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Histogram: Study the shape

A histogram can be created using software such as SQCpack. How would you describe the shape of the histogram?

Bell-shaped: A bell-shaped picture, shown below, usuallypresents a normal distribution.

Bimodal: A bimodal shape, shown below, has two peaks. This shape may show that the data has come from two different systems. If this shape occurs, the two sources should be separated and analyzed separately.

Skewed right: Some histograms will show a skewed distribution to the right, as shown below. A distribution skewed to the right is said to be positively skewed. This kind of distribution has a large number of occurrences in the lower value cells (left side) and few in the upper value cells (right side). A skewed distribution can result when data is gathered from a system with has a boundary such as zero. In other words, all the collected data has values greater than zero.

Skewed left: Some histograms will show a skewed distribution to the left, as shown below. A distribution skewed to the left is said to be negatively skewed. This kind of distribution has a large number of occurrences in the upper value cells (right side) and few in the lower value cells (left side). A skewed distribution can result when data is gathered from a system with a boundary such as 100. In other words, all the collected data has values less than 100.

Uniform: A uniform distribution, as shown below, provides little information about the system. An example would be a state lottery, in which each class has about the same number of elements. It may describe a distribution which has several modes (peaks). If your histogram has this shape, check to see if several sources of variation have been combined. If so, analyze them separately. If multiple sources of variation do not seem to be the cause of this pattern, different groupings can be tried to see if a more useful pattern results. This could be as simple as changing the starting and ending points of the cells, or changing the number of cells. A uniform distribution often means that the number of classes is too small.

Random: A random distribution, as shown below, has no apparent pattern. Like the uniform distribution, it may describe a distribution that has several modes (peaks). If your histogram has this shape, check to see if several sources of variation have been combined. If so, analyze them separately. If multiple sources of variation do not seem to be the cause of this pattern, different groupings can be tried to see if a more useful pattern results. This could be as simple as changing the starting and ending points of the cells, or changing the number of cells. A random distribution often means there are too many classes.

Follow these steps to interpret histograms.

Study the shape.
Calculate descriptive statistics.
Compare the histogram to the normal distribution.

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Run chart

What is it?

A run chart is a line graph of data plotted over time. By collecting and charting data over time, you can find trends or patterns in the process. Because they do not use control limits, run charts cannot tell you if a process is stable. However, they can show you how the process is running. The run chart can be a valuable tool at the beginning of a project, as it reveals important information about a process before you have collected enough data to create reliable control limits.

What does it look like?

Run charts show individual data points in chronological order.

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X-MR chart

What is it?

An individuals and moving range (X-MR) chart is a pair of control charts for processes with a subgroup size of one. Used to determine if a process is stable and predictable, it creates a picture of how the system changes over time. The individual (X) chart displays individual measurements. The moving range (MR) chart shows variability between one data point and the next. Individuals and moving range charts are also used to monitor the effects of process improvement theories.

What does it look like?

The individuals chart, on top, shows each reading. It is used to analyze central location. The moving range chart, on the bottom, shows the difference between consecutive readings. It is used to study system variability.

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Median chart

What is it?

A median chart is a special purpose variation of the X-bar chart. This chart uses the median instead of the subgroup average to show the system’s central location. The median is the middle point when data points are arranged from high to low. The chart shows all the individual readings. Use charts to determine if the system is stable and predictable or to monitor the effects of process improvement theories.

Although median charts show both central location and spread, they are often paired with range charts.

What does it look like?

The median chart shows each reading in a subgroup. Subgroup medians are connected by the data line and are used to analyze the central location.

When is it used?

Use the median chart when you want to plot all the measured values, not just subgroup statistics. This may be the case when subgroup ranges vary a great deal, as showing all the points will emphasize the spread. It shows users that individual data points can fall outside the control limits, while the central location is within the limits.

Use the median chart when you can answer yes to these questions:

Do you need to assess system stability?
Is the data in variables form?
Is the data collected in subgroups larger than one?
Is the time order of subgroups preserved?
Do you want to see individual data points?

Getting the most

Collect as many subgroups as possible before calculating control limits. With smaller amounts of data, the median chart may not represent variability of the entire system. The more subgroups you use in control limit calculations, the more reliable the analysis. Typically, twenty to twenty-five subgroups will be used in control limit calculations.

Median charts have several applications. When you begin to improve a system, use them to assess the system’s stability.

After the stability has been assessed, determine if you need to stratify the data. You may find entirely different results between shifts, among workers, among different machines, among lots of materials, etc. To see if variability on the median chart is caused by these factors, you should collect and enter data in a way that lets you stratify by time, location, symptom, operator, and lots.

You can also use median charts to analyze the results of process improvements. Here you would consider how the process is running and compare it to how it ran in the past. Do process changes produce the desired improvement?

Finally, use median charts for standardization. This means you should continue collecting and analyzing data throughout the process operation. If you made changes to the system and stopped collecting data, you would have only perception and opinion to tell you whether the changes actually improved the system. Without a control chart, there is no way to know if the process has changed or to identify sources of process variability.

How is it used?

Variables data is normally analyzed in pairs of charts which present data in terms of location or central location and spread. Location, usually the top chart, shows data in relation to the process average. It is presented in X-bar, individuals, or median charts. Spread, usually the bottom chart, looks at piece-by-piece variation. Range, sigma, and moving range charts are used to illustrate process spread. Another aspect of these variables control charts is that the sample size is generally constant.

Use the following types of charts and analysis to study variables data:

These charts, and more, can be created easily using software packages such as SQCpack.

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X-bar and sigma

What is it?

An X-bar and s (sigma) chart is a special purpose variation of the X-bar and R chart. Used with processes that have a subgroup size of 11 or more, X-bar and s charts show if the system is stable and predictable. They are also used to monitor the effects of process improvement theories. Instead of using subgroup range to chart variability, these charts use subgroup standard deviation. Because standard deviation uses each individual reading to calculate variability, it provides a more effective measure of the process spread. X-bar and sigma charts To create an X-bar and sigma chart using software, download a copy of SQCpack.

What does it look like?

The X-bar chart, on top, shows the mean or average of each subgroup. It is used to analyze central location. The sigma chart, on the bottom, shows how the data is spread and used to study system variability.

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X-bar and range chart

What is it?

An X-bar and R (range) chart is a pair of control charts used with processes that have a subgroup size of two or more. The standard chart for variables data, X-bar and R charts help determine if a process is stable and predictable. The X-bar chart shows how the mean or average changes over time and the R chart shows how the range of the subgroups changes over time. It is also used to monitor the effects of process improvement theories. As the standard, the X-bar and R chart will work in place of the X-bar and s or median and R chart. To create an X-bar and R chart using software, download a copy of SQCpack.

What does it look like?

The X-bar chart, on top, shows the mean or average of each subgroup. It is used to analyze central location. The range chart, on the bottom, shows how the data is spread. It is used to study system variability.

When is it used?

You can use X-bar and R charts for any process with a subgroup size greater than one. Typically, it is used when the subgroup size falls between two and ten, and X-bar and s charts are used with subgroups of eleven or more.

Use X-bar and R charts when you can answer yes to these questions:

Do you need to assess system stability?
Is the data in variables form?
Is the data collected in subgroups larger than one but less than eleven?
Is the time order of subgroups preserved?

Getting the most

Collect as many subgroups as possible before calculating control limits. With smaller amounts of data, the X-bar and R chart may not represent variability of the entire system. The more subgroups you use in control limit calculations, the more reliable the analysis. Typically, twenty to twenty-five subgroups will be used in control limit calculations.

X-bar and R charts have several applications. When you begin improving a system, use them to assess the system’s stability.

After the stability has been assessed, determine if you need to stratify the data. You may find entirely different results between shifts, among workers, among different machines, among lots of materials, etc. To see if variability on the X-bar and R chart is caused by these factors, collect and enter data in a way that lets you stratify by time, location, symptom, operator, and lots.

You can also use X-bar and R charts to analyze the results of process improvements. Here you would consider how the process is running and compare it to how it ran in the past. Do process changes produce the desired improvement?

Finally, use X-bar and R charts for standardization. This means you should continue collecting and analyzing data throughout the process operation. If you made changes to the system and stopped collecting data, you would have only perception and opinion to tell you whether the changes actually improved the system. Without a control chart, there is no way to know if the process has changed or to identify sources of process variability.

How is it used?

Use the following types of charts and analysis to study variables data:

These charts, and more, can be created easily using software packages such as SQCpack.

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control chart interpretation, and other quality metrics.

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Variables data (measurements)

What is it?

Variables data is data that is acquired through measurements, such as length, time, diameter, strength, weight, temperature, density, thickness, pressure, and height. With variables data, you can decide the measurement’s degree of accuracy. For example, you can measure an item to the nearest centimeter, millimeter, or micron.

How is it used?

Use the following types of charts and analysis to study variables data:

These charts, and more, can be created easily using software packages such as SQCpack.

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Measurement Systems Analysis – Improving measurement accuracy with gage R&R

What is it?

Variation is inherent to any system, and the data collection process is no exception. However, excessive variation in the data collection process will appear as variation on the control chart and can have a negative effect on process analysis. In addition to using operational definitions to ensure measurement consistency, you should periodically perform repeatability and reproducibility tests and recalibrate gages.

Gage R&R refers to testing the repeatability and reproducibility of the measurement system. Repeatability is the variation found in a series of measurements that have been taken by one person using one gage to measure one characteristic of an item. Reproducibility is the variation in a series of measurements that have been taken by different people using the same gage to measure one characteristic of an item.

Gage R&R studies let you address two major categories of variation in measuring systems: gage variability and operator variability. Gage variability refers to factors that affect the gage’s accuracy, such as its sensitivity to temperature, magnetic and electrical fields and, if it is mounted, how tight or loose the mount is. Operator variability refers to variation caused by differences among people. It can be caused by different interpretations of a vague operational definition, as well as differences in training, attitude, and fatigue level.

Performing gage R&R studies can be made easier by using software such as GAGEpack.

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Operational definition

What is it?

An operational definition, when applied to data collection, is a clear, concise detailed definition of a measure. The need for operational definitions is fundamental when collecting all types of data. It is particularly important when a decision is being made about whether something is correct or incorrect, or when a visual check is being made where there is room for confusion.

For example, data collected will be erroneous if those completing the checks have different views of what constitutes a fault at the end of a glass panel production line. Defective glass panels may be passed and good glass panels may be rejected. Similarly, when invoices are being checked for errors, the data collection will be meaningless if the definition of an error has not been specified.

When collecting data, it is essential that everyone in the system has the same understanding and collects data in the same way. Operational definitions should therefore be made before the collection of data begins.

When is it used?

Any time data is being collected, it is necessary to define how to collect the data. Data that is not defined will usually be inconsistent and will give an erroneous result. It is easy to assume that those collecting the data understand what and how to complete the task. However, people have different opinions and views, and these will affect the data collection. The only way to ensure consistent data collection is by means of a detailed operational definition that eliminates ambiguity.

What does it look like?

Example 1 Attributes

Characteristic of interest:	Number of black spots per radiator grill.

Measuring instrument:	The observation will be performed with the naked eye (or with corrective lenses if normally worn), under the light available in the work station (in 100% working order, i.e., no burned-out bulbs).

Method of test:	The number of black spots per radiator grill will be counted by taking samples at the work station. The sample should be studied at a distance of 18 inches (roughly half an arm’s length) from the eye. Only the top surface of the grill is to be examined.

Decision criteria:	Wipe the top surface of the grill with the palm of your hand and look for any black specks embedded in the plastic. Any observed black speck of any size counts as a black spot.

Example 2 Variables

Characteristic of interest:	Diameter of 48 inch rod

Measuring instrument:	Micrometer

Method of test:	The sample size is n=3. Measure 3 rods every hour. When the grinder releases the rod, take one measurement each at 8″ down, 24″ down, and 40″ down from the notched end. Tighten the micrometer as much as possible. Record to 4 decimal points. If the fifth number to the right of the decimal point is 5 or higher, round the fourth number up one.

How is it made?

Identify the characteristic of interest.
Identify the characteristic to be measured or the defect type of concern.
Select the measuring instrument.
The measuring instrument is usually either a physical piece of measuring equipment such as a micrometer, weighing scale, or clock; or alternatively, a visual check. Whenever a visual check is used, it is necessary to state whether normal eyesight is to be used or a visual aid such as a magnifying glass. In the example, normal eyesight is sufficient. On some occasions, it may also be necessary to state the distance the observer should be from the item being checked. In general, the closer the observer, the more detail will be seen. In the example, a clear visual indication is given of acceptable and unacceptable, so the observer needs to be in a position where the decision can be made. When completing a visual check, the type of lighting may also need to be specified. Certain colors and types of light can make defects more apparent.
Describe the test method.
The test method is the actual procedure used for taking the measurement. When measuring time, the start and finish points of the test need to be specified. When taking any measurement, the degree of accuracy also needs to be stated. For instance, it is important to know whether time will be measured in hours, minutes, or seconds.
State the decision criteria.
The decision criteria represents the conclusion of the test. Does the problem exist? Is the item correct? Whenever a visual check is used, a clear definition of acceptable versus unacceptable is essential. Physical examples or photographs of acceptable and unacceptable, together with written support, are the best definitions.
Document the operational definition.
It is important that the operational definition is documented and standardized. Definitions should be included in training materials and job procedure sheets. The results of steps 1 through 4 should be included in one document. The operational definition and the appropriate standards should be kept at the work station.
Test the operational definition.
It is essential to test the operational definition before implementation. Input from those that are actually going to complete the tests is particularly important. The operational definition should make the task clear and easy to perform. The best way to test an operational definition is to ask different people to complete the test on several items by following the operational definition. Watch how they perform the test. Are they completing the test as expected? Are the results consistent? Are the results correct?

The above article is an excerpt from the “Operational definition” chapter of Practical Tools for Continuous Improvement: Volume 1 – Statistical Tools. The full chapter provides more details on creating operational definition.