Consider the following set of three​ distributions, all of which are drawn to the same scale. Identify the two distributions that are normal. Of the two normal​ distributions, which one has the larger​ variation?
x
y



Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins at the origin, rises sharply to a maximum and falls sharply to the x-axis.
x
y



Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins just above the origin, rises gradually to a maximum and falls gradually to just above the x-axis.
x
y



Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins at the origin, rises sharply to a maximum and falls gradually to just above the x-axis.
​(a)
​(b)
​(c)
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Part 1
The two normal distributions are 
▼ 
(a) and (c)
(b) and (c)
(a) and (b)
​, where 
▼ 
(b)
(c)
(a)
 has the larger standard deviation.

To determine which of the given curves represent normal distributions and which has the larger variation, let's break down the descriptions:

1. **(a) Curve Description**:
   - Starts at the origin, rises sharply to a maximum, then falls sharply to the x-axis.
   - This curve does not seem to resemble a normal distribution because it is too "sharp" on both sides. Normal distributions are smoother with gradual tails.

2. **(b) Curve Description**:
   - Starts just above the origin, rises gradually to a maximum, and then falls gradually, stopping just above the x-axis.
   - This curve looks like a normal distribution. It has a smooth, symmetric rise and fall, which is characteristic of a normal distribution.

3. **(c) Curve Description**:
   - Starts at the origin, rises sharply to a maximum, then falls gradually to just above the x-axis.
   - This curve may also be considered a normal distribution because it still follows a general bell-shaped curve, although it is skewed slightly.

### Identifying the Two Normal Distributions
The two normal distributions are **(b)** and **(c)**, as they exhibit the smooth and symmetric rise and fall associated with normal curves.

### Determining the Larger Standard Deviation
Standard deviation measures the spread or variation in a distribution. A distribution with a **gradual rise and fall** typically has a larger spread, indicating a larger standard deviation.

- **(b)** rises and falls gradually, implying a larger standard deviation.
- **(c)** rises sharply and falls gradually, indicating a smaller variation.

Thus, **(b)** has the larger standard deviation.

### Final Answer:
- The two normal distributions are **(b) and (c)**.
- Of these, **(b)** has the larger standard deviation.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is a normal distribution, and why is it important in statistics?
2. How does standard deviation affect the shape of a normal distribution?
3. Can a normal distribution be skewed, or must it always be symmetric?
4. How would the height of a normal distribution curve relate to its standard deviation?
5. How can we calculate standard deviation given a dataset?

**Tip:** A higher standard deviation in a normal distribution results in a wider and flatter curve, while a lower standard deviation produces a narrower and taller curve.

Consider the following set of three distributions, all of which are drawn to the same scale. Identify the two distributions that are normal. Of the two normal distributions, which one has the larger variation? The curves are described as follows: 
(a) A curve that begins at the origin, rises sharply to a maximum, and falls sharply to the x-axis. 
(b) A curve that begins just above the origin, rises gradually to a maximum, and falls gradually to just above the x-axis. 
(c) A curve that begins at the origin, rises sharply to a maximum, and falls gradually to just above the x-axis. Which curves are normal, and which has the larger standard deviation?

Learn how to identify normal distributions and compare standard deviations in three curve graphs. This problem focuses on understanding the characteristics of normal distributions and the effect of standard deviation on their shape. Suitable for high school students in Grades 9-12.

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