Chapter 2 The Normal Distributions

Section 2.1 Density curves and the normal distributions

Activity Roll one die 20 times and record the results Graph Roll two dice 20 times and record the sum of the two die Graph Page (using a calculator for a simulation)

We now have a clear strategy for exploring data on a single quantitative variable: ◦ Always plot your data (usually a histogram or a stemplot) ◦ Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers ◦ Calculate a numerical summary to briefly describe the center and spread We now add a new step: ◦ Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve  Called a Density Curve

Density Curves A density curve is a mathematical model for the distribution ◦ A mathematical model is an idealized description Gives a compact picture of the overall pattern of the data Ignores minor irregularities as well as any outliers

Density Curves Is always on or above the horizontal axis Has area exactly 1 underneath it Describes the overall pattern of a distribution The area under the curve and between any range of values is the proportion of all observations that fall in that range

Standard Normal Distributions

Mean and median of a density curve Median of a density curve is the equal-areas point (hard to find with skewed curves) In a symmetric distribution the mean and median are in approximately the same place Mean is the balance point of a density curve In a skewed distribution the mean is pulled towards the tail Can basically locate the mean, median, and quartiles of any density curve (not the standard deviation)

Normal Distributions Normal curves describe normal distributions All normal distributions have the same overall shape Symmetric, single-peaked, bell-shaped The exact density curve for a particular normal distribution is described by giving its mean μ and its standard deviation σ Changing μ without changing σ moves the normal curve along the horizontal axis without changing its spread The standard deviation σ controls the spread of a normal curve Inflection points are the points at which the change of the curvature takes place and are located at a distance σ on either side of the mean ◦ They change as σ changes

Points of Inflection

68% within one Standard deviation 95% within two Standard Deviations 99.7% within three Standard Deviations Empirical Rule:

Normal Distribution Notation N( μ, σ ) ◦ Ex: distribution of women’s heights  N(64.5, 2.5) Give good descriptions for some distributions of real data ◦ Ex: SAT scores Give good approximations to the results of many kinds of chance outcomes ◦ Ex: tossing a coin Statistical inference procedures are based on normal distributions

Percentiles Used when we are most interested in seeing where an individual observation stands relative to the other individuals in the distribution In practice the number of observations is quite large so can talk about the distribution as a density curve Median score is the 50 th percentile First quartile is the 25 th percentile Third quartile if the 75 th percentile

Section 2.2 Standard Normal Calculations

Normal distributions vary from one another in two main ways: ◦ The mean μ may be located anywhere on the x-axis ◦ The bell shape may be more or less spread according to the size of the standard deviation σ We can standardize our distribution in order to find proportions and compare to other distributions

z-score The standardized value, or z value, or z score tells us the number of standard deviations the original measurement is from the mean, and in what direction ◦ observations larger than the mean are positive when standardized ◦ observations smaller than the mean are negative when standardized

Calculating the Z-score This says that she is 1.4 standard deviations above the mean.

Calculating the Z-score How about a woman who is 5 feet (60 inches) tall? What would be her standardized height? This says that she is 1.8 standard deviations below the mean.