1.4 Density Curves and Normal Distrubutions

Exploring Quantitative Data:

1. Always plot your data: make a graph

2. Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers

3. Calculate a numerical summary to briefly describe center and spread

4. Sometimes, the overall pattern of a large number of observations
   is so regular that we can describe it by a smooth curve

Density Curve:

• a curve that:

  • is always on or above the horizontal axis

  • has an area of exactly 1 underneath it

• A density curve describes the overall pattern of a distribution

• The area under the curve and above any range of values on the horizontal axis is the proportion of all observations that fall in that range

Distinguishing between Median and Mean of a Density Curve:

• The

• of a density curve is the “equal-areas” point―the point that
  divides the area under the curve in half

• The mean of a density curve is the balance point—that is, the point at which the curve would balance if it were made of solid material

• The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve

• The mean of a skewed curve is pulled away from the median in the direction of the long tail

• The mean and standard deviation computed from actual observations (data) are denoted by 𝑥 and s, respectively

• The mean and standard deviation of the distribution represented by the density curve are denoted by μ (“mu”) and s (“sigma”), respectively

Normal Distributions:

• One particularly important class of density curves is the class of Normal curves, which describe Normal distributions

  • All Normal curves are symmetric, single-peaked, and bell shaped

  • A specific Normal curve is described by giving its mean μ and
    standard deviation σ

  • The mean of a Normal distribution is the center of the symmetric Normal curve

  • The standard deviation is the distance from the center to the
    change-of-curvature points on either side

  • We abbreviate the Normal distribution with mean μ and
    standard deviation σ as N(μ, σ)
    

The 68-95-99.7 Rule:

In the Normal distribution with mean μ and standard deviation σ:

• § approximately 68% of the observations fall within σ of μ.

• § approximately 95% of the observations fall within 2σ of μ.

• § approximately 99.7% of the observations fall within 3σ of μ.

• Sketch the Normal density curve for this distribution

Standardized Value / z-score:

• If a variable x has a distribution with mean μ and standard deviation σ,
  then the standardized value of x, or its z-score, is

• 𝑧 = (𝑥 − 𝜇) / 𝜎

• All Normal distributions are the same if we measure in units of size σ
  from the mean μ as center

• The standard Normal distribution is the Normal distribution with mean
  0 and standard deviation 1. That is, the standard Normal distribution is
  N(0, 1)

How to Solve Problems Involving Normal Distributions:

1. Express the problem in terms of the observed variable x

2. Draw a picture of the distribution, and shade the area of interest
   under the curve

3. Perform calculations

   1. Standardize x to restate the problem in terms of a standard Normal variable z

   2. Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve

4. Write your conclusion in the context of the problem