Unit 5: Intermediate Statistics

Based on the Council for Six Sigma Certification Body of Knowledge

Exponential Distribution

Creates a histogram or trend line that is exponential in nature. It decreases exponentially as you travel across the x-axis. In an exponential distribution, the data is not organized around the mean but at one end of the x-axis. Often used with data sets that include arrival times, mean time between failures, or wait-line theories.

Exponential Distribution Description

Never appears symmetrical; is described statistically by the mean of the data and a value known as lambda.

Lognormal Distribution

Asymmetrical, with the trend line appearing more as a wave that moves across the page. The location and size of the wave varies with each data set. Often used with data sets that describe time durations or distribution of assets or wealth. Anytime data has a positive skew (tail to the right), a lognormal distribution might be possible.

Lognormal Distribution Description

Typically describes a data set with values in large ranges; can be described with both mean and standard deviation; always has a positive skew.

Weibull Distribution

Probabilities can fall into families of distributions, such as the Weibull distributions. Data that fits a Weibull distribution might also fit another distribution. Weibull distributions relate to continuous data. Can describe many data types and are often used when working with reliability applications and failure probabilities that change or vary with time.

Central Limit Theorem

States that the distribution of the mean of a large, identically distributed number of independent variables will approximate the normal curve. Because of this, you'll be able to use software like Minitab to apply statistical analysis to a number of data sets.

Cauchy Distribution

Often looks like an elongated normal curve with a tighter peak. The distribution doesn't have a defined mean or variance, which makes it not useful for many common statistical analysis tools used in Six Sigma.

Logistic Distribution

Appears to approximate the normal curve, and is used in some science and math functions to approximate other symmetrical distributions because the CDF is more consistently calculable.

Laplace Distribution

Often referred to as the bilateral exponential distribution or the double-exponential distribution. The Laplace distribution does seem to pair exponential distributions back to back.

Uniform Distribution

Occurs when data points are divided evenly among bins. Exact uniform distributions are almost never going to occur in randomly-sampled data, which means Six Sigma teams that see such distributions should check the authenticity of the data. Uniformity often points to an error of measurement or a non-random sample.

Beta Distribution

Like Weibull distributions, Beta distributions can take on a number of shapes. Beta distributions are considered extremely flexible, and can become stand-ins for other distributions given certain statistical parameters.

Gamma Distribution

Similar to Beta and Weibull distributions, in that they take on a number of shapes. Gamma distributions are, however, always skewed to the right.

Triangular Distribution

Formed using the mode and the upper and lower limits of a data set.

Binomial Distribution

Used when you are dealing with discrete data and there are only two outcomes for each trial or sample. Binomial distributions relate to attribute data

Binomial Distribution Probability of Success

Notated as p (probability of success) and probability of failure (notated as 1 – p).

Binomial Distribution Probability Function

p(x) = n! / (x!(n-x)!) * p^x * q^(n-x), where x is the number of successes desired, n is the number of trials, p is the probability of success in each trial, and q is equal to 1 – p.

Poisson Distribution

Often used when dealing with data that is distributed randomly within time, distance, or other unit of measurement. The word "per" in a metric is a good indication that the Poisson distribution might be useful. The Poisson distribution is discrete, not continuous.

Poisson Distribution Applicability

Data describes events that occur at a random interval measurement; sample size is at least 16; population size is 10 times the sample size or bigger; event occurrences are independent; probability of occurrences is less than 0.1; the rate of occurrence per interval is constant.

Poisson Equation

P(x) = e^(-μ) * μ^x / x!, where e equals the Poisson constant (2.71828), μ is the mean or average of the process, and X is the specific case or data point.

Poisson and Exponential Relationship

The Poisson distribution relates closely to the exponential distribution. If the process is following a Poisson distribution, then the time between each event in the process when looked at continuously yields an exponential distribution.

Geometric Distribution

Used when there are two outcomes for a trial, trials are independent, and there is a waiting time before the first occurrence. A real-world use might include the number of inspections before finding a defective part.

Negative Binomial Distribution

Also used with attribute data – fail/pass and other situations where there are only two outcomes for each trial. Often used when teams are trying to determine the probability of a certain number of passes or fails before reaching the sth pass or fail.

Hypothesis Testing Categories

Testing whether the data you have fits a data model; comparing a statistic to a hypothesis about the data or population; answering the question whether something changed within the data.

Null Hypothesis (H0)

Abbreviated as H0 and is usually a statement about the data that reflects no effect or no difference. Typically an equal statement of some type. The null hypothesis is like the innocent verdict in a criminal court

Alternative Hypothesis (Ha)

Abbreviated as Ha and is usually a statement that is likely to be true if the null hypothesis is not true. Typically written as a not equals, a greater than, or a less than statement.

Type I Error (Producer Risk)

The null hypothesis is rejected when it is actually true. Also called producer risk. The probability of the risk is measured by alpha (α), where α is a probability between 0 and 1.

Type II Error (Consumer Risk)

The null hypothesis is accepted when it is actually false. Also called the consumer risk. The probability of the risk is measured by beta (β), where β is a probability between 0 and 1.

Confidence Level

The most common confidence level used is 95 percent, or α = 0.05. Typically, the confidence level is set with the Type I error in mind, so you use alpha for the confidence level. The value of β then contributes to the sample size requirements and the power.

Selecting the Right Hypothesis Test

You must know

1-Proportion Test

Used when there is only one factor for x, one level of interest for x, and when comparing proportions (percentage, rates) between samples and a target. Hypothesis

2-Proportion Test

Used when there is only one factor for x, two levels of interest for x, and when comparing proportions between samples. Hypothesis

1-Sample T Test (or Paired T Test)

Used when comparing means. Typically used when dealing with smaller samples or when standard deviation is known. 1-Sample T Test compares the mean of a sample to a target mean. Paired T Test compares the mean of a sample against the mean of another sample for the same factor of X. Hypothesis

Chi Square Test (or 1-Variance Test)

Used when comparing standard deviation or variance. Compares the standard deviation or variance between two samples of the same x factor or compares to a target. Hypothesis

2-Sample T Test

Used when comparing means. Compares the means between two samples of the different x factors. The 2-Sample T Test compares two samples from different populations. Hypothesis

One Sample Wilcox

Used when comparing medians. Compares the medians between a sample and a hypothesized sample or a new sample to a previous sample before changes were made. Typically used when data is somewhat symmetrical. Hypothesis

Mann-Whitney Test

Used when comparing medians. Compares the medians between samples of two factors of x. Hypothesis

P-Value (Hypothesis Testing)

The p-value is returned for each test, and we compare the p-value to the alpha value we set before we ran the test. If the p-value returned is less than the alpha value, then you reject the null hypothesis and accept the alternative hypothesis. If the p-value is more than the alpha value, then you fail to reject the null hypothesis.

Hypothesis Testing Steps

1. State null and alternative hypotheses; 2. Set the confidence level for alpha; 3. Decide which hypothesis test to use; 4. Decide whether sample size is fixed or can be selected; 5. Run the test in Minitab; 6. Interpret the p-value; 7. Translate the statistical analysis into business-relevant language.

Sample Size Importance

Accuracy in inferential statistics requires that you have the right sample size. When we sample the population and draw inferences from those samples, there is always some risk of error. In many cases, a larger sample size helps reduce the uncertainty of our conclusions.

Confidence Interval (Margin of Error)

The uncertainty associated with sampling. For example

Information Required for Choosing Sample Size

Alpha, Beta, Delta (the practical difference the experimenter wants to detect), Standard deviation (estimated population standard deviation), Type of Data (discrete or continuous), Type of Test (which hypothesis test).

Guidelines for Testing Means (Continuous Data)

Set alpha at 0.05; set beta at 0.10 or 0.20 (power of 0.90 or 0.80); set delta as a function of standard deviation (.5 σ, 1 σ, and 2 σ).

Guidelines for Testing Proportions (Discrete/Binomial Data)

Set alpha at 0.05; set beta at 0.10 or 0.20 (power of 0.90 or 0.80); set delta logically according to business needs or as a function of standard deviation.

Sample Size Calculation Methods (Minitab)

1-Sample T Test (Means, comparing to target, have sample statistics); 1-Sample Z Test (Means, comparing to target, no sample statistics); 2-Sample T Test (Means, comparing means from two sets); 2-Sample Variance Test (Variance, comparing variance from two sets); 1-Sample Variance Test (Variance, comparing to target); 1-Sample Proportion Test (Proportion, comparing to target); 2-Sample Proportion Test (Proportion, comparing from two sets); ANOVA (Means, more than 2 sets, one factor); DOE (Means, more than 2 sets, more than one factor).

Minitab Power and Sample Size

Found under Stat > Power and Sample Size. Minitab lets you calculate sample sizes or back into power values if a sample size is fixed.

Power = 1 - beta

Power is the probability of correctly rejecting a false null hypothesis (avoiding a Type II error).

X-bar & R Chart

Data is variable (continuous); data can be grouped into subgroups; data count in each subgroup is less than 8; presents two charts

X-bar & S Chart

Data is variable (continuous); data can be grouped into subgroups; data count in each subgroup is more than 8; sigma can be easily calculated; presents two charts

I & MR Chart (Individual and Moving Range)

Data is variable (continuous); data cannot be easily grouped into reasonable subgroups, so you must track individual data points instead of subgroup means. Used when data is difficult or expensive to obtain, production is very slow, or products have a low cycle time.

P-Chart

Data is discrete; data is presented as a percent (percent defective) rather than a count; doesn't require a constant sample/subgroup size; plots the proportion of units that are nonconforming.

NP-Chart

Data is discrete; data is presented as a count; does require a constant sample size; plots the number of units that are nonconforming in each sample size.

U-Chart

Data is discrete; data is presented as a count (number of defects); doesn't require a constant sample/subgroup size; plots the number of defects per unit. Use when data is about the defects themselves, not the overall defective product.

C-Chart

Data is discrete; data is presented as a count (number of defects); does require a constant sample/subgroup size; plots the number of defects per sample. Use when data is about the defects themselves, not the overall defective product.

Common Cause Variation

The normal variation within a process – the variation that is caused by the process itself. For example, a phone operator working at a call center is going to make an occasional speaking or typing error while performing job duties. That variation is accounted for on a control chart.

Special Cause Variation

Variation that is caused by something outside of the normal expectations within a process. For example, an operator who is working too many hours, has a bad headset, or is new on the job might make more than the normal amount of errors.

Common Challenge - Presenting Statistical Analysis

Most people don't care about which hypothesis test you used or what your p-value was. You have to present your information in a way that ties in with what they care about and doesn't rely solely on statistical concepts and language.

Presenting Statistics - Tailor to Audience

When presenting or communicating about a Six Sigma project, tailor the information to reflect your audience's level of concern and knowledge. High-level leadership typically wants to know whether the project was a success, what savings can be expected, and how you are controlling the process.

Presenting Statistics - Tell a Story

Use PowerPoint or another program to combine your information in a way that tells a story that the audience can understand and care about. Ask a question as the title of the slide, include an image that answers the question, include some basic statistical information, and finally answer the question posed in the title in a short, text-based conclusion.

Presenting Statistics - Be Clear and Concise

Make sure each slide – and each element on the slide – is clear and concise. Write in active language, avoid long text explanation, ensure everything is spelled correctly, and don't use the wrong word by accident.

Presenting Statistics - Don't Misuse Tools

Don't misuse PowerPoint; don't misuse Minitab or your other statistical analysis software; don't misuse your data analysis skills. You can never allow biases to force statistical conclusions that aren't supported by the data.