COURSE: JUST 3900INTRODUCTORY STATISTICS

FOR CRIMINAL JUSTICE

Instructor:Dr. John J. Kerbs, Associate Professor

Joint Ph.D. in Social Work and Sociology

Chapter 6: Chapter 6: ProbabilityProbability

ProbabilityProbability

Probability is a method for measuring and Probability is a method for measuring and quantifying the likelihood of obtaining a quantifying the likelihood of obtaining a specific sample from a specific population. specific sample from a specific population.

We define probability as a fraction or a We define probability as a fraction or a proportion. proportion.

The probability of any specific outcome is The probability of any specific outcome is determined by a ratio comparing the determined by a ratio comparing the frequency of occurrence for that outcome frequency of occurrence for that outcome relative to the total number of possible relative to the total number of possible outcomes. outcomes.

Probability (Continued)Probability (Continued)Inferential statistics use sample data to answer questions about

populations

Probability is used to predict what kinds of samples are likely

to be obtained from a population.

Whenever the scores in a population are variable, it Whenever the scores in a population are variable, it is impossible to predict with perfect accuracy exactly is impossible to predict with perfect accuracy exactly which score(s) will be obtained when you take a which score(s) will be obtained when you take a sample from the population. sample from the population.

In this situation, researchers rely on probability to In this situation, researchers rely on probability to determine the relative likelihood for specific samples. determine the relative likelihood for specific samples.

Thus, although you may not be able to predict exactly Thus, although you may not be able to predict exactly which value(s) will be obtained for a sample, it is possible which value(s) will be obtained for a sample, it is possible to determine which outcomes have high probability and to determine which outcomes have high probability and which have low probability.which have low probability.

Probability (Continued)Probability (Continued)

Probability (Continued)Probability (Continued) Definition: For a situation in which several different outcomes Definition: For a situation in which several different outcomes

are possible, the probability for any specific outcome is are possible, the probability for any specific outcome is defined as a fraction or a proportion of all possible outcomes. defined as a fraction or a proportion of all possible outcomes. If the possible outcomes are identified as A, B, C, D, and so If the possible outcomes are identified as A, B, C, D, and so on, then the following formula applies:on, then the following formula applies:

Other Notation: Other Notation: pp(outcome) = . . .(outcome) = . . . Remember: You can express probabilities as Remember: You can express probabilities as

1. 1. FractionsFractions 2. 2. PercentagesPercentages 3. 3. Decimals - - Decimals - - most commonly used approachmost commonly used approach

Probability and SamplingTo assure that the definition of probability is To assure that the definition of probability is

accurate, the use of random sampling is accurate, the use of random sampling is necessary.necessary.Random sampling Random sampling requires that each member of requires that each member of

a population has an equal chance of being a population has an equal chance of being selected.selected.If more than one person is being selected, the If more than one person is being selected, the

probabilities must stay constant from one selection to probabilities must stay constant from one selection to the next - - Thus, the next requirement below.the next - - Thus, the next requirement below.

Independent random sampling Independent random sampling includes the includes the conditions of random sampling and further requires conditions of random sampling and further requires that the probability of being selected remains that the probability of being selected remains constant for each selectionconstant for each selection

Probability and SamplingTwo main approaches to random samplingTwo main approaches to random sampling

Approach #1: Keeps probabilities from changing Approach #1: Keeps probabilities from changing from one selection to anotherfrom one selection to anotherRandom Sampling with ReplacementRandom Sampling with Replacement: This approach to : This approach to

sampling requires that you return each person to the sampling requires that you return each person to the population before you make another selection. population before you make another selection. aka - - aka - - Independent Random SamplingIndependent Random Sampling, , which keeps a which keeps a constant probability of selectionconstant probability of selection

Approach #2: Selections are random, but there is Approach #2: Selections are random, but there is no maintenance of constant probability.no maintenance of constant probability.Random Sampling without ReplacementRandom Sampling without Replacement: This approach : This approach

does not maintain constant probability from one does not maintain constant probability from one selection to the next selectionselection to the next selection

Probability and SamplingCard example for Card example for sampling sampling withwith replacement replacement

with constant probabilitywith constant probabilitypp(ace of diamonds) = 1/52, replace card(ace of diamonds) = 1/52, replace cardpp(ace of diamonds) = 1/52, replace card…(ace of diamonds) = 1/52, replace card…

Card example for Card example for sampling sampling withoutwithout replacementreplacement and no assumption of constant and no assumption of constant probabilityprobabilitypp(ace of diamonds) = 1/52, then next draw(ace of diamonds) = 1/52, then next drawpp(ace of diamonds) = 1/51, then next draw(ace of diamonds) = 1/51, then next drawpp(ace of diamonds) = 1/50, then next draw(ace of diamonds) = 1/50, then next draw

NOTICE HOW PROBABILITY OF SELECTION IS NOT NOTICE HOW PROBABILITY OF SELECTION IS NOT CONSTANT WITHOUT REPLACEMENTCONSTANT WITHOUT REPLACEMENT

Probability (Continued)Probability (Continued)

When a population of scores is When a population of scores is represented by a frequency distribution, represented by a frequency distribution, probabilities can be defined by proportions probabilities can be defined by proportions of the distribution. of the distribution.

Probability values are expressed by a Probability values are expressed by a fraction or proportion. fraction or proportion.

In graphs, probability can be defined as a In graphs, probability can be defined as a proportion of area under the curve. proportion of area under the curve.

Probability and the Normal Probability and the Normal DistributionDistribution

Probability and the Normal Probability and the Normal DistributionDistribution

If a vertical line is drawn through a normal If a vertical line is drawn through a normal distribution, several things occur.distribution, several things occur.The line divides the distribution into two The line divides the distribution into two

sections. The larger section is called the body sections. The larger section is called the body and the smaller section is called the tail.and the smaller section is called the tail.

The exact location of the line can be specified The exact location of the line can be specified by a by a zz-score.-score.

Probability and the Normal Probability and the Normal Distribution (Continued)Distribution (Continued)

The The unit normal tableunit normal table lists several lists several different proportions corresponding to different proportions corresponding to each each zz-score location. -score location. Column A of the table lists Column A of the table lists zz-score values. -score values. For each For each zz-score location, columns B and C list the -score location, columns B and C list the

proportions in the body and tail, respectively. proportions in the body and tail, respectively. Finally, column D lists the proportion between the Finally, column D lists the proportion between the

mean and the mean and the zz-score location. -score location.

Because probability is equivalent to Because probability is equivalent to proportion, the table values can also be proportion, the table values can also be used to determine probabilities. used to determine probabilities.

The Normal Distribution ExampleThe Normal Distribution Example

.633

1

.3669

B + C = 1.00

z = 0.34

Probability and the Normal Probability and the Normal Distribution (cont'd.)Distribution (cont'd.)

To find the probability corresponding to a To find the probability corresponding to a particular score (particular score (XX value), you first transform value), you first transform the score into a the score into a zz-score, then look up the -score, then look up the zz--score in the table and read across the row to score in the table and read across the row to find the appropriate proportion/probability. find the appropriate proportion/probability.

To find the score (To find the score (XX value) corresponding to a value) corresponding to a particular proportion, you first look up the particular proportion, you first look up the proportion in the table, read across the row to proportion in the table, read across the row to find the corresponding find the corresponding zz-score, and then -score, and then transform the transform the zz-score into an -score into an XX value. value.

Probability and the Normal Probability and the Normal Distribution (cont'd.)Distribution (cont'd.)

4 Key Facts about Unit Normal 4 Key Facts about Unit Normal (z)(z) Tables Tables1. The 1. The body (column B) body (column B) always represents the larger always represents the larger

part of the distribution and the part of the distribution and the tail (column C) tail (column C) is always is always the smaller section, whether on the right or left side.the smaller section, whether on the right or left side.

2. The 2. The normal distribution is symmetricalnormal distribution is symmetrical; therefore, the ; therefore, the proportions will be the same for the positive and proportions will be the same for the positive and negative values of a specific negative values of a specific zz-score .-score .

3. 3. Proportions are always positiveProportions are always positive, even if the , even if the corresponding corresponding zz-score is negative. To find proportions -score is negative. To find proportions for negative z-scores, look up the corresponding for negative z-scores, look up the corresponding proportion for the positive value of z.proportion for the positive value of z.

4. A 4. A negative negative zz-score means that the tail of the -score means that the tail of the distribution is on the left side and the body is on the distribution is on the left side and the body is on the rightright, and vice versa for a positive z-score., and vice versa for a positive z-score.

Probability and the Normal Probability and the Normal Distribution (cont'd.)Distribution (cont'd.)

Note: z = + 0.25 here.Your book had a typo on p. 174 and listed it as a negative z-score.

Please correct in your book

Note: z = - 0.25 here.Book is correct for

this side of the graph

See Column C in Unit Normal Table on page 699

Percentiles and Percentile RanksPercentiles and Percentile Ranks

The The percentile rank percentile rank for a specific for a specific XX value is value is the percentage of individuals with scores at or the percentage of individuals with scores at or below that value. below that value.

When a score is referred to by its rank, the When a score is referred to by its rank, the score is called a score is called a percentilepercentile. The percentile . The percentile rank for a score in a normal distribution is rank for a score in a normal distribution is simply the proportion to the left of the score. simply the proportion to the left of the score.

Percentiles and Percentile RanksPercentiles and Percentile Ranks

0 +0.25

P(z>1.00)Tail = 0.1587Or 15.87%

P(z<1.50)Body = 0.9332

Or 93.32%

P(z<-0.50)Tail = 0.3085Or 30.85%

Look up 10% in Tail Tail = 0.1003For z = 1.28

Think Symmetry!!!!!

30% on each side of μ = 0

Look up z for .30 or 30% between μ=0 and z

(see Column D)

Z=+/- 0.84 for proportion .2995

Percentiles and Percentile RanksPercentiles and Percentile RanksPercentile ranks Percentile ranks represent specific scores as a represent specific scores as a

percentage of individuals in the distribution who percentage of individuals in the distribution who have scores that are less than or equal to the have scores that are less than or equal to the specific score. specific score.

For example, if 80% of all JUST3900 students For example, if 80% of all JUST3900 students had term grades that were less than or equal to had term grades that were less than or equal to 87, then a score of 87 has a percentile rank of 87, then a score of 87 has a percentile rank of 80%. 80%. Thus, a score of 87 puts students at the Thus, a score of 87 puts students at the

8080thth percentile percentile..

Percentiles and Percentile RanksPercentiles and Percentile Ranks Imagine that the population of all drug abusing Imagine that the population of all drug abusing

offenders are assessed for their drug cravings offenders are assessed for their drug cravings on a scale from 0 (no cravings) to 150 (intense on a scale from 0 (no cravings) to 150 (intense cravings).cravings).

The assessment finds The assessment finds μμ = 100 and = 100 and σσ = 15 and = 15 and we need to determine what proportion of drug we need to determine what proportion of drug abusing offenders have cravings that fall abusing offenders have cravings that fall between scores of 115 and 130.between scores of 115 and 130.

Percentiles and Percentile RanksPercentiles and Percentile RanksThe assessment finds The assessment finds μμ = 100 and = 100 and σσ = 15 and = 15 and

we need to determine what proportion of drug we need to determine what proportion of drug abusing offenders have cravings that fall abusing offenders have cravings that fall between scores of 115 and 130.between scores of 115 and 130.

Step 1: Find z-scores for two valuesStep 1: Find z-scores for two values For x = 115: z = (x-For x = 115: z = (x- μ μ)/)/ σ σ = (115-100)/15 = 1.00 = (115-100)/15 = 1.00 For x = 140: z = (x-For x = 140: z = (x- μ μ)/)/ σ σ = (130-100)/15 = 2.00 = (130-100)/15 = 2.00

Step 2: Find corresponding proportion between Step 2: Find corresponding proportion between the two z-scoresthe two z-scores p(1.00<z<2.00) = .1587 - .0228 = 0.1359 or 13.59%p(1.00<z<2.00) = .1587 - .0228 = 0.1359 or 13.59%

Percentiles and Percentile RanksPercentiles and Percentile RanksPlease note that you must be able to convertPlease note that you must be able to convert

raw scores (i.e., x values) into z-scores raw scores (i.e., x values) into z-scores Use the z-score formula z =Use the z-score formula z = (x- (x- μ μ) / ) / σ σ

z-scores into proportions and probabilitiesz-scores into proportions and probabilitiesproportions and probabilities into z-scoresproportions and probabilities into z-scoresZ-scores into raw scores (i.e., x values)Z-scores into raw scores (i.e., x values)

Use the z-score formula z =Use the z-score formula z = (x- (x- μ μ) / ) / σ σ

Please remember that it is impossible to directly Please remember that it is impossible to directly transform an x value into a proportion or transform an x value into a proportion or probability without first converting the x value probability without first converting the x value into a z-score and then into a probability or into a z-score and then into a probability or proportion proportion

Probability and the Binomial Probability and the Binomial Distribution for Two OutcomesDistribution for Two Outcomes

Binomial distributions are formed by a series Binomial distributions are formed by a series of observations (for example, 100 coin tosses) of observations (for example, 100 coin tosses) for which there are exactly two possible for which there are exactly two possible outcomes (heads and tails)outcomes (heads and tails)

The two outcomes are identified as A and B, The two outcomes are identified as A and B, with probabilities of with probabilities of pp((AA) = ) = pp and and pp((BB) = ) = qq. .

pp + + qq = 1.00 = 1.00

The distribution shows the probability for each The distribution shows the probability for each value ofvalue of X X, where , where XX is the number of is the number of occurrences of occurrences of AA in a series of in a series of nn observations. observations.

Probability and the Binomial Probability and the Binomial Distribution (cont'd.)Distribution (cont'd.)

When When pnpn and and qnqn are both equal to or greater than 10 are both equal to or greater than 10, the , the binomial distribution is closely approximated by a normal binomial distribution is closely approximated by a normal distribution with a mean of μ = distribution with a mean of μ = pnpn and a standard deviation of and a standard deviation of σ = σ = npqnpq. .

In this situation, a In this situation, a zz-score can be computed for each value of -score can be computed for each value of XX and the unit normal table can be used to determine and the unit normal table can be used to determine probabilities for specific outcomes. probabilities for specific outcomes.

Within the normal distribution, each value of X has a Within the normal distribution, each value of X has a corresponding z-score as follows:corresponding z-score as follows:

Binomial DistributionsBinomial Distributions

Probability and Inferential Probability and Inferential StatisticsStatistics

Probability is important because it establishes Probability is important because it establishes a link between samples and populations. a link between samples and populations.

For any known population, it is possible to For any known population, it is possible to determine the probability of obtaining any determine the probability of obtaining any specific sample. specific sample.

In later chapters, we will use this link as the In later chapters, we will use this link as the foundation for inferential statistics. foundation for inferential statistics.

Probability and Inferential Probability and Inferential Statistics (cont'd.)Statistics (cont'd.)

The general goal of inferential statistics is to The general goal of inferential statistics is to use the information from a sample to reach a use the information from a sample to reach a general conclusion (inference) about an general conclusion (inference) about an unknown population. unknown population.

Typically a researcher begins with a sample. Typically a researcher begins with a sample.

Probability and Inferential Probability and Inferential Statistics (cont'd.)Statistics (cont'd.)

If the sample has a high probability of being If the sample has a high probability of being obtained from a specific population, then the obtained from a specific population, then the researcher can conclude that the sample is researcher can conclude that the sample is likely to have come from that population. likely to have come from that population.

If the sample has a very low probability of If the sample has a very low probability of being obtained from a specific population, being obtained from a specific population, then it is reasonable for the researcher to then it is reasonable for the researcher to conclude that the specific population is conclude that the specific population is probably not the source for the sample.probably not the source for the sample.

Research Study - - Likelihood of Predicting a Card’s Suit 15 Research Study - - Likelihood of Predicting a Card’s Suit 15 Times in a Row for in 48 TrialsTimes in a Row for in 48 Trials

Actual Score Value

Real Limits

Thus, z = 1.17 and p = .1210 or 12.10%

μ = pn

μ = (1/4)*48 = 12

qn = (3/4)*48

Research StudyResearch StudyThe goal of this study is to determine whether the treatment has an effect.

As a primer for the next chapter, extreme effects are considered those that are defined by scores that are very unlikely to be obtained from the original population by random chance, thus providing evidence of treatment effects.

Cutoff scores for 1-tail tests:Scores with p<0.05 (z = 1.65)Scores with p<0.01 (z = 2.33)Scores with p<0.001 (z = 3.11)

Cutoff scores for 2-tail tests:Scores with p<0.05 (z = +/-1.96) w/0.025 per tailScores with p<0.01 (z = +/-2.58) w/0.005 per tailScores with p<0.001 (z = +/-3.30) w/0.0005 per

tailValue as discussed in book on page 190-191

If the x-score is 440, is this an extreme value as defined by

the book?