Introductory Statistics and Data Analysis MAT 135 Introductory Statistics and Data Analysis Adjunct Instructor Kenneth R. Martin Lecture 11 November 9, 2016

Confidential - Kenneth R. Martin Agenda Housekeeping Exam #2 Readings Chapter 1, 14, 10, 2, 3, 4 & 5 Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Housekeeping Exam #2 Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Housekeeping Read, Chapter 1.1 – 1.4 Read, Chapter 14.1 – 14.2 Read, Chapter 10.1 Read, Chapter 2 Read, Chapter 3 Read, Chapter 4 Read, Chapter 5 Confidential - Kenneth R. Martin

Continuous vs. Discrete vs. Attribute Data infinite # of possible measurements in a continuum 1 2 3 4 5 6 7 8 9 10 Discrete: Count Discrete: Ordinal 1 2 3 4 5 6 7 8 9 10 “low”/“small”/“short” “high”/”large”/”tall” “medium” / “mid” Discrete: Nominal or Categorical defines several groups - no order Group A Group B Group C Group D Group E Group F Attribute: Binary “bad”/“no-go”/”group #1” “good”/“go”/”group #2 defines TWO groups - no order Confidential - Kenneth R. Martin

Discrete Probability Distribution Examples: Confidential - Kenneth R. Martin

Discrete Probability Distribution Examples: Confidential - Kenneth R. Martin

Discrete Probability Distribution Examples: Confidential - Kenneth R. Martin

Discrete Probability Distribution Theorem 1: Probability of 1.000 means an event is certain to occur Probability of 0 means the event is certain to NOT occur. Therefore: 0  P(E)  1 Confidential - Kenneth R. Martin

Discrete Probability Distribution Theorem 5: The total (sum) of the probabilities, for any discrete distribution, of all situations equals to 1.000 Confidential - Kenneth R. Martin

Discrete Probability Distribution Theoretical Mean: Confidential - Kenneth R. Martin

Discrete Probability Distribution Theoretical Mean - Example: Confidential - Kenneth R. Martin

Discrete Probability Distribution Variance: Confidential - Kenneth R. Martin

Discrete Probability Distribution Variance - Example: Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution: Used for discrete single point (Integer) probabilities. A Binomial probability distribution occurs when there’s a fixed number of “trials” or where there’s a steady stream of items coming from a source. Used for data with two outcomes, (pass / fail, head / tail, etc.); the events are independent, and probability of outcomes do not change. Uses Combination and Simple Multiplication Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution: P(d) = Prob. of d nonconforming or target units in sample size n n = # units in sample d = # nonconforming or target units in a sample p0 = proportion nonconforming / targets in population (lot) q0 = proportion conforming / not a target (1-p0) in population (lot) Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution – Mean / Var. & SD: Confidential - Kenneth R. Martin

Discrete Probability Distributions Binomial Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Hypergeometric Distribution: Used for discrete single point (Integer) probabilities. A Hypergeometric probability distribution occurs when the population is finite, two outcomes are possible, and the random sample is taken without replacement (trials are not Independent). Uses three Combinations and Simple Multiplication. Confidential - Kenneth R. Martin

Discrete Probability Distributions Hypergeometric Distribution: P(d) = Prob. of d nonconforming / target units in sample size n N = # units in the lot (population) n = # units in the sample D = # nonconforming / target units in the lot d = # nonconforming / target units in the sample N-D = # conforming / not a target in the lot n-d = # conforming / not a target in the sample C = Combinations Confidential - Kenneth R. Martin

Discrete Probability Distributions Hypergeometric Distribution (example): = 3 * 20 126 Confidential - Kenneth R. Martin

Discrete Probability Distributions Hypergeometric Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution: Use for discrete single point (Integer) probabilities. A Poisson probability distribution occurs when n is large and p0 is small. Used for applications of observations per time, or observations per quantity. Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution: X = occurrences of events occurring in a sample. λ = average count of events occurring per unit. e = 2.718281 Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution Table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution (alternate): C = count of events occurring in a sample, i.e. count of non-conformities. np0 = average count of events occurring in population. e = constant = 2.718281 Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution: The Poisson distribution formula can be used directly to find probability estimates, or Table C can be used. The table gives point values, and cumulative (parenthesis from top - down) Mean = np0 SD = (np0)1/2 Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution (example): Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution Table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution Table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution Table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution Table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution Table: Confidential - Kenneth R. Martin

Discrete Probability Distributions Poisson Distribution (example): Confidential - Kenneth R. Martin

Continuous vs. Discrete vs. Attribute Data infinite # of possible measurements in a continuum 1 2 3 4 5 6 7 8 9 10 Discrete: Count Discrete: Ordinal 1 2 3 4 5 6 7 8 9 10 “low”/“small”/“short” “high”/”large”/”tall” “medium” / “mid” Discrete: Nominal or Categorical defines several groups - no order Group A Group B Group C Group D Group E Group F Attribute: Binary “bad”/“no-go”/”group #1” “good”/“go”/”group #2 defines TWO groups - no order Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Probability - Review Theorem 5: The total (sum) of the probabilities, for any discrete distribution, of all situations equals to 1.000 Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Probability - Review Definition, Theorem 5: Correspondingly, the total area under a continuous probability distribution (normal curve) is equal to 1.000 also. The tails of the curve never touch the x-axis. Thus, area can be used to estimate probabilities. Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Histogram – by increasing data and thus bins, the fitted line becomes smoother and more accurate Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Histogram – by increasing data and thus bins, the fitted line becomes smoother and more accurate Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Histogram – by increasing data and thus bins, the fitted line becomes smoother and more accurate Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics By increasing data, you approach the population, and get a smooth polygon. Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Area Under Curve We can find the area under any curve by 2 methods. We can make a large quantity of really narrow bins, find each individual bin area / rectangle area (under the curve), and add them all up. We can integrate under the curve, to find the area bound by the curve and the X-axis. This method is simpler, and gives more accurate results. Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Equation of a Normal Distribution Y = Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Area Under Curve We may wish to find the area under the curve when, for example: We want to find the number of students whose final semester grade falls between standard grade lettering schemes, and we have a collection of student scores. Or if we want to find the number of people who arrive at a fast food restaurant chain after 11 am, and we have the associated data. Etc. Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Continuous Probability Distribution (aka. CRV) A function of a Continuous Random Variable that describes the likelihood the variable occurs at a certain value within a given set of points by the integral of its density (prob. density) function (i.e. corresponding area under f(x) curve). We shall calculate CRV over ranges Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Continuous Probability Distribution (aka. CRV) So we are seeking to find the area under some curve, y=f(x), bounded by the X-axis, between some values along the x-axis. Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Probability Density Function (cont. prob. dist.) f ( X ) = PDF f ( X) a b X Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Cumulative Density Function – Cross Section f ( X ) = PDF +∞ ∫f(X) dx = 1.0 -∞ f ( X) Sum under entire curve = 1.0 X Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Probability Density Function (cont. prob. dist.) = p(x≤b) - p(x≤a) = F(b) - F(a) = Entire area under curve to section(b) minus Entire area under curve to section(a) Sum under entire curve = 1.0 Curve typically read left to right Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Cumulative Density Function f ( X ) = PDF t P(X<t)=∫f(X) dx = F(t) -∞ f(X) t F(t) X Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Cumulative Density Function F(t) + R(t) = 1.0 Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Normal Curve AKA, Gaussian distribution of CRV. Mean, Median, and Mode have the approx. same value. Associated with mean () at center and dispersion () X  N(,) [when a random variable x is distributed normally] Observations have equal likelihood on both sides of mean *** When normally distributed, Mean is used to describe Central Tendency The graph of the associated probability density function is called “Bell Shaped” Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Normal Curve Developed from a frequency histogram, with  sample size,  intervals (bin width), the associated curve becomes smooth. Typical of much data and distributions in reality. The basis for most quality control techniques, formulas, and assumptions. However, different Normal Distributions (pdf’s) can have varying means and SD’s. The means and SD’s are independent (i.e. the mean does not effect the SD, and vice versa) Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Various Normal Curves (Different means, common SD) Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Various Normal Curves (Different SD’s, common means) Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Various Normal Curves Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standardized Normal Value There are an infinite combination of mean and SD’s for normal curves. Thus, the shapes of any two normal curves will be different. To find the area under any normal curve, we can use the two methods previously described (rectangles and integration). Or, we can use the Standard Normal Approach, thus using tables to find the area under the curve, and thus probabilities. Standard Normal Distribution: N (0,1) Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standardized Normal Value Standard Normal Distribution has a Mean=0 and a SD=1 Standard Normal Transformation (z-Transformation), converts any normal distribution with any mean and any SD to a Standard Normal Distribution with mean 0 and SD 1 Standard Normal Distribution is distributed in “z-score” units, along the associated x-axis. Z-score specifies the number of SD units a value is above or below the mean (i.e. z = +1 indicates a value 1 SD above the mean). A formula is used to convert your mean and SD to a z-score. Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Normal Curve - Distribution of Data Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standard Normal Curve - Distribution of Data (z-scores) Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Normal Curve - Distribution of Data Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standard Normal Distribution (z-scores) Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standardized Normal Value Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Normal distribution example Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standard Normal Distribution example Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standardized Normal Table Confidential - Kenneth R. Martin

Confidential - Kenneth R. Martin Statistics Standardized Normal Table Confidential - Kenneth R. Martin