© 2004 Prentice-Hall, Inc.Chap 1-1 Basic Business Statistics (11 th Edition) Review of Stat 1 Key Concepts -Descriptive Statistics and Normal Probabilities-

© 2004 Prentice-Hall, Inc. Chap 1-2 Why a Manager Needs to Know About Statistics To Know How to Properly Present Information To Know How to Draw Conclusions about Populations Based on Sample Information To Know How to Improve Processes To Know How to Obtain Reliable Forecasts

Basic Business Statistics, 10e © 2006 Prentice- Hall, Inc.Chap 1-3 What is Statistics?? Statistics consists of a set of rules, formulas, and theorems which can be applied to numerical information (data) in order to address problems or answer questions dealing with uncertainty. or Statistics is a tool box with tools that we can apply to numerical information in order to organize, describe, and apply those data to problem solve.

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 1-4 Basic Concepts of Statistics Statistics is concerned with: data Collecting, Organizing, and Analyzing numerical information (i.e. data) Presenting, and Interpreting data to assist decision makers in Answering Questions Solving Problems Making Decisions

© 2004 Prentice-Hall, Inc. Chap 1-5 Some Important Definitions VARIABLE A variable is a characteristic of an item or individual. DATA Data are the different values associated with a variable (numerical information) OPERATIONAL DEFINITIONS Data values are meaningless unless their variables have operational definitions, universally accepted meanings that are clear to all associated with an analysis.

© 2004 Prentice-Hall, Inc. Chap 1-6 Some Important Definitions Population A Population (Universe) is the Whole Collection of Things Under Consideration Sample A Sample is a Portion of the Population Selected for Analysis Parameter A Parameter is a Summary Measure Computed to Describe a Characteristic of the Population Statistic A Statistic is a Summary Measure Computed to Describe a Characteristic of the Sample

© 2004 Prentice-Hall, Inc. Chap 1-7 From Business Stat 1, Remember How To: Construct Frequency Distributions and Histograms Calculate Measures of Central Tendency/Location Mean, Median, Mode, Quartiles Calculate Measures of Dispersion Range, Variance, Standard Deviation, Coefficient of Variation Calculate Normal Probabilities (using PhStat)

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 1-8 Population vs. Sample PopulationSample Measures used to describe the population are called parameters Measures computed from sample data are called statistics

© 2004 Prentice-Hall, Inc. Chap 1-9 Population and Sample PopulationSample Use parameters to summarize features Use statistics to summarize features Inference on the population from the sample

© 2004 Prentice-Hall, Inc. Chap 1-10 Reasons for Drawing a Sample Less Time Consuming Than a Census TIME Less Costly to Administer Than a Census COST Less Cumbersome and More Practical to Administer Than a Census of the Population CONVENIENCE

© 2004 Prentice-Hall, Inc. Chap 1-11 Statistical Methods Descriptive Statistics Collecting, presenting, and characterizing data Inferential Statistics Drawing conclusions and/or making decisions concerning a population based only on sample data Estimation and Hypothesis Testing

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 3-12 Using Excel Use menu choice: tools / data analysis / descriptive statistics

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 3-13 Enter dialog box details Check box for summary statistics Click OK Using Excel (continued)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 3-14 Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000, , , , ,000

© 2004 Prentice-Hall, Inc. Chap 1-15 From Business Stat 1, Remember How To: Construct Frequency Distributions and Histograms Calculate Measures of Central Tendency/Location Mean, Median, Mode, Quartiles Calculate Measures of Dispersion Range, Variance, Standard Deviation, Coefficient of Variation Calculate Normal Probabilities (using PhStat)

© 2004 Prentice-Hall, Inc. Chap 1-16 Measures of Central Tendency (Location) Mean Median - The median is the middle value of data once the data are ordered from lowest to highest. If the sequence contains an odd number of observations, the median is the middle value in the ordered sequence. If there are an even number of observations, the median is the average of the two center values. Mode - The mode of a data set is the value that occurs with the greatest frequency.

© 2004 Prentice-Hall, Inc. Chap 1-17 Population and Sample Mean Population Mean (“Mu”) Sample Mean (“X-Bar”)

© 2004 Prentice-Hall, Inc. Chap 1-18 Data: Thirteen CEO’s salaries ($1000’s) 141, 152, 154, 165, 171, 127, 241, 132, 138, 177, 144, 146, 192 n X X n   1= i i = …+ 192 = 2080 = Example of Sample Mean Calculation

© 2004 Prentice-Hall, Inc.Chap 1-19 The median is the middle value of data once the data are ordered from lowest to highest. If the sequence contains an odd number of observations, the median is the middle value in the ordered sequence. If there are an even number of observations, the median is the average of the two center values. The mode of a data set is the value that occurs with the greatest frequency. Median and Mode

© 2004 Prentice-Hall, Inc.Chap 1-20 Arrange the data from lowest salary to highest: 127, 132, 138, 141, 144, 146, 152, 154, 165, 171, 177, 192, 241 Note: 152 is the 7th or middle value when = 13. If a fourteenth lawyer joined the sample with a salary of $148 (000), then the median would become: /2 = 150. Example of Median Calculation Data: Thirteen CEO salaries ($1000’s) 141, 152, 154, 165, 171, 127, 241, 132, 138, 177, 144, 146, 192

© 1998 Brooks/Cole Publishing Co./ITP© 2004 Prentice-Hall, Inc. Chap 1-21 Relationships Between the Mean, Median, and Mode Mean Median ModeMean Median Mode Mode Median Mean Leftward SkewedSymmetricRightward Skewed

© 2004 Prentice-Hall, Inc. Chap 1-22 Midrange A Measure of Central Tendency Average of Smallest and Largest Observation: Affected by Extreme Value Midrange Midrange = 5

© 2004 Prentice-Hall, Inc. Chap 1-23 Quartiles Not a Measure of Central Tendency Split Ordered Data into 4 Quarters Position of i-th Quartile: position of point 25% Q1Q1 Q2Q2 Q3Q3 Q i(n+1) i  4 Data in Ordered Array: Position of Q 1 = 2.50 Q1Q1 =12.5 = 1(9 + 1) 4

© 2004 Prentice-Hall, Inc. Chap 1-24 Midhinge A Measure of Central Tendency The Middle point of 1st and 3rd Quarters Not Affected by Extreme Values Midhinge = Data in Ordered Array: Midhinge =

© 2004 Prentice-Hall, Inc.Chap 1-25 range The range of a set of observations is the difference between the largest value and the smallest value. Measures of Dispersion: The Range

© 2004 Prentice-Hall, Inc.Chap 1-26 Data: Thirteen CEO salaries ($1000’s) 141, 152, 154, 165, 171, 127, 241, 132, 138, 177, 144, 146, 192 Range = 241 – 127 = $114 (000) Measures of Dispersion: The Range Example of Range Calculation

© 2004 Prentice-Hall, Inc. Chap 1-27 Deviations from the Mean For any value x, the deviation from the mean is the difference between the x value and the mean… (x -  ) for populations and (x – X ) for samples.

© 2004 Prentice-Hall, Inc.Chap 1-28 The variance is the average squared deviation from the mean. Measures of Dispersion: The Variance The Variance  2 =  (X -  ) 2 /N Population Variance (“sigma-squared”) S 2 =  (X - X) 2 /n – 1 Sample Variance (“s-squared”)

© 2004 Prentice-Hall, Inc.Chap 1-29 The following formula may be used for calculating the sample variance because of its convenience: Measures of Dispersion: The Variance (cont.) The Variance (cont.)

© 2004 Prentice-Hall, Inc.Chap 1-30 The standard deviation is simply the square root of the variance…for both populations and for samples: Measures of Dispersion: The Standard Deviation

© 2004 Prentice-Hall, Inc.Chap 1-31 Data: Thirteen CEO salaries ($1000’s) 141, 152, 154, 165, 171, 127, 241, 132, 138, 177, 144, 146, 192 Measures of Dispersion: Example of Sample Variance Calculation

© 2004 Prentice-Hall, Inc.Chap 1-32 Data: Thirteen CEO salaries ($1000’s) Measures of Dispersion Example of Sample Variance Calculation: ALTERNATIVE FORMULA

© 2004 Prentice-Hall, Inc.Chap 1-33 The coefficient of variation (CV) is a measure of variability that overcomes the difficulties that might arise when comparing absolute measures of variation such as the variance and standard deviation. The CV expresses the standard deviation as a percentage of the mean: CV = (  /  ) * 100% or CV = (s/X) * 100% Coefficient of Variation

© 2004 Prentice-Hall, Inc.Chap 1-34 Chebyshev’s Theorem Chebyshev’s theorem: For any set of observations (sample or population), the proportion of the values that lie fewer than c standard deviations of the mean is at least where c is any number greater than 1

© 2004 Prentice-Hall, Inc.Chap 1-35 Chebyshev’s Theorem 2 standard deviations At least 75% of the data lie inside this range

© 2004 Prentice-Hall, Inc.Chap % 95% 99.7% Empirical Rule: When the distribution of a population or sample of data is approximately bell-shaped, approximately 68% of the observations will lie within 1 standard deviation of the mean; approximately 95% will fall within 2 standard deviations of the mean; and approximately 99.7% will fall within 3 standard deviations of the mean. The Empirical Rule

Pareto Diagram Used to portray categorical data (nominal scale) A bar chart, where categories are shown in descending order of frequency A cumulative polygon is often shown in the same graph Used to separate the “vital few” from the “trivial many”

Pareto Diagram Example cumulative % invested (line graph) % invested in each category (bar graph) Current Investment Portfolio

Data in raw form (as collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Data in ordered array from smallest to largest: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 (continued) The Ordered Array

Stem-and-Leaf Diagram A simple way to see distribution details in a data set METHOD: Separate the sorted data series into leading digits (the stem) and the trailing digits (the leaves)

Example Here, use the 10’s digit for the stem unit: Data in ordered array: 21, 24, 24, 26, 27, 27, 30, 32, 38, is shown as 38 is shown as 41 is shown as Stem Leaf

Graphing Numerical Data: The Histogram A graph of the data in a frequency distribution is called a histogram The class boundaries (or class midpoints) are shown on the horizontal axis the vertical axis is either frequency, relative frequency, or percentage Bars of the appropriate heights are used to represent the number of observations within each class

Class Midpoints Histogram Example (No gaps between bars) Class 10 but less than but less than but less than but less than but less than Frequency Class Midpoint

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-44 Introduction to Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 6-45 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

© 2004 Prentice-Hall, Inc. Chap 1-46 From Business Stat 1, Remember How To: Construct Frequency Distributions and Histograms Calculate Measures of Central Tendency/Location Mean, Median, Mode, Quartiles Calculate Measures of Dispersion Range, Variance, Standard Deviation, Coefficient of Variation Calculate Normal Probabilities (using PhStat)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 3-47 Using Excel Use menu choice: tools / data analysis / descriptive statistics

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 3-48 Enter dialog box details Check box for summary statistics Click OK Using Excel (continued)

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 3-49 Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000, , , , ,000

© 2004 Prentice-Hall, Inc.Chap 1-50 Chebyshev’s Theorem Chebyshev’s theorem: For any set of observations (sample or population), the proportion of the values that lie fewer than c standard deviations of the mean is at least where c is any number greater than 1

© 2004 Prentice-Hall, Inc.Chap 1-51 Chebyshev’s Theorem 2 standard deviations At least 75% of the data lie inside this range

© 2004 Prentice-Hall, Inc.Chap % 95% 99.7% Empirical Rule: When the distribution of a population or sample of data is approximately bell-shaped, approximately 68% of the observations will lie within 1 standard deviation of the mean; approximately 95% will fall within 2 standard deviations of the mean; and approximately 99.7% will fall within 3 standard deviations of the mean. The Empirical Rule

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 5-53 Introduction to Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 6-54 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

© 2004 Prentice-Hall, Inc.Chap 1-55 Many variables have continuous distributions… For a continuous distribution, the area betweena and b represents For a continuous distribution, the area between a and b represents P (a < X < b) a b Continuous Distributions and Probabilities

© 2004 Prentice-Hall, Inc.Chap 1-56 The Normal Distribution  22 Many real-life continuous variables belong in the family of normal probability distributions!

© 2004 Prentice-Hall, Inc.Chap 1-57 Characteristics of the Normal Curve n The curve is bell-shaped and symmetric. n The curve extends from -  to +  n The total area under the curve is equal to 1. n The curve is always above the x-axis. n The mean, median, and mode are all equal to the parameter. n The mean, median, and mode are all equal to the parameter .

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 6-58 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode X f(X) μ σ

© 2004 Prentice-Hall, Inc.Chap 1-59 Two Normal Curves - same mean, different variances 1212 2222 The Normal Distribution (cont.)

© 2004 Prentice-Hall, Inc.Chap 1-60 Two Normal Curves - same variance, different means The Normal Distribution (cont.) 11 22

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 6-61 The Normal Distribution Shape X f(X) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. X ~ N(μ, σ 2 )

© 2004 Prentice-Hall, Inc.Chap 1-62 The Standard Normal Distribution 0 X ~ N(0, 1)  2 = 1 A very special member of the family of normal probabilities…any standard normal variable has a mean of 0 and a variance of 1!

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 6-63 The Standardized Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) Need to transform X units into Z units

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 6-64 Translation to the Standardized Normal Distribution Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation: The Z distribution always has mean = 0 and standard deviation = 1

© 2004 Prentice-Hall, Inc. Chap 1-65 Finding Probabilities Probability is the area under the curve! c d X f(X)f(X)

© 2004 Prentice-Hall, Inc. Chap 1-66 Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up!

© 2004 Prentice-Hall, Inc. Chap 1-67 Page 835

© 2004 Prentice-Hall, Inc.Chap 1-68 Areas & Probabilities of the Standard Normal Distribution 0 P(  Z  1.21) = = Area = Area =.3869

© 2004 Prentice-Hall, Inc.Chap P(1.21  Z  2.15) = = Areas & Probabilities of the Standard Normal Distribution

© 2004 Prentice-Hall, Inc. Chap 1-70 Standardize the Normal Distribution One table! Normal Distribution Standardized Normal Distribution

© 2004 Prentice-Hall, Inc. Chap 1-71 Transforming Normal to Standard Normal P( ≤ Z ≤  )=? Solution: Look up the z-scores and  in the standard normal table and find the area!

© 2004 Prentice-Hall, Inc. Chap 1-72 If Karscig’s exam scores are distributed normally with a mean of 80 and a variance of 25, then what’s the probability of obtaining an exam score between 77 and 83? Normal Probability Example

© 2004 Prentice-Hall, Inc. Chap 1-73 Example: Normal Distribution Standardized Normal Distribution  = 5  = 1 83  = 80  =

© 2004 Prentice-Hall, Inc. Chap 1-74 Normal Distribution in PHStat PHStat | Probability & Prob. Distributions | Normal … Example in Excel Spreadsheet

© 2004 Prentice-Hall, Inc. Chap 1-75 More Examples of Normal Distribution Using PHStat A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam?

© 2004 Prentice-Hall, Inc. Chap 1-76 What percentage of students scored between 65 and 89 inclusively? More Examples of Normal Distribution Using PHStat (continued)

© 2004 Prentice-Hall, Inc. Chap 1-77 Only 5% of the students taking the test scored higher than what grade? ? =86.16 (continued) More Examples of Normal Distribution Using PHStat

© 2004 Prentice-Hall, Inc. Chap 1-78 Standardizing Example Normal Distribution

© 2004 Prentice-Hall, Inc. Chap 1-79 Standardizing Example Normal Distribution

© 2004 Prentice-Hall, Inc. Chap 1-80 Standardizing Example Normal Distribution Standardized Normal Distribution P(5 < X < 6.2) with

© 2004 Prentice-Hall, Inc. Chap 1-81 Obtaining the Probability Standardized Normal Probability Table (Portion) ProbabilitiesProbabilities Shaded area exaggerated

© 2004 Prentice-Hall, Inc. Chap Example: Normal Distribution Standardized Normal Distribution.1179

© 2004 Prentice-Hall, Inc.Chap 1-83 Calculating Areas Under Any Normal Curve Example: If is distributed normally with a mean of 100 and a variance of 400, find the probability that is between 80 and 120: Example: If X is distributed normally with a mean of 100 and a variance of 400, find the probability that X is between 80 and 120:

© 2004 Prentice-Hall, Inc. Chap 1-84 Normal Distribution in PHStat PHStat | Probability & Prob. Distributions | Normal … Example in Excel Spreadsheet

© 2004 Prentice-Hall, Inc. Chap 1-85 More Examples of Normal Distribution Using PHStat A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam?

© 2004 Prentice-Hall, Inc. Chap 1-86 What percentage of students scored between 65 and 89 inclusively? More Examples of Normal Distribution Using PHStat (continued)

© 2004 Prentice-Hall, Inc. Chap 1-87 Only 5% of the students taking the test scored higher than what grade? ? =86.16 (continued) More Examples of Normal Distribution Using PHStat

© 2004 Prentice-Hall, Inc. Chap 1-88 The middle 50% of the students scored between what two scores? More Examples of Normal Distribution Using PHStat (continued)