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The Normal Distribution

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

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The Normal Distribution

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

  2. 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

  3. Mathematical Models • 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.

  4. A histogram with a model

  5. 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?

  6. 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

  7. Different Normal Distributions • Changing the Mean Shifts the Distribution • Changing the Standard Deviation makes the distribution wider or narrower.

  8. Changing the Mean

  9. 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. Mean=100, s.d. = 10

  10. 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-m)/s (m and s are population Mean and s.d.) Mean = 0 s.d. = 1

  11. 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 handy-dandy normal distribution chart. • Where do you get one? • Just open any statistics book, like, say, yours!

  12. Practice with the 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 • 6.21 feet 6.53 feet • 4.38 feet 7.21 feet • 9.77 feet 3.11 feet

  13. A return to sample size • So, how many people should you ask in a survey? • Imagine you want to know the average number of books read each year by everyone over the age of 10 in the United States. • What do you do? • Take a survey of N people, and calculate the mean of your sample. • You calculate the mean, because it is the best estimate of the population mean. • How good an estimate is it?

  14. 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

  15. Lots of Means • This distribution of survey results would follow a normal distribution • Mean =  • standard deviation = /(N)1/2 Sample N=5 Underlying Population Mean = 100 s.d. = 10 This is a distribution of means of samples of size N

  16. 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. Sample N = 1000 Sample N = 10

  17. How big a sample? • The sampling distribution of the mean • Mean =  • standard deviation = /(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

  18. Where will this return? • 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 book provides ways to test the fit of a normal distribution to a set of data • 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.

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