The problem presented asks for the ordering of the standard deviations of three symmetrical distributions labeled (a), (b), and (c) based on their histograms, without performing calculations. The goal is to compare the spread or variability of each distribution.

Standard deviation (σ):

• The standard deviation is a measure of how spread out the values in a dataset are. A greater spread in the data means a larger standard deviation, while a smaller spread means a smaller standard deviation.

Observations:

1. Distribution (a): The values are concentrated between 4 and 11, with most of the data clustered around the center (7–9). This suggests a moderate spread.
2. Distribution (b): The data is similarly clustered between 4 and 11, but it seems slightly more compact compared to (a). This would suggest a lower standard deviation than (a).
3. Distribution (c): This histogram is the most spread out, with data extending from around 5 to 10, but with gaps between the bars. This greater variability suggests that it will have the largest standard deviation.

Ordering of Standard Deviations:

From smallest to largest:

• σ_b < σ_a < σ_c

This is because distribution (b) is more tightly clustered than (a), while distribution (c) has the most spread-out data.

Would you like further details or have any questions? Here are some related questions:

1. How does the concept of standard deviation apply to real-world data?
2. Why does a symmetrical distribution matter when comparing standard deviations?
3. How do gaps in data, like those in distribution (c), affect standard deviation?
4. What happens to standard deviation when all values are identical?
5. Can two distributions with different shapes have the same standard deviation?

Tip: Visualizing a distribution’s spread is often the quickest way to estimate its standard deviation without calculations.