business process improvement approach that seeks to find and eliminate causes of defects and errors, reduce cycle times and cost of operations, improve productivity, better meet customer expectations, and achieve higher asset use and returns on investment in manufacturing and service processes 

• based on DMAIC- Define, Measure, Analyze, Improve, and Control 

• Appeal- focus on measurable results 

• 3.4 errors in a million opportunities (DPMO= defects per million opportunities) 

• all work occurs in a system of interconnected processes 

• variation exists in all processes 

• understanding and reducing variation are keys to success

expected value of the mean

weighted average of possible values

1. The probability of any event is the sum of the probabilities of the outcomes that compose that event 

2. If A and B are mutually exclusive, the P(A or B)= P(A)+P(B) 

3. If 2 events A and B are NOT mutually exclusive, then P(A or B)=P(A)+P(B)-P(A and B)

1. If 0≤f(xsubi)≤1 (the probability of each outcome must be between 0 and 1) 

2. the sum of all probabilities must add to 1

probability mass function of binomial distribution

models n independent replications of a Bernoulli experiment, each with a  probability of p success. X represents the number of successes in these n experiments. Difficult to compute by hand. 

(n over x)= n!/(x!(n-x!))

probability mass function for poisson distribution

discrete distribution used to model the number of occurrences in some unit of measure  

average number of occurrences per unit is a constant= λ

a random variable with two possible outcomes with constant probabilities of p occurence.

Typically, success (x=1) or failure (x=0)

probability mass function for uniform distribution

characterizes a continuous random variable for which all outcomes between some minimum value a and maximum value b are equally likely. (therefore, a graph will have a flat line representing probability)

transformation to standardized normal values

(for use in normal distribution)

standard error of the mean

the standard deviation of the sampling distribution of the mean

a random variable with two possible outcomes with constant probabilities of p occurrence. Typically, success (x=1) or failure (x=0).

The expected value is p, and the variance is p(1-p)

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models n independent replications of a Bernoulli experiment, each with a  probability of p success. X represents the number of successes in these n experiments. Difficult to compute by hand. 

The expected value is np, and the variance is np(1-p)

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discrete distribution used to model the number of occurrences in some unit of measure (ie. The number of events occurring in an interval of time). Assumes no limit on the number of occurrences, occurrences are independent, and that the average number of occurrences per unit is a constant, λ (lower case lambda). 

Expected value is λ, and variance is λ. 

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confidence interval for the mean with a known standard deviation

confidence interval for the mean with an unknown population standard deviation

Where (t sub alpha over 2, n-1) is the value from the t-distribution with n-1 degrees of freedom, giving an upper-tail probability of alpha/2.

confidence interval for proportion

confidence intervals for variance

Chi-square distribution is not symmetric, which means that the confidence interval is not an estimate to the left or right of a point estimate

TO FIND CI FOR SD, take the square root of both values

t test statistic

for one-sample hypothesis tests (for samples with unknown population standard deviation)

where mu sub 0 is the hypothesized value and s/sqrtn is the standard error or the sampling distribution of the mean

Has a t-distribution with n-1 degrees of freedom

z-test statistic for one-sample test for proportions

where pisub0 is the hypothesized value. The denominator represents the standard error for the sampling distribution of the proportion and is shown in the intermediate calculations