To determine which data set has the greatest sample standard deviation, let's analyze the three data sets provided:

Key Concept:

The standard deviation measures the spread or dispersion of a set of data points. A higher standard deviation means the data points are more spread out from the mean, and a lower standard deviation means they are clustered more tightly around the mean.

Now, let's examine each data set:

Data Set (i):

The frequency distribution is as follows:

• $4 \to 1$
• $5 \to 2$
• $6 \to 3$
• $7 \to 4$
• $8 \to 3$
• $9 \to 2$
• $10 \to 1$

The data points are fairly evenly spread between 4 and 10, with the largest concentration at 7. While the frequency of the values changes, the values themselves range fairly evenly.

Data Set (ii):

The frequency distribution is as follows:

• $4 \to 6$
• $5 \to 2$
• $6 \to 0$
• $7 \to 0$
• $8 \to 0$
• $9 \to 2$
• $10 \to 6$

Here, we see two clusters: one at 4 and the other at 10. The data set is heavily skewed towards the extremes (4 and 10) with few data points in between. This suggests a larger spread, which typically results in a higher standard deviation.

Data Set (iii):

The frequency distribution is as follows:

• $4 \to 0$
• $5 \to 0$
• $6 \to 5$
• $7 \to 6$
• $8 \to 5$
• $9 \to 0$
• $10 \to 0$

This set shows two distinct peaks: one around 6 and 7, and another around 8. However, the data is more concentrated around the center (6 to 8) and not spread out across a wide range. As a result, this set has a smaller spread compared to Data Set (ii).

Comparison of Standard Deviation:

• Data Set (i): The data points are fairly evenly spread between 4 and 10.
• Data Set (ii): The data is concentrated in two far-apart clusters (4 and 10), meaning there is a larger spread and likely a higher standard deviation.
• Data Set (iii): While there are two peaks, the data is more tightly packed around the middle range (6-8), resulting in a lower standard deviation.

Conclusion:

Data set (ii) has the greatest standard deviation because it has the most entries that are farther from the mean (due to the large gaps between 4 and 10 with few entries in between).

Thus, the correct answer is: B. Data set (ii), because it has more entries that are farther away from the mean.

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Would you like more details on how to calculate the standard deviation for these sets, or have any other questions about statistics? Here are some related questions you might find useful:

1. How do you calculate the sample standard deviation from a frequency distribution?
2. What does it mean when two data sets have similar means but different standard deviations?
3. How does the presence of outliers affect the standard deviation of a data set?
4. What are some real-world scenarios where a high standard deviation would indicate variability?
5. How do you interpret a standard deviation in the context of risk analysis or data variability?

Tip: When comparing standard deviations, always pay attention to how spread out the data points are from the mean, as this is the main factor that influences the size of the standard deviation!