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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-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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X-MR Chart

What is it?

What does it look like?

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np-chart

What is it?

An np-chart is an attributes control chart used with data collected in subgroups that are the same size. Np-charts show how the process, measured by the number of nonconforming items it produces, changes over time. The process attribute (or characteristic) is always described in a yes/no, pass/fail, go/no go form. For example, the number of incomplete accident reports in a constant daily sample of five would be plotted on an np-chart. Np-charts are used to determine if the process is stable and predictable, as well as to monitor the effects of process improvement theories. Np-charts can be created using software programs like SQCpack.

What does it look like?

The np-chart shows the number of nonconforming units in subgroups of set sizes.

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control chart interpretation, and other quality metrics.

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p-chart

What is it?

A p-chart is an attributes control chart used with data collected in subgroups of varying sizes. Because the subgroup size can vary, it shows a proportion on nonconforming items rather than the actual count. P-charts show how the process changes over time. The process attribute (or characteristic) is always described in a yes/no, pass/fail, go/no go form. For example, use a p-chart to plot the proportion of incomplete insurance claim forms received weekly. The subgroup would vary, depending on the total number of claims each week. P-charts are used to determine if the process is stable and predictable, as well as to monitor the effects of process improvement theories. P-charts can be created using software programs like SQCpack.

What does it look like?

The p-chart shows the proportion of nonconforming units in subgroups of varying sizes.

When is it used?

Use a p-chart when you can answer “yes” to all these questions:

1. Do you need to assess system stability?

2. Does the data consist of counts that can be converted to proportions?

3. Are there only two outcomes to any given check?

4. Has the characteristic being charted been operationally defined prior to data collection?

5. Is the time order of subgroups preserved?

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c-chart

What is it?

A c-chart is an attributes control chart used with data collected in subgroups that are the same size. C-charts show how the process, measured by the number of nonconformities per item or group of items, changes over time. Nonconformities are defects or occurrences found in the sampled subgroup. They can be described as any characteristic that is present but should not be, or any characteristic that is not present but should be. For example a scratch, dent, bubble, blemish, missing button, and a tear would all be nonconformities. C-charts are used to determine if the process is stable and predictable, as well as to monitor the effects of process improvement theories. C-charts can be created using software products like SQCpack.

What does it look like?

The c-chart shows the number of nonconformities in subgroups of equal size.

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u-chart

What is it?

A u-chart is an attributes control chart used with data collected in subgroups of varying sizes. U-charts show how the process, measured by the number of nonconformities per item or group of items, changes over time. Nonconformities are defects or occurrences found in the sampled subgroup. They can be described as any characteristic that is present but should not be, or any characteristic that is not present but should be. For example, a scratch, dent, bubble, blemish, missing button, and a tear are all nonconformities. U-charts are used to determine if the process is stable and predictable, as well as to monitor the effects of process improvement theories. U-charts can be created using software programs like SQCpack.

What does it look like?

The u-chart shows the proportion of nonconformities units in subgroups of varying sizes.

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control chart interpretation, and other quality metrics.

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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, usually presents 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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control chart interpretation, and other quality metrics.

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Histogram: Calculate descriptive statistics

There are several statistics which are useful to describe and analyze a histogram. They are calculated to describe the area under the curve formed by its shape. These descriptive statistics can be calculated using software such as SQCpack.

Central location

The central location of a set of data points is where (on what value) the middle of the data set is located. Central location is commonly described by the mean, the median, and/or the mode. The mean is the average value of the data points. The median is the middle number in the data set when the data points are arranged from low to high. The mode is the value in the data set that occurs most often.

Spread

Both range and the standard deviation illustrate data spread. Range is calculated by subtracting the minimum data value from the maximum data value. The standard deviation is a measure that indicates how different the values are from each other and from the mean. There are two methods of calculating standard deviation using individual data points or using a samples average range. Both formulas are available here.

Skewness

Skewness is the measure of the asymmetry of a histogram (frequency distribution). A histogram with normal distribution is symmetrical. In other words, the same amount of data falls on both sides of the mean. A normal distribution will have a skewness of 0. The direction of skewness is “to the tail.” The larger the number, the longer the tail. If skewness is positive, the tail on the right side of the distribution will be longer. If skewness is negative, the tail on the left side will be longer. The formula for skewness is available here.

Kurtosis

Kurtosis is a measure of the combined weight of the tails in relation to the rest of the distribution. As the tails of a distribution become heavier, the kurtosis value will increase. As the tails become lighter the kurtosis value will decrease. A histogram with a normal distribution has a kurtosis of 0. If the distribution is peaked (tall and skinny), it will have a kurtosis greater than 0 and is said to be leptokurtic. If the distribution is flat, it will have a kurtosis value less than zero and is said to be platykurtic. The formula for kurtosis is available here.

Coefficient of variance

The coefficient of variance is a measure of how much variation exists in relation to the mean. It may also be described as a measure of the significance of the sigma in relation to the mean. The larger the coefficient of variance, the more significant the sigma, relative to the mean. For example, if the standard deviation is 10, what does it mean? If the process average (mean) is 1000, a sigma value of 10 is not very significant. However, if the average is 15, a standard deviation of 10 is VERY significant. The formula for coefficient of variance is available here.

Chi-square

In SPC, the chi-square statistic is used to determine how well the actual distribution fits the expected distribution. Chi-square compares the number of observations found in each cell in a histogram (actual) to the number of observations that would be found in an expected distribution. If the differences are small, the distribution fits the theoretical distribution. If the difference are large, the distribution probably does not fit the expected distribution.

Using Chi-square with the assumption of a normal distribution

The calculated chi-square is compared to the value in the table of constants for chi-square based on the number of “degrees of freedom.”
If the calculated chi-square is less than the value in the table, the chi-square test passes, affirming that the process has a normal distribution.
If the chi-square is larger than the value in the table, the chi-square test fails. At this confidence level, you either do not have enough data to judge the process, or you should reject the assumption that the process has a normal distribution.

Note: Theoretical percent outside of specifications may be misleading.

The formula for chi-square is available here along with the degrees of freedom table.

Follow these steps to interpret histograms.

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