To compare the standard deviations $\sigma_a$, $\sigma_b$, and $\sigma_c$ of the three histograms without performing calculations, we observe the spread of each distribution. A larger spread corresponds to a larger standard deviation.

1. Histogram (a): The data points are concentrated at the extremes (3-4 and 11-12), with no values in the middle range. This creates a wide spread.
2. Histogram (b): The data points are spread across the entire range, but they are more centralized (6-7 and 10-11). This is less spread out than (a).
3. Histogram (c): The data points are distributed more evenly across the entire range (4-12), indicating the least spread.

Order of Standard Deviations:

$\sigma_a > \sigma_b > \sigma_c$

Let me know if you’d like more details or explanations.

Related Questions:

1. What is the definition of standard deviation and how is it related to data spread?
2. Why does the concentration of data points near the mean affect the standard deviation?
3. How does symmetry in the distribution affect its variability?
4. How would the standard deviation change if the number of data points increased but the spread stayed the same?
5. Can two distributions with different shapes have the same standard deviation?

Tip:

Visualizing the spread and clustering of data is a quick way to estimate standard deviation without calculations.