Introduction to Statistical Process ControlModule 4

History of Statistical Process ControlQuality Control in IndustryShewhart and Bell TelephonesDeming & Japan after WWIIUse in Health Care & Public Health

The Run Chart

The CountDayCups of Coffee

Sunday12Monday2Tuesday4Wednesday3Thursday5Friday4Saturday2

The MeanThe mean of 4, 7, 8 , and 2 is equal to:4+7+8+24

The Median-Odd Numbers= the middle value in an ordered series of numbers.To take the median of 1, 7, 3, 10, 19, 4, 8Order these numbers: 1,3,4,7,8,10,19The 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 kOrder 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 ProportionYou 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 = .333Proportion= n/N, where n=number of people who got sick andN=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 onDistributionsIn simple statistical process control, we dealwith 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, cant 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 i=1N1N(xi - )2Population Variance =The average squared deviation!

Population Standard Deviation i=1N1N(xi - )2Population Standard Deviation =The square root of the average squared deviation!

Standard Deviation has Nice Properties!.025025

Its Time to Dance

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 i=1N1N-1(xi - x)2S =How is this different from the Population StandardDeviation?

Standard ErrorSE =How is this different from the Sample StandardDeviation?sn

Z-Score for Distribution of Sample MeansZ = x - SEYou can convert any group of numbers to z-scores.X = mean observed in your sample = is the population mean you believe in.Z = number of standard errors x is away from ,

If we kept dancing for hundreds of timesHere 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 toConfidence Intervals P-Values

Lets focus on p-values for now.

Here is our distribution of sample meansWE BELIEVE.025.025

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 curveis probability andit adds up to one.

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?

Lets 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 Chart1 2 3 4 5 6 7 . . . 25120

Indian Health Service, DHHSAnatomy of the control chart:From Amin, 2001