developed at utah state university dept of engr & tech educ — materials and processes 5.6...

Developed at Utah State UniversityDeveloped at Utah State UniversityDept of Engr & Tech Educ — www.ete.usu.eduDept of Engr & Tech Educ — www.ete.usu.edu

Materials and ProcessesMaterials and Processes5.6 calculate the mean and standard5.6 calculate the mean and standard

deviation of a sample or populationdeviation of a sample or population

Materials and Processes

Statistical Process Control (SPC)

5.6 calculate the mean and standard deviation of a sample or population




Performance Objective

Given a data set of a sample or population calculate the mean and standard deviation using an electronic spread sheet.




Enabling Objectives

1. match the following terms with the appropriate definition: statistical process control (SPC), mean, standard deviation (sigma), population, sample, normal distribution (bell curve), tolerance, upper and lower limits

2. calculate a mean3. calculate a standard deviation




Quality Control

• Classical Quality Control is achieved by observing (measuring & testing) important properties of the finished product to determine acceptance or rejection.

• Statistical Process Control uses statistical tools to observe the performance of the production line to predict significant deviations that may result in rejected products.




Normal Distribution

• At the heart of statistical process control is the concept of a normal distribution or bell-shaped curve.

• The frequency distribution of many events in nature and in processes used in manufacturing are found in practice to be approximated closely by the normal curve, and they are said to be normally distributed.




Bell-shape Curve

The frequency curve, which is bell-shaped, is called the normal curve or normal distribution.




Statistical ProcessControl (SPC)

• Statistical Process Control is a method for achieving quality control in manufacturing processes using statistical tools to detect if production is within limits.

• Two of the more common tools used are mean and standard deviation which also describe the shape of the normal distribution.




Sample & Population• Often the size of a group makes it

impractical to perform SPC calculations of the population. When this occurs, a sample is used.

• The population is the entire group.• A sample is a sub-group of the

population with the expectation that it is representative of the larger group.




Mean

Mean is the average of a set of scores or values and is represented by the symbol, . The mean is calculated by adding the set of values then dividing by the number of values.

Example:

x

65

30

5

58854

x




Mean height of female high school students in the USA

• It would be very impractical to measure the height of all female high school students in the USA in order to calculate their mean height.

• It would be more practical to use a sample and only measure the height of the female students in your high school.




Standard Deviation

Standard Deviation is an estimate of the average each item’s value differs from the group’s mean. Standard deviation is represented by the symbol, .




Standard Deviationof a Population

• The formula for calculating the standard deviation of a population is:

• Although this formula may appear complicated, it is relatively easy to solve.

n

xx

2




Example—population data:5, 2, 8, 6, 4

Step 1: List the values of the data in the first column of the table labeled x.

Step 2: Calculate the mean of the data in the first column.

x

5

2

8

6

4

= 5 x

xx 2xx





Step 3: Complete column two by subtracting the mean from each item value,

.

x

5 0

2 -3

8 3

6 1

4 -1

= 5 x

xx 2xx

xx





Step 4: Now square the value in column two and enter it in column three.

Step 5: Sum the values in column three and divide by n.

20/5 = 4

x

5 0 0

2 -3 9

8 3 9

6 1 1

4 -1 1

= 5 = 20x

xx 2xx





Step 6:To obtain the standard deviation of the data, finish by calculating the square root of the value obtained in step 5.

24




Standard Deviationof a Sample

• The formula for calculating the standard deviation of a sample is:

• The only difference between the two standard deviation formulas is the denominator (n vs. n – 1).

1

2

n

xx




Test Your Understanding

Without performing calculations on paper, what is the mean of the following population of data: 20, 40, 50, 60, 80?

A. 5B. 20C. 50D. 60

Hint: Mean is the average




Test Your Understanding

Without performing calculations on paper, what is the standard deviation of the following population of data: 20, 40, 50, 60, 80?

A. 5B. 20C. 50D. 60

Hint: Standard deviation is the estimated average each item deviates from the mean




Electronic Spreadsheet

• Microsoft Excel is an excellent tool to calculate both the mean and standard deviation of a population or sample.

• The command AVERAGE would be used to calculate the mean.

• The command STDEVP would be used to calculate the standard deviation of a population.

• The command STDEV would be used to calculate the standard deviation of a sample.




Excel Spreadsheet

The graphic to the right shows the use of a spreadsheet to calculate the mean and standard deviation of a sample.




Happy Calculating

developed at utah state university dept of engr & tech educ — materials and processes 5.6...

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