Carillon Technologies Limited Process Description (Descriptive Statistics)

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To Fundamentals of Probability Theory

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Process Description

Measure of Central Tendencies (These describe when the process is centered)

• Average (mean)
• Median
• Mode

Measures of Variation (These describe the spread of the process)

• Range
• Standard deviation
• Capability

Example of an SPC data set from a printed sheet from a four color sheet fed printing press Use quality control data of black solid density and specification limits that indicate that the measured density on the OK sheet that was approved by customer is +/- 0.10. Procedure: During a 10,000 sheet run, a sample was taken once every 330 sheets.

Sample Data: (in order of selection)  1.44  1.44  1.42  1.40  1.41  1.40  1.42  1.40  1.43  1.45  1.40  1.41  1.40  1.44  1.40  1.36  1.41  1.41  1.42  1.40  1.38  1.36  1.40  1.38  1.41  1.41  1.42  1.40  1.43  1.42

Measure of Central Tendency: The Average

The Average: Definition: The average (or arithmetic mean) is the sum of all the values in a sample divided by the number of values. The average represents the balancing point of the data. If we consider five black color densities from the sample data (1.44,1.44,1.42,1.40,1.41) and plot them on a number line, the average is the fulcrum.

Calculation Method:

The average is calculated by adding all the data measurements and dividing this sum by the sample size. Mathematically this is represented by the following formula:
Using the sample data one would total the color densities and then divide by the sample size (30). Total = 42.27 n = 30

Measures of Central Tendency: The Median

The Median: Definition: The median is the value that exactly divides the sample in half. One half above and one half below.	For an odd number sample it is the center value. i.e. for a sample of five (1.44,1.44,1.42,1.40,1.41) the median value is 1.42. For an even number example it is the point halfway between the center two sample points. i.e. for a sample of six (1.44,1.44,1.42,1.40,1.41, 1.40) the median value is 1.415.
For the sample data of 30 the first step is to do a frequency tally to determine the median. Since the median is the value that divides the top half from the bottom half we can observe that the median is 1.41 since 15 values are 1.41 or lower and 15 values are 1.41 or higher.	Value Frequency  Cumulative Frequency 1.36  2 2 1.37 0 2 1.38 2 4 1.39 0 4 1.40 9 13 1.41 6 19 1.42 5 24 1.43 2 26 1.44 3 29 1.45 1 30

Measures of Central Tendency: The Mode

The Mode: Defin: The Mode is the value or cell that occurs most frequently. It is possible to have a bimodal distribution ( two peaks) if the process is effected by assignable causes.
Using the frequency tally we look for the highest frequency and its' corresponding value. For the example data we see that the mode is 1.40.	Value Frequency  Cumulative Frequency 1.36  2 2 1.37 0 2 1.38 2 4 1.39 0 4 1.40 9 13 1.41 6 19 1.42 5 24 1.43 2 26 1.44 3 29 1.45 1 30

Important observation:

Since the mean (average), median and mode are nearly the same for this sample data indicates that the underlying distribution is normally distributed.

Measures of Variation: The Range

The Range Definition: The range is simply the difference between the maximum and minimum values in the data. The range is effective for small sample sizes as a measure of spread of data. (10 or fewer) Remember we only use two data points to determine the range value.	Range = largest - smallest
Example from the black solid density data:	Range = R = 1.45 - 1.36 = 0.09

Measures of Variation: The Standard Deviation

The Standard Deviation Definition: The standard deviation is a measure of the data scatter around the sample data's average. The standard deviation is the most important measure of variation. The Greek lettersigma is used to denote the standard deviation of a population.	The mathematical formula for calculating sigma is:
The sample standard deviation is the most commonly used measure. Statisticians use the letter s to denote the standard deviation of a sample.	The mathematical formula for calculating the sample standard deviation is:
Estimating the Standard Deviation Using the Range Recall that the range is the largest value - smallest value of all measurements taken.	An estimate of the standard deviation can be determined using the following formula: where is a fudge factor.

Table where n = number of measurements  From our example data:

n   n 2 1.128 10 3.078 3 1.693 15 3.472 4 2.059 20 3.735 5 2.326 25 3.931 6 2.534 30 4.086 7 2.704 40 4.322 8 2.847 50 4.498 9 2.970 100 5.015

Link to Using Excel to do Data Analysis and Creating Histograms

Further Analysis

Given that we know a measure of the process middle (i.e. average) and of the spread (standard deviation), and the process appears to be normally distributed we can predict the natural (or common cause) variation of the process.

Predicting the natural variation of a stable process: If we have a normal distribution we can predict the expected outcome of our processes: For example: If the process average = 1.409 and the standard deviation is estimated as 0.022 then we can predict that 99.7% of the population will fall between the Average +/- 3 standard deviations or 1.409 +/- 3*0.022 = 1.475 at the high end and 1.343 at the lower end. This "six sigma" spread from 1.343 to 1.475 is also known as the capability of the process.

To more about the Normal distribution

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