$($

change to

$8$

$$oz

$.$

$$bottles

$)$

$$The mean

$\backslash$

mu determines the location of a normal distribution. The dispersion or spread of the distribution is determined by the standard deviation

$\backslash$

sigma

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$$The normal distribution, theoretically, extends to infinity on both sides of the mean but

$9$

$9$

$.$

$7$

$4$

$\%$

$$of the data is contained within three standard deviations around the mean. A normal table gives more information about the probabilistic behavior of normal variables. Some commonly used probabilistic information is shown as follows. Probability

$($

$$

z

$<$

$$X

$<$

$$z

$)$

$$Standard Normal Variable is within z

$\backslash$

sigma deviations around the mean Number of

$\backslash$

sigma away from mean

$3$

$-$

$9$

$9$

$.$

$7$

$4$

$2$

$-$

$9$

$5$

$.$

$4$

$4$

$1$

$.$

$9$

$6$

$-$

$9$

$5$

$.$

$0$

$1$

$-$

$6$

$8$

$.$

$2$

$6$

$$TYPES OF ERRORS SPC methods are used to control processes and to detect when deviations, problems, or changes have occurred in the process. If an unusual, assignable, cause is at work, it is expected to show up in certain probabilistic behavior. Management would then stop the process to identify the assignable cause. Thus, the goal of SPC is to identify assignable causes through observing behavior of a quality characteristic. Two types of errors can happen in practice. First, data might fall outside of acceptable limits

$,$

$$which would seem to indicate that some changes have occurred to the process and that some assignable cause is influencing the process data. This might not actually be the case. Even when the process operates under natural causes, a certain number of observations would be expected to fall outside acceptable limits

$.$

$$Thus$,$ the method would indicate an assignable cause and recommend stopping the process to investigate it

$.$

$$In this case, it would be unnecessary, hence would be considered a mistake. This is called as Type I error. The degree of Type I error that management is willing to tolerate is usually indicated by

$\backslash$

alpha

$.$

$$A second kind of mistake happens when an assignable cause is really at work but does not show up as deviant behavior in data. Thus, the method is unable to detect when a change has happened in the process and recommends that management take no action. This is called as Type II error and usually referred to by the symbol

$\backslash$

beta

$($

beta

$)$

$.$

$$The example with

$1$

$6$

$-$

oz

$.$

$$soda cans can illustrate these two errors. Suppose if soda cans have

$1$

$6$

$$ounces of soda on the average with a standard deviation of

$0$

$.$

$0$

$1$

$$ounce$.$ Suppose any can with amount in the

$3$

$\backslash$

sigma limits

$,$

$$which corresponds to

$\backslash$

alpha as

$0$

$.$

$0$

$3$

$\%$

$,$

$$would be considered as being a

$1$

$6$

$-$

oz can. Then all soda cans with amounts

$1$

$5$

$.$

$9$

$7$

$$and

$1$

$6$

$.$

$0$

$3$

$$oz

$.$

$$would be accepted. Finding cans in the process with

$1$

$6$

$.$

$0$

$4$

$$and

$1$

$5$

$.$

$9$

$6$

$$oz

$.$

$$would indicate that there are assignable causes and that the process should be investigated. However, a certain percent of time cans would have more than

$1$

$6$

$.$

$3$

$$oz

$.$

$$or less than

$1$

$5$

$.$

$9$

$7$

$$oz

$.$

$,$

$$and this variation would fall in the realm of being random. Now suppose that the soda filling machine is filling slowly so that actually less than

$1$

$6$

$$oz

$.$

$$is being filled in many cans. Even if the average has slipped below

$1$

$6$

$$oz

$.$

$,$

$$so long as data points are between the limits

$,$

$$nothing would be thought of as being changed.