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TMAT 103 Chapter 19 Statistics for Process Control TMAT 103 §19.1 Graphic Presentation of Data §19.1 – Graphic Presentation of Data • Collected data is presented graphically to: – Understand distribution of data – Identify trends • Current • Future – Draw conclusions §19.1 – Graphic Presentation of Data • An automobile manufacturer is analyzing data gathered in regards to cars coming off an assembly line and not starting on the first try. In the month of April, the following data was collected: 42 25 12 44 17 18 27 18 48 30 36 9 33 46 29 14 26 28 31 39 35 31 32 21 29 20 16 45 23 35 §19.1 – Graphic Presentation of Data • A frequency table can make the data more readable: First-Try Nonstarts Tallies Frequency 9 – 16 17 – 24 25 – 32 33 – 40 41 – 48 |||| |||||| |||||||||| ||||| ||||| 4 6 10 5 5 §19.1 – Graphic Presentation of Data • A histogram can also be used: – Rectangles must be of equal width §19.1 – Graphic Presentation of Data • A frequency polygon can also be used: – X coordinate is center of interval, y coordinate is frequency §19.1 – Graphic Presentation of Data • Example: – Using 10 intervals containing 6 numbers each, construct a frequency table, histogram, and frequency polygon for the following situation: A local restaurant counted the number of hamburgers served on 25 consecutive weekends – the data is given below. 268 252 222 279 234 260 261 220 246 268 253 250 228 273 243 268 245 272 254 225 230 240 250 279 231 TMAT 103 §19.2 Measures of Central Tendency §19.2 – Measures of Central Tendency • Central Tendency – Finding a number which describes a set of data – 3 general methods • Mean • Median • Mode §19.2 – Measures of Central Tendency • The Mean – The Mean of a set of n numbers, a1, a2, …, an is given by: a1 a2 ... an x n §19.2 – Measures of Central Tendency • Examples: – 13 students took an exam, with the following scores. Find the mean score. 98, 92, 90, 85, 85, 82, 77, 76, 75, 74, 74, 68, 52 – 10 people were surveyed for their salary. The following data was collected. Find the mean salary. 35,000,000 99,500 88,300 67,200 60,000 59,500 55,300 30,200 25,400 22,000 §19.2 – Measures of Central Tendency • The Median – The Median of an ordered set of n numbers, a1, a2, …, an is the middle number if n is odd, and the mean of the two middle numbers if n is even. §19.2 – Measures of Central Tendency • Examples: – 13 students took an exam, with the following scores. Find the median score. 98, 92, 90, 85, 85, 82, 77, 76, 75, 74, 74, 68, 52 – 10 people were surveyed for their salary. The following data was collected. Find the median salary. 35,000,000 99,500 88,300 67,200 60,000 59,500 55,300 30,200 25,400 22,000 §19.2 – Measures of Central Tendency • The Mode – The Mode of an set of n numbers, a1, a2, …, an is the number which occurs most often. There may be more than one mode. §19.2 – Measures of Central Tendency • Examples: – 13 students took an exam, with the following scores. Find the mode. 98, 92, 90, 85, 85, 82, 77, 76, 75, 74, 74, 68, 52 – 10 people were surveyed for their salary. The following data was collected. Find the mode. 35,000,000 99,500 88,300 67,200 60,000 59,500 55,300 30,200 25,400 22,000 TMAT 103 §19.3 Measures of Dispersion §19.3 – Measures of Dispersion • Terminology – Range • Difference between largest and smallest values – Population • Collection of all items being considered – Sample • Items selected to be in calculation – Random • When each item has an equal chance to be selected – Sample Standard Deviation • One way to measure dispersion §19.3 – Measures of Dispersion • Sample Standard Deviation – The sample standard deviation of a set of data x1, x2, …, xn is given by: ( x1 x ) 2 ( x2 x ) 2 ... ( xn x ) 2 sx n 1 §19.3 – Measures of Dispersion • Example: – A furniture company manufactures 28-in. table legs. Acceptable lengths are between 27.9375 and 28.0625 in. A random sample of 30 legs were measured each day for a week. The number of acceptable legs produced each day were: 41, 41, 43, 44, 46, 46, 48. Find the range and sample standard deviation for this set. TMAT 103 §19.4 The Normal Distribution §19.4 – The Normal Distribution • Normal distribution – Histogram of sample means with smooth curve drawn through centers of rectangles §19.4 – The Normal Distribution • Important features of normal distribution – – – – Bell shaped Symmetric about a vertical line passing through the mean Smaller sx implies more data is closer to the mean Distribution of data is predictable §19.4 – The Normal Distribution • Smaller sx implies more data is closer to the mean §19.4 – The Normal Distribution • Distribution of data is predictable – 68.2% within 1 standard deviation of mean – 95.4% within 2 standard deviations of mean – 99.7% within 3 standard deviations of means §19.4 – The Normal Distribution • Examples: – Does the following set of numbers meet the criteria for a normal distribution in terms of the percent of values with one standard deviation (allow a 2% margin of error)? 35, 36, 38, 43, 47, 48, 48, 55, 67 – The scores from a test resulted in a mean of 72 and standard deviation of 8.5. Mark scored 89. Assuming the scores were normally distributed, what percent of students can he estimate scored below him? TMAT 103 §19.5 Fitting Curves to Data Sets §19.5 – Fitting Curves to Data Sets • Regression Analysis – Finding an equation which relates to a data set as closely as possible – Allows for analysis and prediction – Advanced regression analysis uses matrix theory and calculus TMAT 103 §19.6 Statistical Process Control §19.6 – Statistical Process Control • Using statistics for quality control – Specifications • Does it meet or exceed established specifications – Durability • Does the item perform as long as expected – Reliability • How often are repairs needed – Service • Is item easy to repair? Are shipping/billing errors rare? – Customer needs • Does item meet expectations and needs of customer? §19.6 – Statistical Process Control • Terminology – Common cause variation • Variations always occur within a product – Process in control • Produced items consistently fall within common cause tolerance limits, and measurements fit a normal curve – Limits • • • • UCL – upper control limit LCL – lower control limit UTL – upper tolerance limit LTL – Lower tolerance limit – Capable process • Control limits within tolerance limits (i.e. specifications) §19.6 – Statistical Process Control • Process in control §19.6 – Statistical Process Control • Capable Process – A capable process is ALWAYS in control §19.6 – Statistical Process Control • Process not in control, but within tolerance limits – Unpredictable, and undesirable §19.6 – Statistical Process Control • Determine which of the processes are capable – Both processes are in control, and all measurements are in centimeters Process Control Limits Order Specifications 1 44.9992 to 45.0008 45 0.001 2 3.9995 to 4.0005 4 0.0001