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## Question

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Which of the distributions has the greatest variation? Part 1A. A normal distribution has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 1 in increments of 0.2. From left to right, the curve rises passing through a point of inflection at (negative 2, 0.1), to a maximum at (0, 0.2) and then falls passing through a point of inflection at (2, 0.1). All coordinates are approximate.B. A normal distribution has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 1 in increments of 0.2. From left to right, the curve rises passing through a point of inflection at (negative 0.5, 0.5), to a maximum at (0, 0.8) and then falls passing through a point of inflection at (0.5, 0.5). All coordinates are approximate.C. A normal distribution has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 1 in increments of 0.2. From left to right, the curve rises passing through a point of inflection at (negative 1, 0.2), to a maximum at (0, 0.6) and then falls passing through a point of inflection at (1, 0.2). All coordinates are approximate.D.

## Answer

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Solution Steps

To determine which distribution has the greatest variation, we need to compare the standard deviations of the given normal distributions. The standard deviation of a normal distribution is related to the distance between the points of inflection. The greater the distance between the points of inflection, the greater the standard deviation and thus the greater the variation.

Step 1: Identify Points of Inflection

We are given the points of inflection for each distribution:

- Distribution A: \((-2, 2)\)
- Distribution B: \((-0.5, 0.5)\)
- Distribution C: \((-1, 1)\)

Step 2: Calculate the Distance Between Points of Inflection

The distance between the points of inflection for each distribution is calculated as follows:

- For Distribution A: \( |2 - (-2)| = 4 \)
- For Distribution B: \( |0.5 - (-0.5)| = 1.0 \)
- For Distribution C: \( |1 - (-1)| = 2 \)

Step 3: Determine the Greatest Variation

The distribution with the greatest distance between points of inflection has the greatest variation. Comparing the distances:

- Distribution A: \( 4 \)
- Distribution B: \( 1.0 \)
- Distribution C: \( 2 \)

The greatest distance is \( 4 \), which corresponds to Distribution A.

Final Answer

The distribution with the greatest variation is \(\boxed{\text{A}}\).

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