.

Median $M$:

1. Arrange the data in ascending order: $\{-9, -8, -5, 1, 1, 3, 5, 6, 7, 9\}$

2. Identify the median: The median is the average of the 5th and 6th numbers in the ordered set: $M = \frac{1 + 3}{2} = 2$

So, the median $M = 2$.

Final Answers:

• (a) Average $A = 1$
• (b) Median $M = 3$
• (c) For the new data set:
  • Average $A = 1$
  • Median $M = 2$

Do you have any questions or want a detailed explanation for any step? Here are some related questions:

1. How is the average calculated for different types of data sets?
2. What happens to the median when the data set size changes?
3. How can we determine if a data set has outliers affecting its average or median?
4. What is the difference between mean, median, and mode?
5. How would adding a large number to the data set affect the average and median?

Tip: Always arrange data in ascending order to accurately find the median.