The Normal Distribution and Control Charts
- Page ID
- 220
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Random Distribution
There is a special distribution that we will use often and is a distribution for a continuous random variable that has the following properties:
- It is symmetric about the mean
- It approaches the horizontal axis on both the left and right side without touching, that is the x-axis is a asymptote.
- It is bell shaped with transition points one standard deviation from the mean.
- Approximately 68% of the data points lie within one standard deviation of the mean.
- Approximately 95% of the data points lie within two standard deviations of the mean.
- Approximately 99.7% of the data points lie within three standard deviations of the mean.
You can play with the graphs by going to
You are the manager at a new toy store and want to determine how many Monopoly games to stock in you store. The mean number of Monopoly games that sell per month is 22 with a standard deviation of 6. Assume that this distribution is Normal.
What is the probability that next month you will sell between 10 and 34 games?
Solution
We notice that
\[ 22 - 2(6) = 10 \nonumber \]
and
\[ 34 = 22 + 2(6) \nonumber \]
We want to know what the probability is that the outcome lies within two standard deviations of the mean. Property 5 says that this percent is about 95%.
If you stock 45 games, should you feel secure about not running out?
Solution
Since three standard deviations above the mean is
\[ 22 + 3(6) = 40 \nonumber \]
and 45 is above that, there is a less than 0.3% chance of running out. You should feel very secure.
Control Charts
We often want to determine if things are beginning to stray from the norm as time goes on.
It has been determined that the mean number of errors that medical staff at a hospital makes is 0.002 per hour with a standard deviation of 0.0003.The medical board wanted to determine if long working hours was related to mistakes. During the day, the medical staff was observed to see when they made mistakes. The table illustrates the finding.
| Hours Worked | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Mistakes per Hour | 0.0019 | 0.0022 | 0.0015 | 0.0017 | 0.0020 | 0.0022 | 0.0018 | 0.0028 | 0.0019 | 0.0027 |
It is difficult to see a trend from just looking at the table and we will create a chart that better illustrates the trends. We call the system out of control if at least one of the following three events occur:
- Out of Control Signal 1: At least one point falls beyond the \(3\sigma\) level
- Out of Control Signal 2: A run of nine consecutive points is on the same side of the center line (usually the mean).
- Out of Control Signal 3: At least two of three consecutive points lie beyond the \(2\sigma\) level on same side of the center line (usually the mean).
For our example we have
\[m + \sigma = 0.002 + 0.0003 = 0.0023 \nonumber \]
\[ m - \sigma = 0.002 - 0.0003 = 0.0017 \nonumber \]
\[m + 2\sigma = 0.002 + 0.0006 = 0.0026 \nonumber \]
\[ m - 2\sigma = 0.002 - 0.0006 = 0.0014 \nonumber \]
\[m + 3\sigma = 0.002 + 0.0009 = 0.0029 \nonumber \]
\[ m - 3\sigma = 0.002 - 0.0009 = 0.0011 \nonumber \]
We now graph the points on a control chart.
We can see that two of the last three data points lie beyond two standard deviations above the mean, which gives out of control warning signals. The information should make the hospital administration weary about long hours.
Contributors
- Larry Green (Lake Tahoe Community College)