Chapter 6

Probability Distributions & Normality1The study of probability begins with understanding the two basic classifications of a random variablediscrete and continuous. In order to apply the correct formula for solving a probability question, one must be able to know whether the variable is discrete or continuous. Rectangular DistributionScores are all about equally frequent or probableRoll a six sided dieBimodal DistributionTwo distinct ranges of scores are more common than any otherNormal DistributionSymmetrical, bell-shapedScores cluster near the mean and become increasingly rare as they diverge from the meanHeight, weight, IQNormal DistributionAbility to describe entire set of scores in the sampleAbility to put a particular score into perspectiveHow many standard deviations above/below the mean?Shape of probability distributions The shape of probability distributions can be very important in statistical analysisScores often only approximate a normal distributionSkewness and Kurtosis are two measures of how scores may deviate from the perfectly normal distribution

SkewnessSkew implies that the shape of a unimodal distribution is asymmetric about its mean-the mean lies towards the direction of the skew (the longer tail) relative to the medianPositive skew scores are shifted towards the left in a positively skewed distribution there tend to be some positive outliersNegative skew scores are shifted towards the right KurtosisHow flat versus sharply peaked a set of scores isHow much Skewness or Kurtosis is too much (or too little)Skewness > 3 is worrisome (> 2 for some)Kurtosis < 2 or 3 usually little concernLarge samples (200+) can offset concernsMultivariate outliers can may be trimmed/deleted based on Mahalanobis distancesSometimes transformations can correct for skew and kurtosisSquare root of raw scoresInverse (1/x) of raw scoresBase 10 Log of raw scoresWinsorizing (trimming) extreme outliers (say z score 3.0)Characteristics of a Normal Probability DistributionIt is bell-shaped and has a single peak. It is symmetrical about the mean.It is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it. The arithmetic mean, median, and mode are equal The total area under the curve is 1.00.The area to the left of the mean = area right of mean = 0.5.

-88The Normal Distribution Graphically

9The Family of Normal Distribution

Equal Means and Different Standard Deviations10The Family of Normal Distribution

Different Means and Standard Deviations11The Family of Normal Distribution

Different Means and Equal Standard Deviations12The Standard Normal Probability DistributionThe standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the z distribution. A z-value is the signed distance between a selected value, designated X, and the population mean , divided by the population standard deviation, . The formula is:

13Areas Under the Normal Curve14

14The Normal Distribution ExampleWhat is the z-value for the income, lets call it X, of a foreman who earns $1,100 per week? For a foreman who earns $900 per week?

The weekly incomes of shift foremen in the glass industry follow the normal probability distribution with a mean of $1,000 and a standard deviation of $100.

15Normal Distribution Finding ProbabilitiesWhat is the likelihood of selecting a foreman whose weekly income is between $1,000 and $1,100?

16Normal Distribution Finding Probabilities

17Normal Distribution Finding Probabilities Using the Normal Distribution Table

18Normal Distribution Finding Probabilities (Example 2)What is the probability of selecting a shift foreman in the glass industry whose income is:Between $790 and $1,000?

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19Normal Distribution Finding Probabilities using the Normal Distribution Table

20Normal Distribution Finding Probabilities (Example 3)What is the probability of selecting a shift foreman in the glass industry whose income is:Less than $790?21

21Normal Distribution Finding Probabilities Using the Normal Distribution Table

22Normal Distribution Finding Probabilities (Example 4)What is the probability of selecting a shift foreman in the glass industry whose income is:Between $840 and $1,200?

23Normal Distribution Finding Probabilities Using the Normal Distribution Table

24Normal Distribution Finding Probabilities (Example 5)What is the probability of selecting a shift foreman in the glass industry whose income is:Between $1,150 and $1,250

25Normal Distribution Finding Probabilities Using the Normal Distribution Table

26The Empirical RuleAbout 68 percent of the area under the normal curve is within one standard deviation of the mean.About 95 percent is within two standard deviations of the mean. Practically all is within three standard deviations of the mean.

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