The Normal Distribution

• Mathematical models
• The Normal Distribution
• Introduction to Sampling Distributions

The Normal Distribution

• Last class we talked about ways to describe distributions
• Central Tendency
• Variability
• Today we will talk about a theoretical distribution important in statistics
• The Normal Distribution
• The Ultimate Well-Behaved Distribution

Mathematical Distributions

• Whenever we have a set of points, we might want to describe them with an equation.
• This provides a formal description or model
• When the set of points are observations from a sample, the model is a distribution
• This model will smooth the curve from a histogram.

Not as good of a fit:

 
 
 
 
 

Why a Model?

• If the mathematical properties are known, then we can use this distribution to reason about the data.
• For example, suppose we wanted to know whether a particular observation we obtained was common or extreme.
• How would we know?

Using the Normal Distribution

• The Normal distribution is one model distribution
• It is defined by an equation that has 2 parameters that determine its shape
• The Mean and the Standard Deviation

Different Normal Distributions

• Changing the Mean Shifts the Distribution

Changing the Standard Deviation makes the distribution wider or narrower.

Area under the curve

• Because the equation that specifies the distribution is known, we know the area under the curve.
• The area under the curve is the proportion of observations that fall between those values.

The standard normal distribution

• All normal distributions are the same, except for a transformation.
• We can change any observation into a standard score (sometimes called a z-score)
• Z = (X-mean)/sd

What is a z-score?

• Using the z-score, you can find the proportion of observations as extreme or more extreme in the normal distribution
• Just use your normal distribution chart.
• Where do you get one?
• The back of your book
• A computer

Practice with z-scores

• Suppose you are scouting for potential Olympic long jumpers.  You observe 5000 sixth graders in the standing broad jump.
• The distribution of the sample looks well-behaved
• Mean = 6.53 feet
• s.d. = 1.14 feet
• What are the z-scores and probabilities for jumps of

Sampling Distributions

• The normal distribution can help.
• If you survey N people, your survey will get some mean response X₁.
• If you took another survey of N people from the same population, this survey would have a mean X₂.
• If you took a bunch of surveys and plotted the means on a histogram, you would find something that looked like a normal distribution
• Even if the data you are sampling is not normally distributed.
• Sampling Distribution of the Mean

Lots of Means

• This distribution of survey results would follow a normal distribution
• Mean = mu
• standard deviation = sd/(N)^1/2

Increasing Sample Size

• As N (sample size) increases, the variability in this distribution decreases substantially.
• By N = 1000, the true mean is quite likely to be very close to the mean obtained in the survey

How big a sample?

• The sampling distribution of the mean
• Mean = mu
• standard deviation = sd/(N)^1/2
• As N gets larger, the standard deviation of the sampling distribution gets smaller.
• Diminishing returns for additional observations
• A cost-benefit analysis

Where is this train taking me?

• The normal distribution is a convenient mathematical construct, but it may not be a good model of your data.
• Poorly behaved distributions deviate from the normal
• The normal distribution will come back when we talk about inferential statistics later in the semester.
• Many of the statistical tests we will talk about assume that the data being tested follow a normal distribution.