Overview of Hypothesis Testing. Photo by Hartwig HDK
You may know all the statistics in the world, but if you jump straight from those statistics to the wrong conclusion, you could make a million-dollar error. That’s where hypothesis testing comes in. It combines tried-and-tested analysis tools, real-world data, and a framework that allows us to test our assumptions and beliefs. This way, we can say how likely something is to be true or not within a given standard of accuracy.
When using hypothesis testing, we create the following:
A null hypothesis (H0): the assumption that the experimental results are due to chance alone; nothing (from 6M) influenced our results.
An alternative hypothesis (Ha): we expect to find a particular outcome.
These hypotheses should always be mutually exclusive: if one is true, the other is false.
Once we have our null and alternative hypotheses, we evaluate them with a sample of an entire population, check our results, and conclude based on those results.
Note: We never accept a NULL hypothesis; we fail to reject it. We are always testing the NULL.
This field uses a lot of specialist terminologies. We’ve collated a list of the most common terms and their meanings for easy lookup. See the hypothesis testing terminology list.
Tailed Hypothesis Tests
These tests are commonly referred to according to their ‘tails’ or the critical regions that exist within them. There are three basic types: right-tailed, left-tailed, and two-tailed. Read more about tailed hypothesis tests.
Errors in Hypothesis Testing
When discussing an error in this context, the word has a very specific meaning: it refers to incorrectly either rejecting or accepting a hypothesis. Read more about errors in hypothesis testing.
P-Values
We use P-values to determine how statistically significant our test results are and how probable we’ll make an error. Read more about p-values.
Types of Hypothesis Tests
One aspect of testing that can confuse the new student is finding which, out of many available tests, is correct to use.
Parametric Tests
You can use these tests when it’s implied that a distribution is assumed for the population or the data is a sample from a distribution (often a normal distribution).
Non Parametric Tests
You use non-parametric tests when you don’t know, can’t assume, and can’t identify what kind of distribution you have.
The Central Limit Theorem is important for inferential statistics because it allows us to safely assume that the sampling distribution of the mean will be normal in most cases. This means that we can take advantage of statistical techniques that assume a normal distribution.
The Central Limit Theorem is one of the most profound and useful statistical and probability results. The large samples (more than 30) from any sort of distribution of the sample means will follow a normal distribution. The central limit theorem is vital in statistics for two main reasons: the normality assumption and the precision of the estimates.
The spread of the sample means is less (narrower) than the spread of the population you’re sampling from. So, it does not matter how the original population is skewed.
The means of the sampling distribution of the mean is equal to the population mean µx̅ =µX
The standard deviation of the sample means equals the standard deviation of the population divided by the square root of the sample size: σ(x̅) = σ(x) / √(n)
Central Limit Theorem allows using confidence intervals, hypothesis testing, DOE, regression analysis, and other analytical techniques. Many statistics have approximately normal distributions for large sample sizes, even when we are sampling from a non-normal distribution.
Control charts and the central limit theorem
In the above graph, subgroup sizes of 2, 8, 16, and 32 were used in this analysis. We can see the impact of the subgrouping. In Figure 2 (n=8), the histogram is not as wide and looks more “normally” distributed than Figure 1. Figure 3 shows the histogram for subgroup averages when n = 16, it is even more narrow and it looks more normally distributed. Figures 1 through 4 show, that as n increases, the distribution becomes narrower and more bell-shaped -just as the central limit theorem says. This means that we can often use well-developed statistical inference procedures and probability calculations that are based on a normal distribution, even if we are sampling from a population that is not normal, provided we have a large sample size.
Z score and Process Capability are used together to get a view of your process. The Z score or value in process capability calculation is the mean distance from specification limits (USL and LSL) measured in standard deviation units.
Z score tells the defects within the system. In other words, Z score tells the number of standard deviation existed between the mean and specification limit.
How Z score impacts process capability in DMAIC project
Six Sigma method can be implemented in any industry with a five-phase structured method: DMAIC. During the measure phase, Six Sigma black belts more focus on Z Score apart from other process performances.
It is very difficult in any organization to define the defects as it varies from process to process. For example, sales, production, and finance departments, defects categories are different in all the three groups. So, it is challenging to compare the process performance. Hence, Z score plays an important role or a common platform to compare all the processes’ performance.
A higher Z score means it has more number of standard deviations within specification limits and the process average is away from the outlying limit. Whereas, lower Z value means it has less number of standard deviation. In other words, there is a variation in the output. If the Z value is higher, most of the values lying near the average and process are capable and also stable.
Benefits of Z score in DMAIC projects
Z score is a flexible measure to use for both variable and attribute data.
It helps to create the baseline for the Six Sigma projects. Also, use to assess and compare before and after process performance.
Z score mandates the six sigma team to define the process mean, standard deviation, specification limits for continuous data. Similarly, defects, sample size etc for attribute data.
Z score is a universal measurement, that can be applied for different functions within the organization like production, sales, finance, HR, administration etc.
Calculate Z score and process capability one from another
Z scores basically use them to transform a given standard distribution into something that is easy for us to calculate probabilities on. Why? So we can determine the likelihood of some event happening.
Naturally, the greater the spread, the less of a chance a process has of meeting its intended specifications.
Calculating Z value
The area outside of specification for a normal curve can be determined by Z value.
ZUpper =(USL-X̅)/ σ̂
ZLower = (X̅-LSL)/ σ̂
Then, Zmin = The smallest value of ZUpper and ZLower
Where
σ̂ is the estimate of the process standard deviation
and σ̂ = R̅/d2
X̅ is the process mean
If Zmin is negative, theX̅ is outside of the specification limit.
Usually, Zmin is also known as ZST.
Capability Indices
Cpis the distance between the specification limits divided by six times an estimate of the process standard deviation.
Cp =(USL-LSL)/ 6σ̂
As the entire normal distribution is within the ±3 sigma, Cp really tells how many standard deviations fit inside of the specification limits. But Cp does not take into account whether or not the process is centered. Whereas, Cpk is more useful as it considers the process centered or not.
Cpk is the minimum of upper specification limit minus the current process mean. The current process mean minus lower specification limit (which gives the distance to the closest specification limit) divided by three times an estimate of the process standard deviation.
Cpk =min (USL- X̅, X̅-LSL)/ 3σ̂ = Zmin/3 = ZST/3
Hence, If you have a Z value, the equation is very easy;
Cpk = Dividing the Z score by three.
A z score is the same as a standard score; the number of standard deviations above the mean.Z = (X-µ)/σ
The issue with process capability indices is that the standard deviation estimates based on the local variation. The location variation estimate of the process standard deviation is too small. In other words, it doesn’t reflect the long term. This will be addressed by Process performance indices.
Process performance Indices
The long term process capability indices also known as process performance indices Pp and Ppk.
Pp =(USL-LSL)/ 6s
Ppk =min (USL- X̅, X̅-LSL)/ 3s
Where s is the sample standard deviation of all the data.
Example to calculate the Cpk value using Z score
Example: The process average is 10 with an upper control limit of 18, a lower control limit of 6, and a standard deviation of 2. Calculate the Cpk value using Z score.
USL=18
LSL=6
σ̂ = 2
Now, compute ZUpper and ZLower
ZUpper =(USL-X̅)/ σ̂ = (18-10)/2 =4
ZLower =(X̅-LSL)/ σ̂ = (10-6)/2 = 2
So, Zmin = The smallest value of ZUpper and ZLower =2
Cpk = Zmin/3 = 2/3 = 0.667
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Z scores and process sigma help us know what chance a process has of meeting its target. This article discusses the relation between the two concepts.
Z Scores and Process Sigma Overview
We’ve covered the basics of Z scores here. Basically, we use them to transform a given standard distribution into something that is easy for us to calculate probabilities on. Why? So we can determine the likelihood of some event happening.
In practical applications, we often have specification limits for our process. For example, we need to create physical widgets with a length between 5 cm and 5.5 cm. It’s important to know what the percent chance your process has of meeting that kind of target.
Assuming your process follows a normal distribution, the distance between the sample mean of your process and the upper (or lower) bound is the process sigma. The process sigma metric is essentially a Z equivalent.
Remember that a sigma score tells you how many standard deviations can fit between the process mean and specification limit of your process. The better your process, the more sigmas. This is a case of more is better!
When we covered the basics of Z scores, I recommended drawing a picture. That holds here, too.
Let’s look at some scenarios:
What is the Z score of a process that can NEVER meet specification?
That means you have a 100% error rate. We would be looking for an area of zero under the curve. On the chart, you’d have a negative infinity Z score.
A low sigma value means that a significant part of the process output is not inside specification limits.
What is the Z score of a process that meets specification half of the time?
If the process meets specification half of the time, the specification limit sits right at the process mean, so the Z score is 0: half of the output falls on each side. You will often see this called a 1.5 sigma process. That label comes from the sigma-level convention, which adds a 1.5 sigma shift to the long-term Z (0 + 1.5 = 1.5).
What does a low Sigma or Z score mean for your process?
A low sigma (Z) score means that your process has a lot of defects. If we made a graph of this, a significant part of the tail of the distribution extends past the specification limit.
What Z score would your process need to be at a 6 Sigma level of accuracy?
Remember, a z value is a standard deviation. The answer here is simple; a Z score of 6.
The higher the Z score, the fewer the defects there are in your process. Graphically, you would see the variation in your process a safe distance away from the specification limit. 6 Standard deviations has been considered an industry standard as a safe distance – some industries require a greater that 6 sigma performance, though.
What Z score would reflect zero errors in the process?
This would be all of the area under the curve, or a positive infinity Z score.
How do you Calculate a Z Score from DPMO?
Ok. Let’s say you’ve calculated your baseline sigma. Let’s use the 333,333 DPMO we calculated in the example on the baseline sigma page. That’s a ratio we can express as 0.3333.
This page uses a cumulative z table: each value is the area to the left of z. If your handbook’s table shows the area from the mean (0 to z), look up 0.5 – 0.3333 = 0.1667 instead and make the z negative, since that area sits below the mean.
If you look at the Z Score table, you’ll see that 0.3333 falls between a Z score of -0.43 and -0.44 (because of the 0.3330 and 0.3336.) Since it is right in the middle, we can say that a DMPO of 333,333 translates to a Z Score of -0.435.
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The Measure is the second phase of DMAIC. The main activity in the Measure phase is to define the baseline. While we have identified a project in the Define phase of DMAIC, let’s take the lessons learned from the first phase and get the ‘real story’ behind the current state by gathering data and interpreting what the current process is capable of.
“Life moves pretty fast. If you don’t stop and look around once in a while, you could miss it.” – Mathew Broderick as Ferris Bueller, Ferris Bueller’s Day Off.
In the Measure phase, the Six Sigma team checks how the process is performing against the customer expectations, and CTQs are noticed in the Define phase of DMAIC.
Six Sigma Phases
Six Sigma is a systematic problem-solving approach centered around defect elimination and variation reduction, leading to process improvement.
One of the principal tools in Six Sigma is the use of the DMAIC methodology. (Also see DMAIC Overview). Particularly, DMAIC is a logical framework that helps you think through and plan improvements to a process to achieve a Six Sigma level of excellence.
There are five phases that are used in the DMAIC method.
The purpose of the Measure phase is to understand the extent of the problem with the help of data. In other words, measure the process performance in its current state in order to understand the problem.
Goals of Measure Phase
Establish baseline performance of the process
Identification of process performance indicators
Develop a data collection plan and then collect data.
Validating the measurement system
Determine the process capability
Measure Phase of DMAIC Overview
The Measure phase is approximately 2 to 3 weeks process based on the project inputs. In particular, all the relevant stakeholders’ involvement is key to getting quality data.
The measure phase is about the baseline of the current process, data collection, validating the measurement system, and determining the process capability. Multiple tools and concepts are available in the Measure phase of Six Sigma.
Process Definition & Basic Tools
Process map: A process map is a tool that graphically shows the inputs, actions, and also outputs of a process in a clear, step-by-step map of the process.
The process map illustrates the relationship between inputs (X) and outputs (Y). Firstly, create a process map of all the activities required to convert raw materials into output (Y). Then, identify the Critical to Quality (CTQ) factors in the process.
Process map helps to identify the inefficiencies or wastes in the process. This also helps to determine the critical steps to collect the data.
Value stream mapping: Value stream mapping provides a visual representation of the organization’s flow of materials and information. Value stream mapping constitutes all the value-added and non-added values required to make the product. It consists of the process flows starting from the raw materials to making the product finally available in the hands of the customers.
Spaghetti Diagram: The spaghetti diagram, also known as the spaghetti chart, represents the basic flow of people, products, and process documents or papers.
In fact, the measure phase is all about collecting as much data as possible to get the actual picture of the problem. Hence, the team must ensure the data collection measurement process is accurate and precise.
Data is a set of values of qualitative or quantitative variables. It may be numbers, measurements, observations, or even just descriptions of things. Below are the types of Quantitative Data
Discrete data: The data is discrete if the measurements are integers or counts. For example, the number of customer complaints, weekly defects data, etc.
Continuous data: The data is continuous if the measurement takes on any value, usually within some range, for example, Stack height, distance, cycle time, etc.
Coding Data
Sometimes it is more efficient to code data by adding, subtracting, multiplying, or dividing by a factor.
Types of Data Coding
Substitution: ex. Replace 1/8ths of an inch with + / 1 deviations from the center in integers.
Truncation: ex. Data set of 0.5541, 0.5542, 0.5547, you might just remove the 0.554 portions.
A data collection plan is a useful tool to focus your data collection efforts on. This directed approach helps to avoid locating & measuring data just for the sake of doing so.
Data collection form: In general, a data collection form is a way of recording the approach to obtaining the data that need to perform the analysis. Additionally, the data should be recorded by trained operators with a calibrated instrument and a standard data collection form.
Data Collection check sheets: A Check Sheet is a data collection tool that usually identifies where and how often problems appear in a product or service. It’s specifically designed for the kind of process being investigated.
Measurement System Analysis
Measurement System Analysis (MSA) is an experimental and mathematical method of determining how much variation within the measurement process contributes to overall process variability.
Accuracy: It is the difference between the true average and the observed average. If the average value differs from the true average, this is an indication of an inaccurate system.
Precision: Precision refers to how close the data points fall in relation to each other. In other words, a high-precision process will have little variance between the individual measurement points.
Gage R&R
The Gage Repeatability and Reproducibility is a method to assess the measurement system’s repeatability and reproducibility. Furthermore, Gage R&R measures the amount of variability in measurements caused by the measurement system itself.
Gage R&R focuses on two key aspects of measurement:
Repeatability: Repeatability is the variation between successive measurements of the same part, same characteristic, by the same person using the same gage.
Reproducibility: Reproducibility is the difference in the average of the measurements made by different people using the same instrument when measuring the identical characteristic on the same part.
Six Sigma Statistics
Basic Six Sigma statistics is the foundation for Six Sigma projects. It allows us to numerically describe the data that characterizes the process Xs and Ys.
Statistics is the science of gathering, classifying, arranging, analyzing, interpreting, and presenting numerical data to make inferences about the population from the sample drawn. There are basically two categories: Analytical (aka Inferential statistics) and Descriptive (aka Enumerative statistics).
Inferential statistics: It is used to determine whether a particular sample or test outcome is representative of the population from the sample originally drawn.
Descriptive statistics: A descriptive statistic is basically organizing and summarizing the data using numbers and graphs. Descriptive statics is to describe the characteristics of the sample or population.
The shape of data distribution is depicted by its number of peaks and symmetry possession, skewness, or uniformity. Skewness is a measure of the lack of symmetry. In other words, skewness is the measure of how much the probability distribution of a random variable deviates from the Normal Distribution.
Data Organization / Data Display / Data Patterns
The graphical analysis creates pictures of the data, which will help to understand the patterns and also the correlation between process parameters. Moreover, graphical analysis is the starting point for any problem-solving method. Hence select the right tool to identify the data patterns and display the data.
Control Chart: The control chart is a graphical display of quality characteristics that have been measured or computed from a sample versus the sample number or time.
Frequency Plots: Frequency plots allow you to summarize lots of data in a graphical manner making it easy to see the distribution of that data and process capability, especially when compared to specifications.
Box Plot: A box plot is a pictorial representation of continuous data. In other words, the Box plot shows the maximum, minimum, median, interquartile range Q1, Q3, and outlier.
Main Effects plot: The main effects plot is the simplest graphical tool to determine the relative impact of various inputs on the output of interest.
Histogram: Histogram is the graphical representation of a frequency distribution. In fact, it is in the form of a rectangle with class intervals as bases and the corresponding frequencies as heights.
Scatter plot: A Scatter Analysis is used when you need to compare two data sets against each other to determine a relationship.
Pareto Chart: Pareto chart is a graphical tool to map and grade business process problems from the most recurrent to the least frequent.
Basic Probability & Hypothesis tests
Basic Six Sigma Probability terms like independence, mutually exclusive, compound events, and more are the necessary foundations for statistical analysis.
Additive law: Additive law is the probability of the union of two events. There are two scenarios in additive law:
When events are not mutually exclusive
When events are mutually exclusive
Multiplication law: It is a method to find the probability of events occurring simultaneously. There are two scenarios in multiplication law:
When events are not independent
When events are dependent
Compound Event: It is an event that has more than one possible outcome of an experiment. In other words, compound events are composed of two or more events.
Independent Event: Events can be independent events when the outcome of one event does not influence another event’s outcome.
Hypothesis Testing
Hypothesis testing is a key procedure in inferential statistics used to make statistical decisions using experimental data. It is basically an assumption that we make about the population parameter.
When using hypothesis testing, we create:
A null hypothesis (H0): the assumption that the experimental results are due to chance alone; nothing (from 6M) influenced our results.
An alternative hypothesis (Ha): we expect to find a particular outcome.
Determine the process capability
Process Capability Analysis tells us how well a process meets a set of specification limits based on a sample of data taken from a process. The process capability study helps to establish the process baseline and measure the future state performance. Revisit the operational definitions and specify what are defects and which are opportunities.
The value in making a sigma calculation is that it abstracts your level of quality enough so that you can compare levels of quality across different fields (and different distributions.) In other words, the sigma value (or even DPMO) is a universal metric that can help you with the industry benchmark/competitors.
Baseline Sigma for discrete data
Calculate the process capability through the number of defects per opportunity. The acceptable number to achieve Six Sigma is 3.4 Defects Per Million Opportunities (DPMO).
DPO = Defects/(Units * Opportunity)
DPMO =(Defects / Units * Opportunities) * Total 1,000,000
Yield = 1-DPO (It is the ability of the process to produce defect-free units).
Baseline Sigma for Continuous data
Process Capability is the determination of the adequacy of the process concerning the customer needs. Process capability compares the output of an in-control process to the specification limits. Cp and Cpk are considered short-term potential capability measures for a process.
Cpk is a measure to show how many standard deviations the specification limits are from the center of the process.
Cplower = (Process Mean – LSL)/(3*Standard Deviation)
Cpupper = (USL – Process Mean)/(3*Standard Deviation)
Cpk is the smallest value of the Cpl or Cpu: Cpk= Min (Cpl, Cpu)
Six Sigma derives from the normal or bell curve in statistics, where each interval indicates one sigma or one standard deviation. Moreover, Sigma is a statistical term that refers to the standard deviation of a process about its mean. In a normally distributed process, 99.73% of measurement will fall within ±3σ, and 99.99932% will fall within ±4.5σ.
