Questions: Three distributions, labeled (a), (b), and (c) are represented below by their histograms. Each distribution is symmetrical and is made of 10 measurements. Without performing any calculations, order their respective standard deviations σa, σb, and σc.

Three distributions, labeled (a), (b), and (c) are represented below by their histograms. Each distribution is symmetrical and is made of 10 measurements. Without performing any calculations, order their respective standard deviations σa, σb, and σc.
Transcript text: Three distributions, labeled (a), (b) rand (c) are represented below by their histograms. Each distribution is symmetrical and is made of 10 measurements. Without performing any calculations, order their respective standard deviations $\sigma_{a}, \sigma_{b}$, and $\sigma_{c}$. \[ { }^{\sigma} \square^{<\sigma} \square^{<\sigma} \square \]
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Solution

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Solution Steps

Step 1: Analyze the spread of each distribution

Standard deviation measures the spread or dispersion of data around the mean. A larger spread indicates a higher standard deviation. Visually, distributions with data points further from the center have higher standard deviations.

Step 2: Compare the distributions

Distribution (a) has most of its data concentrated around the center (values 3 and 9). Distribution (b) has data slightly more spread out than (a), with some values further from the center (values 3, 4, 9, and 10). Distribution (c) is the most spread out, with data points distributed more evenly across the range (3, 4, 6, 7, 9 and 10).

Step 3: Order the standard deviations

Since distribution (a) has the smallest spread, it has the smallest standard deviation. Distribution (c) has the largest spread, and thus the largest standard deviation. Distribution (b) falls in between.

Final Answer: σa < σb < σc

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