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## Introduction to Statistical Process Control

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**History of Statistical Process Control**• Quality Control in Industry • Shewhart and Bell Telephones • Deming & Japan after WWII • Use in Health Care & Public Health**The Count**Cups of Coffee Day**The Mean**The mean of 4, 7, 8 , and 2 is equal to: 4+7+8+2 4**The Median-Odd Numbers**• = the middle value in an ordered series of numbers. • To take the median of 1, 7, 3, 10, 19, 4, 8 • Order these numbers: 1,3,4,7,8,10,19 • The median is zth number up the series where z=(k+1)/2 and k=number of numbers. • What is the median in this case?**The Median-Even k**• Order the numbers, i.e., 1,7,10,14,15, 17. • Find the middle values, i.e., 10 and 14. • Take the average between these two values. • What is the median?**The Proportion**You have these 10 values representing 10 people: 0,0,0,1,0,1,0,0,1,0. Zero means person did not get sick. One means person did get sick. What is the mean of these 10 values? (0+0+0+1+0+1+0+0+1+0)/10 = .333 Proportion= n/N, where n=number of people who got sick and N=total number of people. n=numerator, N=denominator.**What is a population?**• A group of people? • A group of people over time? • Hospital visits? • Motor vehicle crashes? • Ambulance Calls? • Vehicle-Miles? • X-Rays Read? • Other?**Populations take onDistributions**In simple statistical process control, we deal with 4 distributions.**From central tendency to variation.**The Normal Distribution**How do we describe variation about the red line in the**normal curve? • In other words, how fat is that distribution? • How about the average difference between each observation and the mean? • Oops, can’t add those differences, some are positive and some are negative. • How about adding up the absolute values of those differences? • Bad statistical properties. • How about the average squared difference? • Now we are talking! **Population Variance**N ∑ 1 Population Variance = (xi - µ)2 N i=1 The average squared deviation!**Population Standard Deviation**N Population Standard Deviation = ∑ √ 1 (xi - µ)2 N i=1 The square root of the average squared deviation!**How do I estimate the standard deviation of the means of**repeated samples? • Estimate the standard deviation of the population with your sample using the sample standard deviation. • Estimate the standard deviation of the mean of repeated samples by calculating the standard error.**Sample Standard Deviation**N ∑ √ 1 S = (xi - x)2 N-1 i=1 How is this different from the Population Standard Deviation?**Standard Error**s SE = √ n How is this different from the Sample Standard Deviation?**Z-Score for Distribution of Sample Means**x - μ Z = SE X = mean observed in your sample μ = is the population mean you believe in. Z = number of standard errors x is away from μ, You can convert any group of numbers to z-scores.**If we kept dancing for hundreds of times**Here is the distribution of our sample means (standardized)**Wait a minute!**• When you do a survey, you only have one sample, not hundreds of repeated samples. • How confident can you be that the mean of your one sample represents the mean of the population? • If you think reality is a normal distribution with mean y and standard deviation s, how likely is your observed mean of x?**Welcome to**• Confidence Intervals • P-Values • Let’s focus on p-values for now.**Here is the mean we observed**(1.96 ≈ 2) Here is our distribution of sample means—WE BELIEVE What is the probability of observing a mean at least as far away as zero (on either side) as 1.96 standard errors? 2.72? Area under curve is probability and it adds up to one. .025 .025**Remember the p-value question?**• If you think reality is a normal distribution with mean y and standard deviation s, how likely is your observed mean of x?**Let’s ask it again.**• We have systolic blood pressure measurements on a sample of 50 patients for each of 25 months. For each of those months, we a mean blood pressure and a sample standard deviation. • “You think reality for each month should be a normal distribution with a mean blood pressure that equals the average of the 25 mean blood pressures. • You also think that for each month, this normal distribution should have a standard error based on the average sample standard deviation across the 25 months. • How likely is your observed mean in month 4 of 220 if the average mean across the 25 months was 120? • How many standard errors is 220 away from 120? What is the probability of being at least that many standard errors away from 120?**Welcome to the Shewart Control Chart**1 2 3 4 5 6 7 . . . 25 120**Anatomy of the control chart:**From Amin, 2001 Indian Health Service, DHHS