Ted Hessing

Histogram

Posted by Ted Hessing

A Histogram is a bar graph that shows the frequency with which data occurs within a certain interval. In other words, it is a bar chart showing the number of times an outcome is repeated. In Six Sigma, we can use a histogram to visualize what is going on. It can reflect the voice of the process.

A Histogram is a popular statistical method for showing a visual model of a continuous series or a class interval frequency distribution. A bar graph is used for categorical data.

Why Use a Histogram?

A histogram is one of the 7 Quality tools that are often used to assess process behavior and demonstrate if the data follows a normal distribution. The purpose of a histogram is to graphically summarize the distribution of a univariate data set. It shows the center, the spread, the skewness of the data, the presence of outliers, and the presence of multiple modes in the data.

While a normal histogram tells you information about the data, a properly made histogram tells you more than the raw data. A normal histogram helps you understand the normal distribution, dispersion, and central tendencies of the data. In contrast, a great histogram helps to visualize the VOC. In addition, you can include the specification limits to reflect the needs of the customer.

When to use a Histogram

Six Sigma uses histograms when the data is numerical and continuous. Histograms are used to view the overall traits of the data distribution and peaks of the distribution in the data set. Furthermore, it can be used to identify whether the distribution is skewed or symmetric and to show the outliers. Histograms help to determine if the process meets the needs of the customer, and it also helps to identify the process changes over time.

Differences between frequency distributions and Histogram

Frequency distributions set up raw data, also known as observations, in a way that is easy to read, and they represent the data with x’s or check marks, whereas a histogram is a continuous relative frequency graph. You can think of it as the bar graph version of a frequency diagram.

How to analyze the Histogram

Six Sigma users can apply the pattern seen in the histogram to discern a process variation. It is a frequency distribution graph in which the class interval is plotted on the X- axis and their respective frequencies on the Y- axis. It is a two-dimensional diagram.

  • The width of each bar displays the interval size
  • The height of each bar displays the frequency of the interval
  • There are no gaps between bars that are next to one another
  • Continues nature of quantitate data

The height or length of each rectangle shows the frequency of the class, and the breadth shows the size of the class interval. Therefore, the total area covered by the histogram represents the total number of times the data point occurs.

Different shapes of Histogram

When reading a histogram, we want to ask: How many peaks are there? Are there any outliers in the data? Does it look near symmetrical and bell-shaped?

Histogram shapes
Histogram shapes

Bell-Shaped :

If there is a bell shape, your data is normally distributed, and hence, no variation (or influence from other factors like the 6Ms).

Spiked

If there are multiple spikes in the chart, there is likely variation in the process.

Even

On the other hand, if all of the bars are at the same level, we are not likely correctly measuring the process.

Doing statistical analysis usually includes the following:

  1. Measuring the central tendency.
  2. Measuring the variation.

Steps to Create a Histogram

The construction of a histogram starts with the division of a frequency distribution into equal classes, and each class is a vertical bar. It is used to plot data density, especially continuous data like weight or height.

  • Collect the data
  • Arrange the data points in order and assign categories
  • Find the lowest and highest value
  • Find the number of intervals
  • Use 2^k = N where k = # of categories (smallest positive integer) and N = # of data.
The number of categories in your histogram depends on the sample size.
  • Compute the intervals with covering the entire range of the data set
  • Find the frequency of occurrence for each of the data points as per the interval
  • Draw the bar chart preserving counts and categories

Example

Ex. Delay times in minutes for an airline

1) Get Data

Use continuous data– data collection from a frequency distribution check sheet.

Histogram data set

Step 2: Order it and Assign Categories

Histogram ordered categories
  • Note: The number of cells (columns) affects the shape – so choose wisely!
  • Could use 2^k = N where k = # of categories (smallest positive integer) and N = # of data.
Number of categories to use in a histogram depends on the sample size.
The number of categories to use in a histogram depends on the sample size.

Step 3: Create a Bar Chart Preserving Counts and Categories

Advantages

  • You can visualize the distribution of the data.
  • It summarizes the large data set graphically.
  • It helps to make future decisions based on data.
  • You can use it to assist in decision making.
  • Simple to understand and also easy to communicate with the team.

Disadvantages

  • It uses only continuous data.
  • More difficult to compare two data sets.
  • Difficult to read values because data is grouped into data sets.

How to Create a Histogram in Minitab

Useful Videos

Ted Hessing

Central Limit Theorem

Posted by Ted Hessing

The Central Limit Theorem states that the distribution of the sample means approaches normal regardless of the shape of the parent population.

Sample means (s) will normally be more distributed around (µ) than the individual readings (Xs). As n, the sample size, increases, the sample averages (Xs means) will approach a normal distribution with mean (µ).

So, don’t worry if your samples are all over the place. The more sample sets you have, the sooner the averages of those sets will approach a normal distribution with a mean of (µ).

Significance of Central Limit Theorem

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 results in all statistics and probability. The large samples (more than 30) from any sort of distribution of the sample means will follow a normal distribution.

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)

Assumptions

  • Samples must be independent of each other
  • Samples follow random sampling
  • If the population is skewed or asymmetric, the sample should be large (for example, a minimum of 30 samples).

Why Central Limit Theorem is important

Central Limit Theorem allows the use of 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 distribution that is non-normal.

Central Limit Theorem
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.

Unimodal Distribution

Mode is one of the measures of central tendency. Mode is the value that appears most often in a set of data values or a frequent number. The Unimodal distribution will have only one peak or only one frequent value in the data set.  In other words, Unimodal will have only one mode, the values increase first and reach to peak (i.e. the mode or the local maximum) and then decreases.

The normal distribution is the best example of a Unimodal distribution. Similarly, Bimodal distribution means there are two different modes, and multimodal means more than two different modes.

Central Limit Theorem Examples

Case 1: Less than

Example: A population of 65 years male patients with blood sugar was 100 mg/dL with a standard deviation of 15 mg/DL. If a sample of 4 patients’ data were drawn, what is the probability of their mean blood sugar being less than 120 mg/dL?

  • µ = 100
  • x̅ = 120
  • n=4
  • σ =15

Compute P(X<120): z= x̅- µ/ σ/√n = 120-100/15/√4=20/7.5=2.66

P(x<120) = z(2.66) = 0.9961 = 99.61%

Hence, the probability of mean blood sugar is less than 120 mg/dL is 99.61%

Case 2: Between

Example: A population of 65 years male patients with blood sugar was 100 mg/dL with a standard deviation of 20 mg/DL. If a sample of 9 patients’ data were drawn, what is the probability of their mean blood sugar being between 85 and 105 mg/dL?

First, compute P(x<105)

  • µ = 100
  • x̅ = 105
  • n=9
  • σ =20

Compute the Z score z= x̅- µ/ σ/√n = 105-100/20/√9=5/6.67=0.75

Central Limit Theorem

P(x<105) = z(0.75) = 0.7734 = 77.3%

Then, compute P(x<85)

  • µ = 100
  • x̅ = 85
  • n=9
  • σ =20

Compute the Z score z= x̅- µ/ σ/√n = 85-100/20/√9=-15/6.67=-2.24

P(x<85) = z(-2.24) = 0.0125 = 1.25%

Since we are looking for blood sugar between 85 and 105 mg/dL P(85<x<105) = 77.3-1.25 = 76.05%

Hence, the probability of mean blood sugar is between 85 and 105 mg/dL is 76.05%

Case 3: Greater than

Example: A population of 65 years male patients with blood sugar was 100 mg/dL with a standard deviation of 20 mg/DL. If a sample of 16 patients’ data were drawn, what is the probability that their mean blood sugar is more than 90 mg/dL?

  • µ = 100
  • x̅ = 90
  • n=16
  • σ =20

Compute the Z score z= x̅- µ/ σ/√n = 90-100/20/√16=-10/5=-2

P(x<90) = z(-2.0) = 0.0228 = 2.28%

Central Limit Theorem

Since we are looking for blood sugar of more than 90 mg/dL =100%-2.28%=97.72%

Hence, the probability of mean blood sugar greater than 90 mg/dL is 97.72%

Jarque- Bera analysis.

Wikipedia has a good article here:

Central Limit Theorem Videos

Ted Hessing

Quincunx Demonstration

Posted by Ted Hessing
quincunx
quincunx

The Quincunx demonstrates that adjusting a process to “correct” the controls actually makes a process more out of control. I always remember it by recalling Bob Barker’s Plinko board!

While I’m amused by grown adults dropping things down a board with a pin in it, the chief intent of Quincunx is not amusement. Sir Francis Galton, an English polymath, invented the Quincunx to demonstrate the Central Limit Theorem. In particular, the Quincunx board demonstrates that the normal distribution is an approximation to the binomial distribution.

A Quincunx helps you to:

The Quincunx is a wonderful teaching tool as it is a great way visually to show the concept of randomness and with enough repetition, a bell-shaped curve. Obviously it helps you see the concept of the central tendency.

What we learn from it:

What isn’t obvious at first is that it helps you understand the danger of adjusting your process while in the moment to the center. Watch the demonstration below. Doesn’t that make it clear how specification limits are used to monitor and then control variation?

The Quincunx demonstration reveals several things:

First, you can’t judge the process by sampling a part every so often. That would be like dropping just one chip down the board and then speaking to how all would fall.

Adjusting your process – moving where you drop the chip from – gives you a very different possible landing zone. It’s for that reason that the more you adjust your process to the center (nominal), the more variation you will create in your results. If you want less variation, keep a consistent process!!!

The counter to that is if you see a process generating a nice bell-shaped curve, it is probably not being influenced by outside forces like the six Ms.

Tanner Zornes

Target Values

Posted by Tanner Zornes
Target Values
Photo by Mike Mozart

Have you ever had to wait in line for something for what felt like forever? Not too long ago, my wife and I waited over 15 minutes for fast food when we were the first car in line. We also had a fussy baby in the back of the car, so you can imagine it was not a pleasant wait. Around the 10-minute mark, I saw cars piled up behind me. A few cars even decided the food wasn’t worth their time and drove off to find something else to eat.

Because those cars up and left the drive-thru, that fast food chain lost revenue. But even worse, their perception of quality was hurt. Even though my burger and fries were made to perfection, I’d rather wait 15 minutes in a sit-in diner compared to a cramped car with a side of a french-crys (pun intended).

My story illustrates that for every process, whether it is waiting in line for fast food or welding material as part of an assembly, there is a target value that must be achieved in order to create quality. for the consumer.

Back to my fast food example, a 15-minute wait is too long since fast food is called FAST for a reason. I might be willing to wait as long as 3 minutes before I feel impatient. But would I be concerned if my food was ready to eat 20 seconds after ordering?

What Are Target Values?

Target values are the qualitative or quantitative parameters required to achieve quality for customers. These values take on a number of different forms depending on what is being measured and how it’s measured.

The customer sets the target value. Customers will often include tolerances or “wiggle room” that determines the allowable differences in target value. Depending on the process, customers will expect different kinds of target values.

Kinds of Target Values

There are primarily three types of target values:

  • Smaller is Better
  • Larger is Better
  • Nominal is Best

Target Value: Smaller is Better

A good example of Smaller is Better is to wait time in a call queue. In the case of sports, the best athletes are the ones that can run the fastest times. When considering Smaller is Better, the process will always have an upper specification limit. Since the client wants the target value to be as close to zero as possible, the lower specification needs no mention.

Sample distributions or population distributions that follow Smaller is Better Target Values will always appear right-skewed. This is also known as positive skewness

Right Skewed Distribution

Other examples include:

  • The late departure times from the airport.
    • The target value for late departures is zero. Since airports do account for some variation, most connecting flights will not depart in less than 30-60 minutes from the first landing.
  • Ambulance response times
  • A game of golf – the lower the score, the better!
  • The amount of waste generated, non-conformances, defects, or rejected parts.
    • Each needs to stay as close to zero to deliver quality to the customer.

Target Value: Larger is Better

Left Skewed Distribution

On the other hand, the target value for Larger is Better is skewed to the left, which creates a negative distribution. Larger is better, like aiming for a high score in an arcade. No one picks up an arcade game with the intent to drop dead and quit the second it starts.

This target value will always have a lower specification limit. If a process falls below the target value, the customer will feel they received less than expected.

Other examples include:

  • Test scores
    • The target is a 100% for each student who takes the test
  • The length of a movie at the theaters
    • Most movies will extend past 45 minutes. Anything less, and you may feel like you wasted your ticket
  • Old school Nintendo game cartridge-breaking force.
    • If that wafer broke, you were out of a game! Larger is better!
  • Net Promoter Scores
  • Pressure needed to explode a tire.
  • Cell phone battery life

Target Value: Nominal is Best

Normal Distribution

Nominal is Best often appears in the manufacturing world where the process has pre-defined parameters that the business must build to. For example, Kellogg’s® may make a box of cereal where the target weight of its content is 20 ounces with a tolerance of ± 3 ounces. Less than 17 ounces and cereal lovers like me will feel cheated. 23 ounces or more, and Kellogg’s® is losing revenue

When Target Values have a nominal value, both lower and upper specification limits are included. The distribution may appear more normal than the other target values previously discussed.

  • Examples:
    • Customer Service Phone Calls
      • Too little time and they are missing things; too much time and they are not being efficient.
    • The story of Goldilocks
      • Not too hot and not too cold – just right!
    • Oil in the car
      • Too much is a waste and causes smoke when hitting the engine (or exploding parts). Too little, and the engine locks and dies.

When do you use Target Values?

The concept of Target Values is considered during the DEFINE Phase and MEASURE Phase. As previously mentioned, the customer or client sets the target value. The target value is determined when applying Six Sigma tools like CTQ, Kano Diagrams, or surveys. After a deep and thorough dive into the DEFINE Phase, you will have a keen sense of the target values your customer expects.

Now that you have the Target Values defined by the customer, you can begin to measure your Process Capability as found in the MEASURE Phase. This is where you establish Upper Specification Limits and Lower Specification Limits around the target value or mean of the process. Finally, by calculating your Cp/Cpk for short-term data or Pp/Ppk for long-term data, you will know to what sigma level your process can deliver to customer target values.

Other Lean Six Sigma Concepts That Apply to Target Values

Critical to Quality (CTQ) – A breakdown of customer expectations that results in a finer understanding of the drivers that lead to Critical Quality Requirements. Target Values are found in the CTQ Requirements.

Control Charts – Control Charts are the voice of the process implemented during the CONTROL Phase of DMAIC. Here the upper and lower control limits establish a range of target values to determine whether or not special cause variation is found in the process.

Kano Model – A great tool for determining Target Values of a good or service as exciters, wants, and musts. Each of these has a target threshold to hit in order to achieve the desired outcome from customers.

Takt Time – A target value metric in and of itself. Take Time determines how much time is available to meet each individual unit of customer demand. Fulfilling demand below target takt time, and you risk delivering late to the customer.

Examples of Using Target Values in a Six Sigma Project

George is a purchasing manager who oversees the operations of several purchasing agents. George wants to see how he can improve his team, so he begins a six sigma project investigation.

DEFINE:

During the define phase, George interviews several production supervisors to understand how his team can better support their operations. These surveys reveal that employees often complain about receiving parts to complete jobs late by a few days. George uses this information to target the lead time for production to receive the required goods.

MEASURE:

George’s investigations led him to measure the time it took buyers to react to system-driven messages for materials to buy. He carefully measured how long each of his buyers had to respond to their purchase suggestions. He discovers that it takes seven days on average to create purchase orders for system-driven messages, 2 days more than the target max of 5 days. George builds a histogram and quickly discovers his buyers are outside of tolerance.

ANALYZE:

Being the good Six Sigma Professional he is, George wanted to prove that his data was statistically significant, so he built a hypothesis test and proved that his buyers have room for improvement.

IMPROVE:

George held a number of trainings with his buyers to improve their accuracy to stay within 5 days of reacting to messages. After monitoring their progress over the course of a month, he found they were on target.

CONTROL:

In order to maintain gains, George created control charts to track purchase suggestions that went untouched for longer than 5 days. If a suggestion went 4 days untouched, George would help expedite the purchase with his buyer to ensure production received material on time.

Other Useful Resources

https://www.isixsigma.com/dictionary/quality-target/

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Ted Hessing

How to Determine Baseline Project Sigma

Posted by Ted Hessing

How to determine baseline project sigma is one of the key aspects of the Measure phase. Baseline sigma calculation is part of the Data collection plan.

Sigma, denoted by the symbol σ (a Greek alphabet), represents the standard deviation of a population. Primarily, it characterizes the dispersion of a set of data values with respect to the mean. It refers to the mathematical concept of standard deviation.

Six Sigma is derived from the normal or bell curve in statistics, where each interval shows 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.

How to Determine Baseline Project Sigma.

The 68-95-99.7 rule, also known as an empirical rule used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of one, two, and three standard deviations, respectively.

Why Would You Want Your Baseline Sigma to be 1, 2, or 3?

You would want your baseline sigma to be 1, 2, or 3 because those often reveal a bad process, and you would like your team to be able to see an improvement in the process at the end of the project.

What is the Reason behind Calculating the Sigma Value?

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.

Does the Data Type (Discrete/Continuous Data) Have Any Effect on Calculating the Sigma Value?

Data type does NOT have an effect on the sigma value so long as we are counting the total number of opportunities and defects properly.

How to Determine the Baseline Project Sigma for Discrete Data

You can determine 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).

  • Unit: the item produced or created.
  • Defect: anything that causes a failure (i.e. misses the requirements of the customer.)
  • Opportunity: the number of critical quality measures we are counting on each opportunity in defects. If there are 4 types of defects, this value is 4.
  • 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).

Determine if Zero defects are needed or if there is partial credit.

  • If the process is only considered correct if there are no defects at all (100% correct), then use the DPMU calculation (defects per million units). DPMU = (Defects / Units) * 1,000,000
  • If partial credit is received for meeting some of the requirements: use the DPMO calculation (defects per million opportunities). DPMO = (Defects / Units * Opportunities) * Total 1,000,000

Examples of Baseline Sigma for Discrete Data

Example: XYZ is a commercial flight carrier operating 10,000 flights a day. There are three defect opportunities like late arrival, lost luggage, and poor in-flight experience. Let’s assume 10,000 defects have been identified. Calculate process sigma level.

  • Unit or sample size = 10,000 flights a day
  • Defects types = 3 (could be a late arrival, lost luggage, poor in-flight experience).
  • Opportunities = 10,000 flights * 3 kinds of defect opportunities = 30,000
  • Defects: 10,000 defects
  • DPMO = (Defects / Units * Opportunities) * 1,000,000
  • DPMO= (10000 /10000*3) * 1,000,000 = (1/3) * 1M. = 333,333 defects per million opportunities.
  • From the below chart, 333,333 DPMO translates to a sigma between 1.95 and 1.9.
How to Determine Baseline Project Sigma.
  • Or in excel use the formula = NORMSINV (1- (defects/ sample size * Number of defect opportunities per unit in the sample) + 1.5

Excel Template Download to Determine Baseline Sigma for Discrete Data

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Question: A company is currently operating at a 2 sigma level of quality. What will be the number of defects expected if they are able to improve to a 3 sigma, 4 sigma and 5 sigma level of quality?

Answer: A sigma of 2 equates to 308,538 defects per million opportunities or 69.2% yield. Further, A Sigma of 3 = 66,807, a sigma of 4 = 6,210, and a sigma of 5 = 233. (refer above table).

How to Determine Baseline Project Sigma for Continuous Data

Process Capability determines if a process meets the needs of the customer. Process capability compares the output of an in-control process to the specification limits. We can say the process is capable when almost all the measurements fall within the specification limits. Cp and Cpk are considered short-term potential capability measures for a process.

In Six Sigma, we want to describe the quality of a process in terms of sigma because this gives us an easy way to talk about how capable a process is at using a common mathematical framework.

How to Determine Baseline Project Sigma.

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)

The main purpose of Cpk is to determine how well a process performs when compared to its specification limits and the natural variability of the process. A larger Cpk is always better. It infers a lower probability of any item being outside the specification limits.

How to Determine Baseline Project Sigma.

Process sigma = 3* Cpk. Hence, we generally want a  Cpk of at least 1.33 [4 sigmas] or higher to satisfy most customers.

Examples of Baseline Sigma for Continuous Data

Example: The specification limit of rubber sheets is 5±1cm. The operator randomly recorded 4 subgroups of values every half an hour in three shifts. While the average mean is 4.7, and the short-term pooled standard deviation is 0.2 cm. Calculate the process sigma level.

  • USL=6cm
  • LSL = 4cm
  • Standard deviation σR= 0.2 cm
  • Process mean = 4.7cm
  • Cplower = (Process Mean – LSL)/(3*Standard Deviation) = 1.16
    Cpupper = (USL – Process Mean)/(3*Standard Deviation) = 2.16
  • Cpk = min (Cpu, Cpl) = 1.16
  • Sigma = 3 * 1.16 = 3.5

Excel Template Download to Determine Baseline Sigma for Continuous Data

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Why Use Sigma Instead of Percent?

Why can’t we just use the percentage reduction in the defects to assess the process rather than the sigma value?

The answer is that, of course, you can. Many people do. But it has to make sense for your business.

However, measuring the change in terms of sigma allows you to judge improvements and opportunities in a more consistent manner.

Important Videos for How to Determine Baseline Project Sigma

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