The question in the image provides two series of numbers and asks for the comparison of their averages and standard deviations.

Series A:

6, 7, 8, 9

Series B:

3, 5, 7, 9, 11

Step 1: Calculate the average (mean) of both series

1. For Series A: $\text{Average of A} = \frac{6 + 7 + 8 + 9}{4} = \frac{30}{4} = 7.5$

2. For Series B: $\text{Average of B} = \frac{3 + 5 + 7 + 9 + 11}{5} = \frac{35}{5} = 7$

Step 2: Calculate the standard deviation of both series

1. Standard deviation for Series A:
   • Mean is 7.5
   = \frac{(1.5)^2 + (0.5)^2 + (0.5)^2 + (1.5)^2}{4} = \frac{2.25 + 0.25 + 0.25 + 2.25}{4} = \frac{5}{4} = 1.25$$ $$\text{Standard deviation of A} = \sqrt{1.25} \approx 1.118$$
2. Standard deviation for Series B:
   • Mean is 7
   = \frac{16 + 4 + 0 + 4 + 16}{5} = \frac{40}{5} = 8$$ $$\text{Standard deviation of B} = \sqrt{8} \approx 2.828$$

Conclusion:

• Averages: The averages of the two series are different (7.5 for Series A and 7 for Series B).
• Standard deviations: The standard deviations are also different (approximately 1.118 for Series A and 2.828 for Series B).

Thus, the correct answers would be:

• The averages are different.
• The standard deviations are different.

Would you like to explore these calculations in more depth?

Related Questions:

1. What happens to the standard deviation if we increase all numbers in a series by a constant?
2. How do you calculate the variance of a data set in general?
3. What is the relationship between variance and standard deviation?
4. Why is the standard deviation of Series B larger than that of Series A?
5. How would the calculations change if Series A and Series B had the same number of elements?

Tip: The standard deviation is sensitive to the spread of the numbers—wider ranges result in larger deviations!