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Quality Control Chapter 4- Fundamentals of Statistics PowerPoint presentation to accompany Besterfield Quality Control, 8e PowerPoints created by Rosida Coowar Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Outline Introduction Frequency Distribution Measures of Central Tendency Measures of Dispersion Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Outline-Continued Other Measures Concept of a Population and Sample The Normal Curve Tests for Normality Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Introduction Definition of Statistics: 1. A collection of quantitative data pertaining to a subject or group. Examples are blood pressure statistics etc. 2. The science that deals with the collection, tabulation, analysis, interpretation, and presentation of quantitative data Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Introduction Two phases of statistics: Descriptive Statistics: Describes the characteristics of a product or process using information collected on it. Inferential Statistics (Inductive): Draws conclusions on unknown process parameters based on information contained in a sample. Uses probability Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Collection of Data Types of Data: Attribute: Discrete data. Data values can only be integers. Counted data or attribute data. Examples include: How many of the products are defective? How often are the machines repaired? How many people are absent each day? Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Collection of Data – Cont’d. Types of Data: Attribute: Discrete data. Data values can only be integers. Counted data or attribute data. Examples include: How many days did it rain last month? What kind of performance was achieved? Number of defects, defectives Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Collection of Data Types of Data: Variable: Continuous data. Data values can be any real number. Measured data. Examples include: How long is each item? How long did it take to complete the task? What is the weight of the product? Length, volume, time Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Precision and Accuracy Precision The precision of a measurement is determined by how reproducible that measurement value is. For example if a sample is weighed by a student to be 42.58 g, and then measured by another student five different times with the resulting data: 42.09 g, 42.15 g, 42.1 g, 42.16 g, 42.12 g Then the original measurement is not very precise since it cannot be reproduced. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Precision and Accuracy Accuracy The accuracy of a measurement is determined by how close a measured value is to its “true” value. For example, if a sample is known to weigh 3.182 g, then weighed five different times by a student with the resulting data: 3.200 g, 3.180 g, 3.152 g, 3.168 g, 3.189 g The most accurate measurement would be 3.180 g, because it is closest to the true “weight” of the sample. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Precision and Accuracy Figure 4-1 Difference between accuracy and precision Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Describing Data Frequency Distribution Measures of Central Tendency Measures of Dispersion Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Frequency Distribution Ungrouped Data Grouped Data Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Frequency Distribution 2-7There are three types of frequency distributions Categorical frequency distributions Ungrouped frequency distributions Grouped frequency distributions Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Categorical 2-7Categorical frequency distributions Can be used for data that can be placed in specific categories, such as nominal- or ordinal-level data. Examples - political affiliation, religious affiliation, blood type etc. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Categorical 2-8 Example :Blood Type Frequency Distribution Class Frequency Percent A 5 20 B 7 28 O 9 36 AB 4 16 Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Ungrouped 2-9Ungrouped frequency distributions Ungrouped frequency distributions - can be used for data that can be enumerated and when the range of values in the data set is not large. Examples - number of miles your instructors have to travel from home to campus, number of girls in a 4-child family etc. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Ungrouped 2-10 Example :Number of Miles Traveled Class Frequency 5 24 10 16 15 10 Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Grouped 2-11 Grouped frequency distributions Can be used when the range of values in the data set is very large. The data must be grouped into classes that are more than one unit in width. Examples - the life of boat batteries in hours. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Grouped 2-12 Example: Lifetimes of Boat Batteries Class limits Class Frequency Cumulative Boundaries frequency 24 - 37 23.5 - 37.5 4 4 38 - 51 37.5 - 51.5 14 18 52 - 65 51.5 - 65.5 7 25 Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Frequency Distributions Number non conforming Frequency Relative Frequency Cumulative Frequency Relative Frequency 0 15 0.29 15 0.29 1 20 0.38 35 0.67 2 8 0.15 43 0.83 3 5 0.10 48 0.92 4 3 0.06 51 0.98 5 1 0.02 52 1.00 Table 4-3 Different Frequency Distributions of Data Given in Table 4-1 Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Frequency Histogram Frequency Histogram 25 Frequency 20 15 10 5 0 0 1 2 3 4 5 Number Nonconforming Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Relative Frequency Histogram Relative Frequency Histogram 0.45 Relative Frequency 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 Number Nonconforming Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Cumulative Frequency Histogram Cumulative Frequency Histogram Cumulative Frequency 60 50 40 30 20 10 0 0 1 2 3 4 5 Number Nonconforming Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved The Histogram The histogram is the most important graphical tool for exploring the shape of data distributions. Check: http://quarknet.fnal.gov/toolkits/ati/histograms.html for the construction ,analysis and understanding of histograms Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Constructing a Histogram The Fast Way Step 1: Find range of distribution, largest smallest values Step 2: Choose number of classes, 5 to 20 Step 3: Determine width of classes, one decimal place more than the data, class width = range/number of classes # classes n Step 4: Determine class boundaries Step 5: Draw frequency histogram Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Constructing a Histogram Number of groups or cells If no. of observations < 100 – 5 to 9 cells Between 100-500 – 8 to 17 cells Greater than 500 – 15 to 20 cells Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Constructing a Histogram For a more accurate way of drawing a histogram see the section on grouped data in your textbook Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Other Types of Frequency Distribution Graphs Bar Graph Polygon of Data Cumulative Frequency Distribution or Ogive Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Bar Graph and Polygon of Data Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Cumulative Frequency Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Characteristics of Frequency Distribution Graphs Figure 4-6 Characteristics of frequency distributions Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Analysis of Histograms Figure 4-7 Differences due to location, spread, and shape Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Analysis of Histograms Figure 4-8 Histogram of Wash Concentration Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Measures of Central Tendency The three measures in common use are the: Average Median Mode Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Average There are three different techniques available for calculating the average three measures in common use are the: Ungrouped data Grouped data Weighted average Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Average-Ungrouped Data n Xi X i 1 n Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Average-Grouped Data h fi X i X i 1 n f1 X 1 f 2 X 2 ... f h X h . f1 f 2 ... f h h = number of cells Xi=midpoint Besterfield: Quality Control, 8th ed.. fi=frequency © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Average-Weighted Average Used when a number of averages are combined with different frequencies w X i1 i i n Xw n w i 1 Besterfield: Quality Control, 8th ed.. i © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Median-Grouped Data M d Lm n cf m 2 fm i Lm=lower boundary of the cell with the median N=total number of observations Cfm=cumulative frequency of all cells below m Fm=frequency of median cell i=cell interval Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Example Problem Boundaries Midpoint Frequency Computation 23.6-26.5 25.0 4 100 26.6-29.5 28.0 36 1008 29.6-32.5 31.0 51 1581 32.6-35.5 34.0 63 2142 35.6-38.5 37.0 58 2146 38.6-41.5 40.0 52 2080 41.6-44.5 43.0 34 1462 44.6-47.5 46.0 16 736 47.6-50.5 49.0 6 294 320 11549 Total Table 4-7 Frequency Distribution of the Life of 320 tires in 1000 km Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Median-Grouped Data M d Lm n 2 cf m fm i Using data from Table 4-7 320 154 2 Md 35.6 3 35.9 58 Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Mode The Mode is the value that occurs with the greatest frequency. It is possible to have no modes in a series or numbers or to have more than one mode. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Relationship Among the Measures of Central Tendency Figure 4-9 Relationship among average, median and mode Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Measures of Dispersion Range Standard Deviation Variance Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Measures of Dispersion-Range The range is the simplest and easiest to calculate of the measures of dispersion. Range = R = Xh - Xl Largest value - Smallest value in data set Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Measures of Dispersion-Standard Deviation Sample Standard Deviation: S Besterfield: Quality Control, 8th ed.. i 1 ( Xi X ) 2 n 1 2 Xi / n i 1 Xi i 1 n 1 n S n n 2 © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Standard Deviation Ungrouped Technique n i 1 Xi (i 1 Xi ) n S Besterfield: Quality Control, 8th ed.. 2 n 2 n(n 1) © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Standard Deviation Grouped Technique s h n i 1 ( f i X ) ( fi X i ) h Besterfield: Quality Control, 8th ed.. 2 i 2 i 1 n(n 1) © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Relationship Between the Measures of Dispersion As n increases, accuracy of R decreases Use R when there is small amount of data or data is too scattered If n> 10 use standard deviation A smaller standard deviation means better quality Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Relationship Between the Measures of Dispersion Figure 4-10 Comparison of two distributions with equal average and range Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Other Measures There are three other measures that are frequently used to analyze a collection of data: Skewness Kurtosis Coefficient Besterfield: Quality Control, 8th ed.. of Variation © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Skewness Skewness is the lack of symmetry of the data. For grouped data: a3 Besterfield: Quality Control, 8th ed.. h f ( X X ) / n i i i 1 3 s 3 © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Skewness Figure 4-11 Left (negative) and right (positive) skewness distributions Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Kurtosis Kurtosis provides information regrading the shape of the population distribution (the peakedness or heaviness of the tails of a distribution). For grouped data: a4 Besterfield: Quality Control, 8th ed.. h f ( X X ) / n i i i 1 4 s 4 © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Kurtosis Figure 4-11 Leptokurtic and Platykurtic distributions Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Coefficient of Variation Correlation variation (CV) is a measure of how much variation exists in relation to the mean. s (100%) CV X Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Population and Sample Population Set of all items that possess a characteristic of interest Sample Subset of a population Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Parameter and Statistic Parameter is a characteristic of a population, i.o.w. it describes a population Example: average weight of the population, e.g. 50,000 cans made in a month. Statistic is a characteristic of a sample, used to make inferences on the population parameters that are typically unknown, called an estimator Example: average weight of a sample of 500 cans from that month’s output, an estimate of the average weight of the 50,000 cans. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved The Normal Curve Characteristics of the normal curve: It is symmetrical -- Half the cases are to one side of the center; the other half is on the other side. The distribution is single peaked, not bimodal or multi-modal Also known as the Gaussian distribution Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved The Normal Curve Characteristics: Most of the cases will fall in the center portion of the curve and as values of the variable become more extreme they become less frequent, with "outliers" at the "tail" of the distribution few in number. It is one of many frequency distributions. Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Standard Normal Distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula: Z Besterfield: Quality Control, 8th ed.. X i © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Relationship between the Mean and Standard Deviation Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Mean and Standard Deviation Same mean but different standard deviation Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Mean and Standard Deviation Same mean but different standard deviation Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Normal Distribution IF THE DISTRIBUTION IS NORMAL Then the mean is the best measure of central tendency Most scores “bunched up” in middle Extreme scores are less frequent, therefore less probable Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Normal Distribution Percent of items included between certain values of the std. deviation Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Tests for Normality Histogram Skewness Kurtosis Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Tests for Normality Histogram: Shape Symmetrical The larger the sampler size, the better the judgment of normality. A minimum sample size of 50 is recommended Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Tests for Normality Skewness (a3) and Kurtosis (a4)” Skewed to the left or to the right (a3=0 for a normal distribution) The data are peaked as the normal distribution (a4=3 for a normal distribution) The larger the sample size, the better the judgment of normality (sample size of 100 is recommended) Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Tests for Normality Probability Plots Order the data from the smallest to the largest Rank the observations (starting from 1 for the lowest observation) Calculate the plotting position 100(i 0.5) PP n Where i = rank PP=plotting position Besterfield: Quality Control, 8th ed.. n=sample size © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Probability Plots Procedure: Order the data Rank the observations Calculate the plotting position Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Probability Plots Procedure cont’d: Label the data scale Plot the points Attempt to fit by eye a “best line” Determine normality Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Probability Plots Procedure cont’d: Order the data Rank the observations Calculate the plotting position Label the data scale Plot the points Attempt to fit by eye a “best line” Determine normality Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Chi-Square Goodness of Fit Test Chi-Square Test 2 ( O E ) i i 2 Ei i 1 k Where 2 Chi-squared Oi Observed value in a cell E i Expected value for a cell Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Scatter Diagram The simplest way to determine if a cause and-effect relationship exists between two variables Figure 4-19 Scatter Diagram Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Scatter Diagram Supplies the data to confirm a hypothesis that two variables are related Provides both a visual and statistical means to test the strength of a relationship Provides a good follow-up to cause and effect diagrams Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved Straight Line Fit xy [( x )( y ) / n m x [( x ) / n] a y / n m( x / n ) 2 2 y a mx Where m=slope of the line and a is the intercept on the y axis Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458. All rights reserved