To rank the standard deviations ($\sigma_a$, $\sigma_b$, and $\sigma_c$) of the three distributions based on their histograms without performing calculations, we consider the spread of each distribution. The standard deviation is a measure of how spread out the data is around the mean, and the wider the distribution, the larger the standard deviation.

• Histogram (a): The data points are centered and relatively close to each other, with only a small spread around the center. This suggests a small standard deviation.
• Histogram (b): The data is more spread out compared to (a), with the bars distributed further from the center. This suggests a larger standard deviation than (a).
• Histogram (c): The data is the most spread out, with gaps between clusters of data points. This distribution has the largest spread, indicating the highest standard deviation.

Thus, the order of standard deviations from smallest to largest is: $\sigma_a < \sigma_b < \sigma_c$

Would you like further details on this explanation or any related concepts?

Here are 5 related questions:

1. What is the definition of standard deviation?
2. How can we calculate the exact standard deviation for a dataset?
3. How does the shape of a histogram relate to its variance and standard deviation?
4. What are other ways to visually estimate the spread of data besides using histograms?
5. How does a symmetrical distribution impact the mean and standard deviation?

Tip: A larger spread in data means more variability, which directly affects the standard deviation.