Statistics 2a

Probability

• relative  frequency = Number of times an event occurs/number of replications
• probability = relative frequency on an event after an indefinitely long number of trials

Probability distributions

• pattern of probabilities for a set of events is called  a probability distribution
• two important elements:
• the probability of each event or combination of events must range from 0 to 1
• the sum of the probabilities of all possible events must be equal to 1
• discrete probability distribution => probabilities associated with series of discrete outcomes
• e.g. for a six sided die: p(y) = 1/6, y=1,2,3,4,5,6
  
• continuous probability distribution => probabilities associated with continuous values
• we cannot assign a positive probability to a specific value
• continuous probability distributions are calculated using a probability density function
• the probability associated with a range of values is equal to the area under the PDF (probability density function) curve
• Distributions  tutorial, we'll focus on the Normal distribution

Random Variables and random samples

• a random variable is a variable whose values occur at random, following a probability distribution
• when a random variable attains a value that value is called an observation
• a collection of observations is called a random sample
• Random sample tutorial (Marksman/target)
• larger samples result in histograms resembling the PDF more closely
  

The Normal Distribution

• the distribution used in the above example is the Normal distribution:


• µ is the true mean, σ the true standard deviation, x the random variable
  
• Probability tutorial  => normal distribution
• last pageof tutorial: find the probability between ±1σ and  ±2σ. (~0.68 and 0.95)
• the normal curve is symmetric about the average, and the total area under it is 100% (fig).
• Standard units say how many sigma’s a value is, above (+) or below (-) the average.
• How to normalize the histogram: x (height in inches)  becomes (x-µ)/σ, and the y-axis changes from %/inch to %/standard unit (fig).
• many histograms have roughly the same shape as the normal curve.
• the main reason why the normal distribution is a good approximation to many experiments in the sciences is the Central Limit Theorem: "As the sample size, n, increases, the sampling distribution of the mean approaches a normal distribution, no matter how the population being sampled is distributed".
• if a list of numbers follows the normal curve, the percentage of entries falling in a given interval can be estimated by converting the interval to standard units and then finding the corresponding area under the normal curve. As already mentioned above, about 68% of the observations in a large sample within one σ of the average, and about 95% are within two σ's of the average.

Parameters and estimators

• the normal distribution is described by two parameters, µ and σ
• these parameters can be estimated by the mean and standard deviation SD
• how large must the sample be to properly estimate µ?
• example: 50 samples with 100 observations each sampled from a normal distribution (µ = 0, σ = 1), the mean of means close to 0, and the SD of the means is ~1/10 of the σ of the original  normal distribution
  
• (Population parameters tutorial)
• how to find out if data are normally distributed?
• use StatPlus to generate random normal data
• display data in two ways (using StatPlus)
  
1. as histogram with normal distribution overlay
   
2. as normal probability plot (normal score is the value you would expect if your sample came from a normal distribution, think of an infinite number of samples, each consisting of 100 observations, and average the lowest number, the second lowest and so forth)
• the SD of  is also referred to as the Standard Error (SE) of 
• by increasing the sample size, we can reduce SE
• Central limit theorem
• the central limit theorem describes the sampling distribution of the sample average , taken from any distribution
• If each observation follows a distribution with a mean µ and standard deviation, σ, the Central Limit theorem states that the distribution of the sample averages approximately folows a Normal distribution with a mean, µ, and standard deviation of σ/n^1/2 (where n is teh smaple size). This theorem is true for any distribution, as long as th mean and standard deviation of the distribution exists and are finite. 
  
• (Central limit theorem tutorial)
  

Standard Error

The standard error of the average (SE) is:

Resources:

• Freedman, D., Pisani, R., Purves, R., and Adhikari, A. (1991) Statistics. WW Norton & Company, New York, 2nd ed. 514pp.
• Fisher, F.E. (1973) Fundamental Statistics Concepts. Canfield Press, San Francisco, 371 pp.
• Berenson, M.L., Levine, D.M., and Rindskopf, D. (1988) Applied statistics - A first course. Prentice Hall, Englewood Cliffs, NJ, 557pp.
• Lyons, L. (1991) A practical guide to data analysis for physical science students. Cambridge University Press, Cambridge, UK, 95p. (Barnard QC 33.L9 1991 c.1)
• Hartwig, F., and Dearing B.E. (1979) Exploratory data anlaysis. Sage University Paper series on Quantitative Applications in the Social Sciences, 16, Sage Publcations, Beverly Hills and London, 83p. (Barnard HA 29.H257 c.1)
• Jaffe, A.J., and Spirer, H.F. (1987) Misused statitistics. Popular statistics, 5. Marcel Dekker, Inc., New York, 236p. (Barnard HA 29.J29 1987)
• Knoke, D., and Bohrnstedt, G.W. (1991) Basic social statistics. F.E. Peacock Publishers, Inc., Itasca, Illinois, 363p. (Barnard HA 29.K735 1991 c.2)
• Levin, J., and Fox, A.J. (1994) Elementary statistics in social research. Harper Collins College Publishers, New York, 508p. (Barnard HA 29.L388 1994 c.1)
• Welkowitz, J, Ewen, R.B., and Cohen, J. (1991) Introductory statistics in the behavioral sciences. Harcourt Brace & Company, Orlando, Florida, 391p.