Chapter 2: The Normal Distribution

Section 1: Density Curves and the Normal Distribution

Density Curves

A density curve is similar to a histogram, but there are several important distinctions.

1. Obviously, a smooth curve is used to represent data rather than bars. However, a density curve describes the proportions of the observations that fall in each range rather than the actual number of observations.

2. The scale should be adjusted so that the total area under the curve is exactly 1. This represents the proportion 1 (or 100%).

Density Curves

3. While a histogram represents actual data (i.e., a sample set), a density curve represents an idealized sample or population distribution.

Density Curves: Mean & Median

Three points that have been previously made are especially relevant to density curves.

1. The median is the "equal areas" point. Likewise, the quartiles can be found by dividing the area under the curve into 4 equal parts.

2. The mean of the data is the "balancing" point.

3. The mean and median are the same for a symmetric density curve.

Greek 101

• Since the density curve represents "idealized" data, we use Greek letters: mu for mean and sigma for standard deviation.

Shapes of Density Curves

• We have mostly discussed right skewed, left skewed, and roughly symmetric distributions that look like this:

Bimodal Distributions

We could have a bi-modal distribution. For instance, think of counting the number of tires owned by a two-person family. Most two-person families probably have 1 or 2 vehicles, and therefore own 4 or 8 tires. Some, however, have a motorcycle, or maybe more than 2 cars. Yet, the distribution will most likely have a “hump” at 4 and at 8, making it “bi-modal.”

Uniform Distributions

We could have a uniform distribution. Consider the number of cans in all six packs. Each pack uniformly has 6 cans. Or, think of repeatedly drawing a card from a complete deck. One-fourth of the cards should be hearts, one-fourth of the cards should be diamonds, etc.

Other Distributions

Many other distributions exist, and some do not clearly fall under a certain label. Frequently these are the most interesting, and we will discuss many of them.

Normal Curves

• Curves that are symmetric, single-peaked, and bell-shaped are often called normal curves and describe normal distributions.

• All normal distributions have the same overall shape. They may be "taller" or more spread out, but the idea is the same.

What does it look like?

Normal Curves: μ and σ

• The "control factors" are the mean μ and the standard deviation σ.

• Changing only μ will move the curve along the horizontal axis.

• The standard deviation σ controls the spread of the distribution. Remember that a large σ implies that the data is spread out.

Finding μ and σ

• You can locate the mean μ by finding the middle of the distribution. Because it is symmetric, the mean is at the peak.

• The standard deviation σ can be found by locating the points where the graph changes curvature (inflection points). These points are located a distance σ from the mean.

The 68-95-99.7 Rule

In a normal distribution with mean μ and standard deviation σ:

• 68% of the observations are within σ of the mean μ.

• 95% of the observations are within 2 σ of the mean μ.

• 99.7% of the observations are within 3 σ of the mean μ.

The 68-95-99.7 Rule

Why Use the Normal Distribution???

1. They occur frequently in large data sets (all SAT scores), repeated measurements of the same quantity, and in biological populations (lengths of roaches).

2. They are often good approximations to chance outcomes (like coin flipping).

3. We can apply things we learn in studying normal distributions to other distributions.

Heights of Young Women

• The distribution of heights of young women aged 18 to 24 is approximately normally distributed with mean = 64.5 inches and standard deviation = 2.5 inches.

The 68-95-99.7 Rule

Use the previous chart...

• Where do the middle 95% of heights fall?

• What percent of the heights are above 69.5 inches?

• A height of 62 inches is what percentile?

• What percent of the heights are between 62 and 67 inches?

• What percent of heights are less than 57 in.?

But...

However, NOT ALL DATA are normal or even close to normal. Salaries, for instance, are generally right skewed. Nonnormal data are common and often interesting.