Let's break down the problem and the options provided.

We have two data sets, both with the same mean ($195195$), but different standard deviations: one is $16$ and the other is $24$. The standard deviation measures how spread out the data points are from the mean. The larger the standard deviation, the more variability (or spread) the data has.

Now, let's analyze the choices:

• Choice A:

  • (a) has a standard deviation of $16$, and
  • (b) has a standard deviation of $24$,
  • Reason: Because the data in (b) have less variability.

• Choice B:

  • (a) has a standard deviation of $24$, and
  • (b) has a standard deviation of $16$,
  • Reason: Because the data in (a) have more variability.

Analysis:

• The standard deviation of $24$ is larger than $16$, meaning the data set with a standard deviation of $24$ has more variability.
• Therefore, the data set with more variability should have the larger standard deviation.

This means that Choice B is correct because the standard deviation of $24$ corresponds to the data set with more variability (a), and the standard deviation of $16$ corresponds to the data set with less variability (b).

Final Answer:

The correct choice is B:

• (a) has a standard deviation of $24$ because it has more variability, and
• (b) has a standard deviation of $16$ because it has less variability.

Would you like further details or have any questions?

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Here are some follow-up questions you might find useful:

1. What is the formula for calculating standard deviation?
2. How does increasing the standard deviation affect the shape of the distribution?
3. Can two data sets have the same mean but different standard deviations? Why?
4. What other statistical measures are influenced by data variability?
5. How do outliers affect the standard deviation?

Tip: When comparing data sets, always check both the mean and standard deviation to fully understand the distribution's characteristics.