Answer: A.

Explanation:1. Understand standard deviation: Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

2. Analyze each frequency table: We are looking for the data set with the *lowest* standard deviation, meaning the data is most concentrated around a central value.

[A] Value | Frequency ------|---------- 4 | 0 5 | 10 6 | 50 (Most data concentrated here) 7 | 10 8 | 0 The data is heavily concentrated at the value $6$. The values $5$ and $7$ are close to $6$. This suggests a low spread.

[B] Value | Frequency ------|---------- 4 | 10 5 | 10 6 | 10 7 | 10 8 | 10 The data is uniformly distributed across all values. This means the data is spread out, suggesting a higher standard deviation than A.

[C] Value | Frequency ------|---------- 4 | 0 5 | 10 6 | 20 7 | 30 8 | 40 The data is skewed towards higher values ($7$ and $8$). There is a significant spread from $5$ to $8$. The peak is at $8$, but there's considerable data at $6$ and $7$ as well.

[D] Value | Frequency ------|---------- 4 | 10 5 | 20 6 | 30 7 | 40 8 | 50 The data is heavily skewed towards $8$, but there are significant frequencies at all other values as well, indicating a wide spread from $4$ to $8$.

3. Compare the distributions: Data set A has most of its values ($50$ out of $70$) at a single point ($6$), with the remaining data points ($10$ at $5$ and $10$ at $7$) very close to this central point. This distribution is the most concentrated. Data set B is uniformly spread. Data sets C and D are spread over a wider range of values with significant frequencies at the extremes or spread out.

The lowest standard deviation will occur when the data points are clustered most tightly around the mean. Distribution A clearly shows the tightest clustering. The values are $5, 6, 7$, with the vast majority being $6$. In B, values are $4, 5, 6, 7, 8$ with equal frequency. In C, values are $5, 6, 7, 8$, with more weight on $7, 8$. In D, values are $4, 5, 6, 7, 8$, with more weight on $6, 7, 8$.

4. Conclusion: Frequency table A represents the data set with the lowest standard deviation because the data is most concentrated around the value $6$.