On this web page, you can explore how the mean and standard deviation impact the bell-shaped normal curve. This will give you a deeper understanding of Lesson 11.3 of Discovering Advanced Algebra: An Investigative Approach.

Sketch

The sketch below originally shows the bell-shaped normal curve with mean 0 and standard deviation 1. The area between one standard deviation above and below the mean is shaded. Drag point mu to change the mean and drag point sigma to change the standard deviation.

Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.

Investigate

1. What transformations happen to the graph when you change the mean? Why?
2. What transformations happen when you change the standard deviation? Why?
3. Where is the vertical line of symmetry for the graph? Is this always true? Why?
4. Does the shape of the shaded area change when you change the mean or standard deviation? Does the amount of shaded area appear to change?

Sketch

This sketch shows how to construct a normal distribution with mean μ and standard deviation σ . First, adjust the sliders for the mean and standard deviation that you want. Then follow these four steps:

• Press Show Parent Function to see the standard normal distribution with mean 0 and standard deviation 1.
• Press Translate Mean to horizontally translate the curve to your mean.
• Press Scale Horizontally to horizontally dilate the curve to your standard deviation.
• Press Scale Vertically to make the area under the curve equal to 1.

The red curve is normal distribution for your mean and standard deviation.

Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.

Investigate

The equation of the parent function is . Again, this standard normal distribution has mean 0 and standard deviation 1.

1. How must the equation be changed to give a horizontal translation by μ?
2. How must the equation be changed to give a horizontal scale factor of σ?
3. How must the equation be changed to give vertical scale factor of σ?
4. All together, what is the equation of the normal distribution with mean μ and standard deviation σ?

Sketch

The red curve in this sketch is the normal distribution with mean μ and standard deviation σ. You can drag the sliders to change the mean or the standard deviation. The green segments show one standard deviation above and below the mean. By dragging point x along the horizontal axis, you can change the position of point (x, y) on the curve. The coordinates of point (x, y) are given. Press Start Over to return to the standard normal distribution.

Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.

Investigate

When answering these questions, try different values for the mean and standard deviation. Then give your answers as generalizations for all normal distributions.

1. For what value of x does y reach its maximum value?
2. Explain how the maximum value of y is related to the the standard deviation.
3. What happens to y when x moves very far away from the mean in either direction? Can you explain why this must be true of every normal distribution?
4. Where is the curve curving upward? Where is it curving downward? Where are the inflection points—the points where the curvature changes?
5. How can the inflection points help you determine the standard deviation?