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< section id="systems-definition-analysis-and-design" class="level1">Systems Definition, Analysis, and Design
< section id="system-analysis-and-design-tools" class="level2">System Analysis and Design Tools
< section id="cause-effect-diagram-fishbone" class="level3">Cause-Effect Diagram (Fishbone)
Pareto Analysis
80% of the items represent 20% of the sales or 20% of the items represent 80% of the cost. This law is a rule of thumb.
Operation Process Chart
The operation process chart only has Operations and Inspections.
< section id="flow-process-chart" class="level3">Flow Process Chart
The flow process chart forces a more detailed look at a system.
Affinity Diagram
organizes a large number of ideas into their natural relationships
< section id="left-hand-right-hand-chart" class="level3">Left Hand Right Hand Chart
Shows when each hand is busy and idle. It is sometimes called a simo chart.
< section id="modeling-techniques" class="level2">Modeling Techniques
< section id="queueing-models" class="level3">Queueing Models
\(Little's \: Law\)
\[L = \lambda W\]
\(M/M/c/K\quad Queue\)
Effective vs. Offered Load:
\[\lambda_{eff} = \lambda(1-\pi_m)\]
Waiting Time Law:
\[L = \sum_{n=0}^Mn\pi_n\] \[L_q = \sum_{n=c+1}^{c-1}(n-c)\pi_n\]
Probability of n arrivals by time t:
\[P[N(t)=n]=\frac{(\lambda t)^ne^{-\lambda t}}{n!}\]\(M/M/1/\infty\quad Queue\)
Probability of customers in system:
\[\pi_0=1-\rho\] \[\pi_n=\rho^n(1-\rho)\] \[L = \frac{\rho}{1-\rho}\] \[L_q = L-\rho=\frac{\rho^2}{1-\rho}\]\(M/G/1/\infty\quad Queue\)
Pollaczek-Khintchine Formula:
\[L = L_q+\rho=\frac{\rho^2+\lambda^2\sigma^2}{2(1-\rho)}+\rho\]\(M/M/c/M\quad Queue\)
\[D = \sum_{n=0}^{c-1}\frac{\rho^n}{n!}+\frac{\rho^c}{c!}[\frac{1-(\rho/c)^{M-c+1}}{1-\rho/c}]\] \[\pi_0=\frac{1}{D}\]
\[\pi_n= \begin{cases} \frac{\rho^n}{n!}\pi_0, \quad \text{n<c} \\ \frac{\rho^n}{c!c^{n-c}}\pi_0, \quad {n\geq c} \end{cases}\]
\[L_q=\pi_0\bigg(\frac{\rho^{c+1}}{(c-1)!(c-\rho)^2}\bigg) \bigg[1-\big(\frac{\rho}{c}\big)^{M-c}-(M-c)\big(\frac{\rho}{c}\big)^{M-c}\big(1-\frac{\rho}{c}\big)\bigg],\quad \rho\neq c\]
\[L_q=\pi_0\bigg(\frac{\rho^c (M-c)(M-c+1)}{2c!}\bigg),\quad \rho=c\]
\[L_q=\pi_0\bigg(\frac{\rho^{c+1}}{(c-1)!(c-\rho)^2}\bigg),\quad c=\infty\]
\(M/M/C/\infty \quad Queue\)
C = 1
\[P_0=1-\rho ,\quad L_q=\frac{\rho^2}{1-\rho}\]
C = 2
\[P_0=\frac{(1-\rho)}{(1=\rho}, \quad L_q = \frac{2\rho^3}{(1-\rho^2)}\]
C = 3
\[P_0 = \frac{2(1-\rho)}{2+4\rho+3\rho^2}, \quad \frac{9\rho^4}{2+2\rho-\rho^2-3\rho^3}\]
< section id="model-verification" class="level3">Model Verification
A model has been verified if a range of models produce similar results on the same situation
< section id="model-validation" class="level3">Model Validation
A model has been validated if a range of results produce similar results on the same situation
< section id="bottleneck-analysis" class="level3">Bottleneck Analysis
Optimize the process that is the bottleneck, then re-evaulate the bottleneck and repeat.
< section id="facilities-engineering-and-planning" class="level1">
Facilities Engineering and Planning
< section id="peopleequipment-requirements" class="level2">People/Equipment Requirements
\[M_j = \sum_{i=1}^{n} \frac{P_{ij}T_{ij}}{C_{ij}}\] \[M_j = \textrm{number of machines/people} \\ P = \textrm{production rate} \\ T = \textrm{production time} \\ C = \textrm{production period} \\ n = \textrm{number of products}\]
< section id="material-handling" class="level2">Material Handling
Euclidian:
\[d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]
To optimize material flow:
\[min\quad \sum_{i=1}^mw_i[(x-a_i)^2+(y-b_i)]^{\frac{1}{2}}\] If there are 4 locations with equal weight, the optimal location is the facility within a triangle of the other facilities. If there is no such facility, the optimal location is at the intersection of two lines.
When the weighted costs are proportional to the square of the Euclidean distance, it is called the ‘gravity’ problem.
\[min\quad \sum_{i=1}^mw_i[(x-a_i)^2+(y-b_i)^2]\]
\[x = \frac{\sum_{i=1}^mw_ia_i}{\sum_{i=1}^mw_i}\] \[y = \frac{\sum_{i=1}^mw_ib_i}{\sum_{i=1}^mw_i}\]
Manhattan:
\[|x_2-x_1| + |y_2-y_1|\]
To optimize flow:
\[min\quad \sum_{i=1}^m w_i(|x-a_1|+|y-b_i|)\]
The x value is the median of the location x-coordinates. The y value is the median of the location y-coordinates.
Chebyshev (simultaneous x and y movement)
\[max(|x_2-x_1|,|y_2-y_1|)\]
< section id="relationship-chart" class="level2">Relationship Chart
| Code | Closeness | Rank |
|---|---|---|
| A | Absolutely Necessary | 0.95 |
| E | Especially Important | 0.85 |
| I | Important | 0.7 |
| O | Ordinary Closeness | 0.5 |
| U | Unimportant | 0 |
| X | Not desirable | – |
< section id="supply-chain-logistics" class="level1">
Supply Chain Logistics
< section id="forecasting-methods" class="level2">Forecasting Methods
< section id="moving-average" class="level3">Moving Average
\[\hat{d}_t = \frac{\sum_{i=1}^{n}d_{t-i}}{n}\]
< section id="exponentially-weighted-moving-average" class="level3">Exponentially Weighted Moving Average
\[\hat{d}_t = \alpha d_{t-1}+(1-\alpha)\hat{d}_{t-1},\quad 0 \leq \alpha( \textrm{smoothing constant})\leq1 \\ d_{t-1} \text{ = actual demand, } \hat{d}_{t-1} \text{ = forecasted demand}\]
< section id="production-planning-methods" class="level2">Production Planning Methods
Systems to compute Master Production and Ordering Plan
Material Requirements Planning (MRP)
Manufacturing Resource Planning (MRPII)
< section id="engineering-economics" class="level2">Engineering Economics
\[\bigg(\frac{F}{P}\bigg)= (1+i)^N, \quad \bigg(\frac{P}{F}\bigg)= \frac{1}{(1+i)^N} \\ \bigg(\frac{F}{A}\bigg)= \frac{(1+i)^N-1}{i}, \quad \bigg(\frac{P}{A}\bigg)= \frac{(1+i)^N-1}{i(1+i)^N} \\ \bigg(\frac{A}{F}\bigg)= \frac{i}{(1+i)^N-1}, \quad \bigg(\frac{A}{P}\bigg)= \frac{i(1+i)^N}{(1+i)^N-1} \\ \bigg(\frac{P}{G}\bigg)=\frac{1}{i}\bigg[\frac{(1+i)^N-1}{i(1+i)^N}-\frac{N}{(1+i)^N} \bigg], \quad \bigg(\frac{A}{G}\bigg)=\frac{1}{i}-\frac{N}{(1+i)^N-1}\]
*Denominator is current value and Numerator is desired conversion
Depreciation
Modified Accelerated Cost Recovery System (MACRS) – See Tables
\[\text{Straight Line (SL) – } \frac{1}{n}\]
< section id="production-scheduling-methods" class="level2">Production Scheduling Methods
Makespan
the time it takes from the start of the first job until the end of the last job
Scheduling Sequence
- Earliest Due Date – order jobs by due date
- Shortest Processing Time – order jobs by processing time
- Critical Ratio – divide time remaining until due date by time left on the machine, order by smallest critical ratio
Johnson’s Optimal Rule for Two Machines
- Find the shortest processsing times and arbitrarily break ties
- If the shortest processing time is on Machine A, schedule immediately. If the shortest processing time is on Machine B, schedule it as late as possible.
- Eliminate the last job scheduled on the list and repeat step 1-2.
Inventory Management and Control
< section id="economic-order-quantity" class="level3">Economic Order Quantity
\[Q^*=\sqrt{\frac{2C_pD}{h}R} \\ R = \frac{1}{1-\frac{D}{P}},\quad\textrm{R=1, when replenishment is instaneous} \\ D=\textrm{demand},P=\textrm{production rate},C_p=\textrm{cost per order},h=\textrm{holding cost}\]
< section id="economic-manufacturing-quantity" class="level3">Economic Manufacturing Quantity
Use the equation above with R not equal to 1.
< section id="with-shortage-costs" class="level3">With shortage costs
\[Q^* = \sqrt{\frac{2C_pD}{h}R\big(\frac{h+z}{z}\big)} \\ z = \textrm{shortage cost}\]
\[M^*=\sqrt{\frac{2C_pD(1-\frac{D}{P})h}{z(h+z)}} \\ M = \textrm{allowed shortage}\]
< section id="carrying-cost" class="level3">Carrying Cost
\[C_T=\frac{hQ}{2}\big(1-\frac{D}{P}\big)+CD+C_p\frac{D}{Q}\]
< section id="probabilistic-inventory-and-production-models" class="level3">Probabilistic Inventory and Production Models
\[F_D(x=y^*)\ge\frac{p-c}{p+h} \\ F_D = \textrm{CDF} \\ x = \textrm{units on hand}, y^*=\textrm{optimal order quantity}, p = \textrm{loss of potential revenue},\\ h = \textrm{loss in value from holding}, c = \textrm{unit acquisition cost}\]
< section id="distribution-methods" class="level2">Distribution Methods
Transhipment:
The intermediary storage
< section id="transportation-problem" class="level3">Transportation Problem
\[min \quad \sum_{i=1}^m \sum_{j=1}^nx_{ij}c_{ij} \\ \sum_{j=1}^nx_{ij}=s_i, n = 1, 2, …, m \\ \sum_{i=1}^mx_{ij}=d_j, m = 1, 2, …, n\]
< section id="storage-and-warehousing-methods" class="level3">Storage and Warehousing Methods
- Dedicated Storage
- easy to retrieve items
- Sum of maximum of each product
- Random Storage
- more efficient use of space
- Maximum of the sums of all products
Transportation Modes
- Variable Path
- truck, vehicle anything that does not have one fixed path
- versatility
- Fixed Path
- conveyor
- tied to one path
Assignment Problem
Hungarian Procedure:
- Subtract the minimum of the row from all elements in the row
- Substract the minimum of the column from all elements in the columns
- Try to make a valid assignment using the zero elements, if all assigments cannot be made proceed to next step
- Cover all zeroes with the minimal number of lines
- From each uncovered element subtract the minimum of the uncovered y, add y to each intersection element. Go to step 3.
- Transfer the assignment plan to the original cost table.
< section id="work-design" class="level1">
Work Design
< section id="controls" class="level2">Controls
An administrative control are training, policies, or procedures.
An engineering control is a physical modification to mitigate hazards.
< section id="noise-dose" class="level2">Noise Dose
Dose
\[D=100*\big(\frac{C_1}{T_1}+\frac{C_2}{T_2}+…+\frac{C_n}{T_n}\big)\le 100\]
Time Weighted Average
\[TWA=16.61log_{10}\big(\frac{D}{100}\big)+90\]
< section id="exposure" class="level2">Exposure
Time Weighted Concentration
\[TWA=\frac{\sum_{i=1}^nC_iT_i}{\sum_{i=1}^nT_i}\]
< section id="taylor-tool-life" class="level2">Taylor Tool Life
\[VT^n=C \\ V = \textrm{speed surface feet per minute} \\ T = \textrm{tool life in minutes} \\ C,n = \textrm{constants that depend on material and tool}\]
< section id="work-sampling" class="level2">Work Sampling
\[D = Z_{\alpha/2}\sqrt{\frac{p(-1-p)}{n}}, \quad Z_{\alpha/2}\sqrt{\frac{1-p}{pn}} \\ p = \textrm{proportion of observed time} \\ D = \textrm{absolute error} \\ R = \textrm{relative error} = \frac{D}{p} \\ n = \textrm{sample size}\]
< section id="sample-size" class="level2">Sample Size
\[E = \frac{z_{\frac{\alpha}{2}}\sigma}{\sqrt{n}}\]
\[n = \bigg( \frac{z_{\frac{\alpha}{2}}\sigma}{E} \bigg)^2\]
< section id="critical-path-method" class="level2">Critical Path Method
\[T = \sum_{(i,j)\in CP}d_{ij}\]
< section id="standard-time" class="level2">Standard Time
\[\textrm{Observed Time * Pace Rating * (1 + personal time allowance) * (1 + fatigue allowance)}\]
< section id="recommended-weight-limit" class="level2">Recommended Weight Limit
Units are pounds and inches.
\[RWL = 51\cdot (\frac{10}{H})\cdot (1-.0075|V-30|)\cdot (.82+\frac{1.8}{D})\cdot (1-.0032A)\cdot FM \cdot CM \\ \textrm{H = horizontal location of the load forward of the midpoint of the ankles} \\ \textrm{V = vertical location of the load} \\ \textrm{D = Vertical travel distance between the origin and the destination} \\ \textrm{A=angle of asymmetry between hands and feet} \\ \textrm{FM = frequency multiplier (from table)} \\ \textrm{CM = coupling mulitiplier (from table)}\]
< section id="learning-curve" class="level2">Learning Curve
\[y=kx^n, n=\frac{log_{e}\phi}{log_{e}(2)} \\ \phi=\textrm{learning ratio}=\frac{T(2N)}{T(N)}, \textrm{T(N) = time to produce Nth unit} \\ \textrm{y= time to produce xth unit, k = time to produce first unit, x = cumulative number of units produced}\]
Total Learning Time:
\[T=k\frac{[(x_2+\frac{1}{2})^{n+1}-(x_1+\frac{1}{2})^{n+1}]}{n+1}\]
Remission Line:
\[y=k+\frac{(k-s)(x-1)}{1-x_s}\]
< section id="quality-control" class="level1">
Quality Control
< section id="statistical-process-control" class="level2">Statistical Process Control
< section id="x-r-chart" class="level3">X & R-Chart
\[UCL = D_4\bar{R} \\ CL = \bar{R} \\ LCL = D_3\bar{R}\]
\[UCL = \bar{\bar{X}}+A_2\bar{R} \\ CL = \bar{\bar{X}} \\ LCL = \bar{\bar{X}}-A_2\bar{R}\]
< section id="x-s-chart" class="level3">X & S-Chart
\[UCL=B_4\bar{S} \\ CL = \bar{X} \\ LCL = B_3\bar{S}\]
\[UCL = \bar{\bar{X}} + A_3\bar{S} \\ CL = \bar{\bar{X}} \\ LCL = \bar{\bar{X}}-A_3\bar{S}\]
< section id="p-chart" class="level3">P-Chart
\[UCL = \bar{p}+3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}} \\ CL = \bar{p} \\ LCL = \bar{p} – 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\]
< section id="c-chart" class="level3">C-Chart
\[UCL = \bar{c}+3\sqrt{\bar{c}} \\ CL = \bar{c} \\ LCL = \bar{c}-3\sqrt{\bar{c}}\]
< section id="tests-for-out-of-control" class="level3">Tests for Out of Control
- A single point falls outside three sigma control limits
- Two out of three successive points fall on the same side of and more than two sigma units from the center line
- Four out of five successive points fall on the same side of and more than one sigma unit from the center line
- Eight successive points fall on the same side of the center line
Control vs. Capability
In control if it is within natural variability
Is capable if it is entirely within specification
< section id="process-capability" class="level2">Process Capability
Actual Capability:
\[C_{pk}=min\bigg(\frac{\mu-LSL}{3\sigma},\frac{USL-\mu}{3\sigma}\bigg)\]
Potential Capability:
\[C_p = \frac{USL-LSL}{6\sigma}\]
< section id="reliability-analysis" class="level2">Reliability Analysis
Series:
\[R = \prod_{i=1}^n P_i\]
Parallel:
\[R = 1-\prod_{i=1}^n (1-P_i) \]
Hazard Function
\[h(x)=\frac{f(x)}{R(x)} \\ f(x) \text{ = density function, } R(x) \text{ = survival function}\]
Exponential \[h(x)=\lambda\]
Weibull \[h(x)=\frac{\beta}{\alpha}\big(\frac{x}{\alpha}\big)^{\beta-1}\]
Mean Time to Failure
\[\frac{1}{\lambda}, \\ \lambda \text{ = constant failure rate}\]
< section id="six-sigma" class="level2">Six Sigma
< !-- html table generated in R 3.4.3 by xtable 1.8-2 package --> < !-- Thu May 17 08:31:19 2018 -->| \(\sigma\) | Defects per Million |
|---|---|
| 1.00 | 158655.254 |
| 1.50 | 66807.201 |
| 2.00 | 22750.132 |
| 2.50 | 6209.665 |
| 3.00 | 1349.898 |
| 3.50 | 232.629 |
| 4.00 | 31.671 |
| 4.50 | 3.398 |
| 5.00 | 0.287 |
| 5.50 | 0.019 |
| 6.00 | 0.001 |
< section id="statistics" class="level1">
Statistics
< section id="normal-distribution" class="level2">Normal Distribution
z-score
\[z=\frac{x-\mu}{\sigma}\]
Confidence Interval
\[\bar{x}\pm\frac{z_{\alpha/2} \sigma}{\sqrt{n}}\]
Two-means comparison:
\[z_0=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\]
< section id="student-t-distribution" class="level2">student-t Distribution
t-score:
\[t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\]
Confidence Interval
\[\bar{x}\pm\frac{t_{\alpha/2,n-1}s}{\sqrt{n}}\]
Two-means comparison:
\[t_0=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]
df for Two Sample t-test:
\[df=\frac{\big(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\big)^2}{\frac{(\frac{s_1^2}{n_1})^2}{n_1-1}+{\frac{(\frac{s_1^2}{n_1})^2}{n_1-1}}}\] Paired t-test:
\[t_0 = \frac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}\]
< section id="hypothesis-testing" class="level2">Hypothesis Testing
| \(H_0 \text{ is true}\) | \(H_0 \text{ is false}\) | |
|---|---|---|
| \(\text{Accept } H_0\) | Correct | Type II Error |
| \(\text{Reject } H_0\) | Type I Error | Correct |
Chi-Squared Goodness of Fit
\[\chi^2=\sum_{j=1}^k\frac{(O_j-E_j)^2}{E_j}\]
< section id="linear-regression" class="level2">Linear Regression
\[SSR=\sum_{i=1}^n(\hat{y}_i-\bar{y})^2\]
\[SSE = \sum_{i=1}^n(y_i-\hat{y}_i)^2\]
\[SST = \sum_{i=1}^n(y_i-\bar{y})^2\]
\[R^2=\frac{SSR}{SST} = 1-\frac{SSE}{SST}\]
< section id="anova" class="level2">ANOVA
\[SSA+SSE=SST\]
One-Way
Given Treatment A:
\[SSA+SSE=SST\]
| SS | df | MS | F |
|---|---|---|---|
| SSA | a-1 | SSA/dfA | MSA/MSE |
| SSE | a(n-1) | SSE/dfE | |
| SST | an-1 |
Two-Way
Given treatment factors A & B:
\[SST=SSA+SSB+SSAB+SSE\]
| SS | df | MS | F |
|---|---|---|---|
| SSA | a-1 | SSA/dfA | MSA/MSE |
| SSB | b-1 | SSB/dfB | MSB/MSE |
| SSAB | (a-1)(b-1) | SSAB/dfAB | MSAB/MSE |
| SSE | ab(n-1) | SSE/dfE | |
| SST | abn-1 |
Bayesian Analysis
Bayes’ Theorem
\[P(A|B)=\frac{P(B|A)P(A)}{P(B)}=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^\prime)P(A^\prime)}\]
< section id="distributions" class="level2">
Distributions
< !-- html table generated in R 3.4.3 by xtable 1.8-2 package --> < !-- Thu May 17 08:31:21 2018 -->| Distribution | pmf | cdf | mean | variance | parameters |
|---|---|---|---|---|---|
| Binomial | \(\binom{n}{x}p^x(1-p)^{n-x}\) | \(\sum_{i=0}^{\lfloor x \rfloor}\binom{n}{i}p^i(1-p)^{n-i}\) | \(np\) | \(np(1-p)\) | \(\text{n = number of trials} \\ \text{p = success probability}\) |
| Discrete Uniform | \(\frac{1}{b-a+1}\) | \(\frac{\lfloor x \rfloor – a + 1}{b-a+1}\) | \(\frac{a+b}{2}\) | \(\frac{(b-a+1)^2-1}{12}\) | \(\text{a = minimum} \\ \text{b = maximum}\) |
| Poisson | \(\frac{\lambda^x e^{-\lambda}}{x!}\) | \(e^{-\lambda}\sum_{i=0}^{\lfloor x \rfloor}\frac{\lambda^i}{i!}\) | \(\lambda\) | \(\lambda\) | \(\lambda\text{ = rate}\) |
| Geometric | \(p(1-p)^{x}\) | \(1-(1-p)^{x+1}\) | \(\frac{1-p}{p}\) | \(\frac{1-p}{p^2}\) | \(\text{k = number of trials} \\ \text{p = success probability}\) |
| Negative Binomial | \(\binom{k+x-1}{x}p^k(1-p)^x\) | \(-\) | \(\frac{k(1-p)}{p}\) | \(\frac{k(1-p)}{p^2}\) | \(\text{k = number of successes}\\ \text{p = success probability}\) |
| Distribution | cdf | mean | variance | parameters | |
|---|---|---|---|---|---|
| Uniform | \(\frac{1}{b-a}\) | \(\frac{x-a}{b-a}\) | \(\frac{a+b}{2}\) | \(\frac{(b-a)^2}{12}\) | \(\text{a = minimum} \\ \text{b = maximum}\) |
| Exponential | \(\lambda e^{-\lambda x}\) | \(1-e^{-\lambda x}\) | \(\frac{1}{\lambda}\) | \(\frac{1}{\lambda^2}\) | \(\lambda \text{ = rate}\) |
| Normal | \(\frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\) | \(\frac{1}{2}\big[1+erf\big(\frac{x-\mu}{\sigma\sqrt{2}}\big)\big]\) | \(\mu\) | \(\sigma^2\) | \(\mu \text{ = mean} \\ \sigma^2 \text{ = variance}\) |
| PERT beta | \(-\) | \(-\) | \(\frac{a+4m+b}{6}\) | \(\frac{(b-a)^2}{36}\) | \(\text{a = 1st percentile} \\ \text{b = 99th percentile} \\ \text{m = mode}\) |
| Triangular | \(\begin{cases} \frac{(x-a)^2}{(b-a)(c-a)},\quad a\le x\le c \\ 1-\frac{(b-x)^2}{(b-a)(b-c)},\quad c<x\le b \end{cases}\) | \(\begin{cases} \frac{(x-a)^2}{(b-a)(c-a)},\quad a\le x\le c \\ 1-\frac{(b-x)^2}{(b-a)(b-c)}, \quad c<x\le b \end{cases}\) | \(\frac{a+b+c}{3}\) | \(\frac{a^2+m^2+b^2-ca-ab-cb}{18}\) | \(\text{a = minimum} \\ \text{b = maximum} \\ \text{c = mode}\) |
| Gamma | \(\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\) | \(\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)\) | \(\alpha\beta\) | \(\alpha\beta^2\) | \(\alpha \text{ = shape} \\ \beta \text{ = scale}\) |
| Weibull | \(\frac{\beta}{\alpha}\binom{x}{\alpha}^{\beta-1}e^{-{(\frac{x}{\alpha}})^\beta}\) | \(1-e^{-(\frac{x}{\alpha})^\beta}\) | \(-\) | \(-\) | \(-\) |
| Lognormal | \(\frac{1}{x\sigma\sqrt{2\pi}}e^{\frac{(ln x-\mu)^2}{2\sigma^2}}\) | \(\frac{1}{2}+ \frac{1}{2} erf \big[ \frac{ln x-\mu}{\sigma\sqrt{2}}\big]\) | \(e^{\mu+\frac{\sigma^2}{2}}\) | \([e^{\sigma^2}-1] e^{2\mu+\sigma^2}\) | \(\mu\text{ = mean} \\ \sigma^2 \text{ = variance}\) |
\(\text{Factors for Control Charts}\)
< !-- html table generated in R 3.4.3 by xtable 1.8-2 package --> < !-- Thu May 17 08:31:21 2018 -->| \(n\) | \(A\) | \(A_2\) | \(A_3\) | \(c_4\) | \(B_3\) | \(B_4\) | \(B_5\) | \(B_6\) | \(d_2\) | \(d_2^{-1}\) | \(d_3\) | \(D_1\) | \(D_2\) | \(D_3\) | \(D_4\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 2.121 | 1.880 | 2.659 | 0.798 | 0.000 | 3.267 | 0.000 | 2.606 | 1.128 | 0.886 | 0.853 | 0.000 | 3.686 | 0.000 | 3.267 |
| 3 | 1.732 | 1.023 | 1.954 | 0.886 | 0.000 | 2.568 | 0.000 | 2.276 | 1.693 | 0.591 | 0.888 | 0.000 | 4.358 | 0.000 | 2.575 |
| 4 | 1.500 | 0.729 | 1.628 | 0.921 | 0.000 | 2.266 | 0.000 | 2.088 | 2.059 | 0.486 | 0.880 | 0.000 | 4.698 | 0.000 | 2.282 |
| 5 | 1.342 | 0.577 | 1.427 | 0.940 | 0.000 | 2.089 | 0.000 | 1.964 | 2.326 | 0.430 | 0.864 | 0.000 | 4.918 | 0.000 | 2.114 |
| 6 | 1.225 | 0.483 | 1.287 | 0.952 | 0.030 | 1.970 | 0.029 | 1.874 | 2.534 | 0.395 | 0.848 | 0.000 | 5.079 | 0.000 | 2.004 |
| 7 | 1.134 | 0.419 | 1.182 | 0.959 | 0.118 | 1.882 | 0.113 | 1.806 | 2.704 | 0.370 | 0.833 | 0.205 | 5.204 | 0.076 | 1.924 |
| 8 | 1.061 | 0.373 | 1.099 | 0.965 | 0.185 | 1.815 | 0.179 | 1.751 | 2.847 | 0.351 | 0.820 | 0.388 | 5.307 | 0.136 | 1.864 |
| 9 | 1.000 | 0.337 | 1.032 | 0.969 | 0.239 | 1.761 | 0.232 | 1.707 | 2.970 | 0.337 | 0.808 | 0.547 | 5.394 | 0.184 | 1.816 |
| 10 | 0.949 | 0.308 | 0.975 | 0.973 | 0.284 | 1.716 | 0.276 | 1.669 | 3.078 | 0.325 | 0.797 | 0.686 | 5.469 | 0.223 | 1.777 |
| 11 | 0.905 | 0.285 | 0.927 | 0.975 | 0.321 | 1.679 | 0.313 | 1.637 | 3.173 | 0.315 | 0.787 | 0.811 | 5.535 | 0.256 | 1.744 |
| 12 | 0.866 | 0.266 | 0.886 | 0.978 | 0.354 | 1.646 | 0.346 | 1.610 | 3.258 | 0.307 | 0.778 | 0.923 | 5.594 | 0.283 | 1.717 |
| 13 | 0.832 | 0.249 | 0.850 | 0.979 | 0.382 | 1.618 | 0.374 | 1.585 | 3.336 | 0.300 | 0.770 | 1.025 | 5.647 | 0.307 | 1.693 |
| 14 | 0.802 | 0.235 | 0.817 | 0.981 | 0.406 | 1.594 | 0.399 | 1.563 | 3.407 | 0.293 | 0.763 | 1.118 | 5.696 | 0.328 | 1.672 |
| 15 | 0.775 | 0.223 | 0.789 | 0.982 | 0.428 | 1.572 | 0.421 | 1.544 | 3.472 | 0.288 | 0.756 | 1.203 | 5.740 | 0.347 | 1.653 |
| 16 | 0.750 | 0.212 | 0.763 | 0.984 | 0.448 | 1.552 | 0.440 | 1.526 | 3.532 | 0.283 | 0.750 | 1.282 | 5.782 | 0.363 | 1.637 |
| 17 | 0.728 | 0.203 | 0.739 | 0.985 | 0.466 | 1.534 | 0.458 | 1.511 | 3.588 | 0.279 | 0.744 | 1.356 | 5.820 | 0.378 | 1.622 |
| 18 | 0.707 | 0.194 | 0.718 | 0.985 | 0.482 | 1.518 | 0.475 | 1.496 | 3.640 | 0.275 | 0.739 | 1.424 | 5.856 | 0.391 | 1.609 |
| 19 | 0.688 | 0.187 | 0.698 | 0.986 | 0.497 | 1.503 | 0.490 | 1.483 | 3.689 | 0.271 | 0.733 | 1.489 | 5.889 | 0.404 | 1.596 |
| 20 | 0.671 | 0.180 | 0.680 | 0.987 | 0.510 | 1.490 | 0.504 | 1.470 | 3.735 | 0.268 | 0.729 | 1.549 | 5.921 | 0.415 | 1.585 |
| 21 | 0.655 | 0.173 | 0.663 | 0.988 | 0.523 | 1.477 | 0.516 | 1.459 | 3.778 | 0.265 | 0.724 | 1.606 | 5.951 | 0.425 | 1.575 |
| 22 | 0.640 | 0.167 | 0.647 | 0.988 | 0.534 | 1.466 | 0.528 | 1.448 | 3.819 | 0.262 | 0.720 | 1.660 | 5.979 | 0.435 | 1.565 |
| 23 | 0.626 | 0.162 | 0.633 | 0.989 | 0.545 | 1.455 | 0.539 | 1.438 | 3.858 | 0.259 | 0.716 | 1.711 | 6.006 | 0.443 | 1.557 |
| 24 | 0.612 | 0.157 | 0.619 | 0.989 | 0.555 | 1.445 | 0.549 | 1.429 | 3.895 | 0.257 | 0.712 | 1.759 | 6.032 | 0.452 | 1.548 |
| 25 | 0.600 | 0.153 | 0.606 | 0.990 | 0.565 | 1.435 | 0.559 | 1.420 | 3.931 | 0.254 | 0.708 | 1.805 | 6.056 | 0.459 | 1.541 |
\(\text{Normal Distribution}\)
| z | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.0 | 0.5000 | 0.5040 | 0.5080 | 0.5120 | 0.5160 | 0.5199 | 0.5239 | 0.5279 | 0.5319 | 0.5359 |
| 0.1 | 0.5398 | 0.5438 | 0.5478 | 0.5517 | 0.5557 | 0.5596 | 0.5636 | 0.5675 | 0.5714 | 0.5753 |
| 0.2 | 0.5793 | 0.5832 | 0.5871 | 0.5910 | 0.5948 | 0.5987 | 0.6026 | 0.6064 | 0.6103 | 0.6141 |
| 0.3 | 0.6179 | 0.6217 | 0.6255 | 0.6293 | 0.6331 | 0.6368 | 0.6406 | 0.6443 | 0.6480 | 0.6517 |
| 0.4 | 0.6554 | 0.6591 | 0.6628 | 0.6664 | 0.6700 | 0.6736 | 0.6772 | 0.6808 | 0.6844 | 0.6879 |
| 0.5 | 0.6915 | 0.6950 | 0.6985 | 0.7019 | 0.7054 | 0.7088 | 0.7123 | 0.7157 | 0.7190 | 0.7224 |
| 0.6 | 0.7257 | 0.7291 | 0.7324 | 0.7357 | 0.7389 | 0.7422 | 0.7454 | 0.7486 | 0.7517 | 0.7549 |
| 0.7 | 0.7580 | 0.7611 | 0.7642 | 0.7673 | 0.7704 | 0.7734 | 0.7764 | 0.7794 | 0.7823 | 0.7852 |
| 0.8 | 0.7881 | 0.7910 | 0.7939 | 0.7967 | 0.7995 | 0.8023 | 0.8051 | 0.8078 | 0.8106 | 0.8133 |
| 0.9 | 0.8159 | 0.8186 | 0.8212 | 0.8238 | 0.8264 | 0.8289 | 0.8315 | 0.8340 | 0.8365 | 0.8389 |
| 1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 | 0.8508 | 0.8531 | 0.8554 | 0.8577 | 0.8599 | 0.8621 |
| 1.1 | 0.8643 | 0.8665 | 0.8686 | 0.8708 | 0.8729 | 0.8749 | 0.8770 | 0.8790 | 0.8810 | 0.8830 |
| 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 | 0.8962 | 0.8980 | 0.8997 | 0.9015 |
| 1.3 | 0.9032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 | 0.9115 | 0.9131 | 0.9147 | 0.9162 | 0.9177 |
| 1.4 | 0.9192 | 0.9207 | 0.9222 | 0.9236 | 0.9251 | 0.9265 | 0.9279 | 0.9292 | 0.9306 | 0.9319 |
| 1.5 | 0.9332 | 0.9345 | 0.9357 | 0.9370 | 0.9382 | 0.9394 | 0.9406 | 0.9418 | 0.9429 | 0.9441 |
| 1.6 | 0.9452 | 0.9463 | 0.9474 | 0.9484 | 0.9495 | 0.9505 | 0.9515 | 0.9525 | 0.9535 | 0.9545 |
| 1.7 | 0.9554 | 0.9564 | 0.9573 | 0.9582 | 0.9591 | 0.9599 | 0.9608 | 0.9616 | 0.9625 | 0.9633 |
| 1.8 | 0.9641 | 0.9649 | 0.9656 | 0.9664 | 0.9671 | 0.9678 | 0.9686 | 0.9693 | 0.9699 | 0.9706 |
| 1.9 | 0.9713 | 0.9719 | 0.9726 | 0.9732 | 0.9738 | 0.9744 | 0.9750 | 0.9756 | 0.9761 | 0.9767 |
| 2.0 | 0.9772 | 0.9778 | 0.9783 | 0.9788 | 0.9793 | 0.9798 | 0.9803 | 0.9808 | 0.9812 | 0.9817 |
| 2.1 | 0.9821 | 0.9826 | 0.9830 | 0.9834 | 0.9838 | 0.9842 | 0.9846 | 0.9850 | 0.9854 | 0.9857 |
| 2.2 | 0.9861 | 0.9864 | 0.9868 | 0.9871 | 0.9875 | 0.9878 | 0.9881 | 0.9884 | 0.9887 | 0.9890 |
| 2.3 | 0.9893 | 0.9896 | 0.9898 | 0.9901 | 0.9904 | 0.9906 | 0.9909 | 0.9911 | 0.9913 | 0.9916 |
| 2.4 | 0.9918 | 0.9920 | 0.9922 | 0.9925 | 0.9927 | 0.9929 | 0.9931 | 0.9932 | 0.9934 | 0.9936 |
| 2.5 | 0.9938 | 0.9940 | 0.9941 | 0.9943 | 0.9945 | 0.9946 | 0.9948 | 0.9949 | 0.9951 | 0.9952 |
| 2.6 | 0.9953 | 0.9955 | 0.9956 | 0.9957 | 0.9959 | 0.9960 | 0.9961 | 0.9962 | 0.9963 | 0.9964 |
| 2.7 | 0.9965 | 0.9966 | 0.9967 | 0.9968 | 0.9969 | 0.9970 | 0.9971 | 0.9972 | 0.9973 | 0.9974 |
| 2.8 | 0.9974 | 0.9975 | 0.9976 | 0.9977 | 0.9977 | 0.9978 | 0.9979 | 0.9979 | 0.9980 | 0.9981 |
| 2.9 | 0.9981 | 0.9982 | 0.9982 | 0.9983 | 0.9984 | 0.9984 | 0.9985 | 0.9985 | 0.9986 | 0.9986 |
| 3.0 | 0.9987 | 0.9987 | 0.9987 | 0.9988 | 0.9988 | 0.9989 | 0.9989 | 0.9989 | 0.9990 | 0.9990 |
| 3.1 | 0.9990 | 0.9991 | 0.9991 | 0.9991 | 0.9992 | 0.9992 | 0.9992 | 0.9992 | 0.9993 | 0.9993 |
| 3.2 | 0.9993 | 0.9993 | 0.9994 | 0.9994 | 0.9994 | 0.9994 | 0.9994 | 0.9995 | 0.9995 | 0.9995 |
| 3.3 | 0.9995 | 0.9995 | 0.9995 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9997 |
| 3.4 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9997 | 0.9998 |
\(t \text{ Distribution}\)
| \(v \big\backslash \alpha\) | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|---|---|---|---|---|
| 1 | 3.0777 | 6.3138 | 12.7062 | 31.8205 | 63.6567 |
| 2 | 1.8856 | 2.9200 | 4.3027 | 6.9646 | 9.9248 |
| 3 | 1.6377 | 2.3534 | 3.1824 | 4.5407 | 5.8409 |
| 4 | 1.5332 | 2.1318 | 2.7764 | 3.7469 | 4.6041 |
| 5 | 1.4759 | 2.0150 | 2.5706 | 3.3649 | 4.0321 |
| 6 | 1.4398 | 1.9432 | 2.4469 | 3.1427 | 3.7074 |
| 7 | 1.4149 | 1.8946 | 2.3646 | 2.9980 | 3.4995 |
| 8 | 1.3968 | 1.8595 | 2.3060 | 2.8965 | 3.3554 |
| 9 | 1.3830 | 1.8331 | 2.2622 | 2.8214 | 3.2498 |
| 10 | 1.3722 | 1.8125 | 2.2281 | 2.7638 | 3.1693 |
| 11 | 1.3634 | 1.7959 | 2.2010 | 2.7181 | 3.1058 |
| 12 | 1.3562 | 1.7823 | 2.1788 | 2.6810 | 3.0545 |
| 13 | 1.3502 | 1.7709 | 2.1604 | 2.6503 | 3.0123 |
| 14 | 1.3450 | 1.7613 | 2.1448 | 2.6245 | 2.9768 |
| 15 | 1.3406 | 1.7531 | 2.1314 | 2.6025 | 2.9467 |
| 16 | 1.3368 | 1.7459 | 2.1199 | 2.5835 | 2.9208 |
| 17 | 1.3334 | 1.7396 | 2.1098 | 2.5669 | 2.8982 |
| 18 | 1.3304 | 1.7341 | 2.1009 | 2.5524 | 2.8784 |
| 19 | 1.3277 | 1.7291 | 2.0930 | 2.5395 | 2.8609 |
| 20 | 1.3253 | 1.7247 | 2.0860 | 2.5280 | 2.8453 |
| 21 | 1.3232 | 1.7207 | 2.0796 | 2.5176 | 2.8314 |
| 22 | 1.3212 | 1.7171 | 2.0739 | 2.5083 | 2.8188 |
| 23 | 1.3195 | 1.7139 | 2.0687 | 2.4999 | 2.8073 |
| 24 | 1.3178 | 1.7109 | 2.0639 | 2.4922 | 2.7969 |
| 25 | 1.3163 | 1.7081 | 2.0595 | 2.4851 | 2.7874 |
| 26 | 1.3150 | 1.7056 | 2.0555 | 2.4786 | 2.7787 |
| 27 | 1.3137 | 1.7033 | 2.0518 | 2.4727 | 2.7707 |
| 28 | 1.3125 | 1.7011 | 2.0484 | 2.4671 | 2.7633 |
| 29 | 1.3114 | 1.6991 | 2.0452 | 2.4620 | 2.7564 |
| 30 | 1.3104 | 1.6973 | 2.0423 | 2.4573 | 2.7500 |
| \(\infty\) | 1.2816 | 1.6449 | 1.9600 | 2.3263 | 2.5758 |
\(\chi^2 \text{ Distribution}\)
| \(v \big\backslash \alpha\) | 0.995 | 0.99 | 0.975 | 0.95 | 0.9 | 0.75 | 0.5 | 0.25 | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.10 | 0.45 | 1.32 | 2.71 | 3.84 | 5.02 | 6.63 | 7.88 | 10.83 |
| 2 | 0.01 | 0.02 | 0.05 | 0.10 | 0.21 | 0.58 | 1.39 | 2.77 | 4.61 | 5.99 | 7.38 | 9.21 | 10.60 | 13.82 |
| 3 | 0.07 | 0.11 | 0.22 | 0.35 | 0.58 | 1.21 | 2.37 | 4.11 | 6.25 | 7.81 | 9.35 | 11.34 | 12.84 | 16.27 |
| 4 | 0.21 | 0.30 | 0.48 | 0.71 | 1.06 | 1.92 | 3.36 | 5.39 | 7.78 | 9.49 | 11.14 | 13.28 | 14.86 | 18.47 |
| 5 | 0.41 | 0.55 | 0.83 | 1.15 | 1.61 | 2.67 | 4.35 | 6.63 | 9.24 | 11.07 | 12.83 | 15.09 | 16.75 | 20.52 |
| 6 | 0.68 | 0.87 | 1.24 | 1.64 | 2.20 | 3.45 | 5.35 | 7.84 | 10.64 | 12.59 | 14.45 | 16.81 | 18.55 | 22.46 |
| 7 | 0.99 | 1.24 | 1.69 | 2.17 | 2.83 | 4.25 | 6.35 | 9.04 | 12.02 | 14.07 | 16.01 | 18.48 | 20.28 | 24.32 |
| 8 | 1.34 | 1.65 | 2.18 | 2.73 | 3.49 | 5.07 | 7.34 | 10.22 | 13.36 | 15.51 | 17.53 | 20.09 | 21.95 | 26.12 |
| 9 | 1.73 | 2.09 | 2.70 | 3.33 | 4.17 | 5.90 | 8.34 | 11.39 | 14.68 | 16.92 | 19.02 | 21.67 | 23.59 | 27.88 |
| 10 | 2.16 | 2.56 | 3.25 | 3.94 | 4.87 | 6.74 | 9.34 | 12.55 | 15.99 | 18.31 | 20.48 | 23.21 | 25.19 | 29.59 |
| 11 | 2.60 | 3.05 | 3.82 | 4.57 | 5.58 | 7.58 | 10.34 | 13.70 | 17.28 | 19.68 | 21.92 | 24.72 | 26.76 | 31.26 |
| 12 | 3.07 | 3.57 | 4.40 | 5.23 | 6.30 | 8.44 | 11.34 | 14.85 | 18.55 | 21.03 | 23.34 | 26.22 | 28.30 | 32.91 |
| 13 | 3.57 | 4.11 | 5.01 | 5.89 | 7.04 | 9.30 | 12.34 | 15.98 | 19.81 | 22.36 | 24.74 | 27.69 | 29.82 | 34.53 |
| 14 | 4.07 | 4.66 | 5.63 | 6.57 | 7.79 | 10.17 | 13.34 | 17.12 | 21.06 | 23.68 | 26.12 | 29.14 | 31.32 | 36.12 |
| 15 | 4.60 | 5.23 | 6.26 | 7.26 | 8.55 | 11.04 | 14.34 | 18.25 | 22.31 | 25.00 | 27.49 | 30.58 | 32.80 | 37.70 |
| 16 | 5.14 | 5.81 | 6.91 | 7.96 | 9.31 | 11.91 | 15.34 | 19.37 | 23.54 | 26.30 | 28.85 | 32.00 | 34.27 | 39.25 |
| 17 | 5.70 | 6.41 | 7.56 | 8.67 | 10.09 | 12.79 | 16.34 | 20.49 | 24.77 | 27.59 | 30.19 | 33.41 | 35.72 | 40.79 |
| 18 | 6.26 | 7.01 | 8.23 | 9.39 | 10.86 | 13.68 | 17.34 | 21.60 | 25.99 | 28.87 | 31.53 | 34.81 | 37.16 | 42.31 |
| 19 | 6.84 | 7.63 | 8.91 | 10.12 | 11.65 | 14.56 | 18.34 | 22.72 | 27.20 | 30.14 | 32.85 | 36.19 | 38.58 | 43.82 |
| 20 | 7.43 | 8.26 | 9.59 | 10.85 | 12.44 | 15.45 | 19.34 | 23.83 | 28.41 | 31.41 | 34.17 | 37.57 | 40.00 | 45.31 |
| 21 | 8.03 | 8.90 | 10.28 | 11.59 | 13.24 | 16.34 | 20.34 | 24.93 | 29.62 | 32.67 | 35.48 | 38.93 | 41.40 | 46.80 |
| 22 | 8.64 | 9.54 | 10.98 | 12.34 | 14.04 | 17.24 | 21.34 | 26.04 | 30.81 | 33.92 | 36.78 | 40.29 | 42.80 | 48.27 |
| 23 | 9.26 | 10.20 | 11.69 | 13.09 | 14.85 | 18.14 | 22.34 | 27.14 | 32.01 | 35.17 | 38.08 | 41.64 | 44.18 | 49.73 |
| 24 | 9.89 | 10.86 | 12.40 | 13.85 | 15.66 | 19.04 | 23.34 | 28.24 | 33.20 | 36.42 | 39.36 | 42.98 | 45.56 | 51.18 |
| 25 | 10.52 | 11.52 | 13.12 | 14.61 | 16.47 | 19.94 | 24.34 | 29.34 | 34.38 | 37.65 | 40.65 | 44.31 | 46.93 | 52.62 |
| 30 | 13.79 | 14.95 | 16.79 | 18.49 | 20.60 | 24.48 | 29.34 | 34.80 | 40.26 | 43.77 | 46.98 | 50.89 | 53.67 | 59.70 |
| 40 | 20.71 | 22.16 | 24.43 | 26.51 | 29.05 | 33.66 | 39.34 | 45.62 | 51.81 | 55.76 | 59.34 | 63.69 | 66.77 | 73.40 |
| 50 | 27.99 | 29.71 | 32.36 | 34.76 | 37.69 | 42.94 | 49.33 | 56.33 | 63.17 | 67.50 | 71.42 | 76.15 | 79.49 | 86.66 |
| 60 | 35.53 | 37.48 | 40.48 | 43.19 | 46.46 | 52.29 | 59.33 | 66.98 | 74.40 | 79.08 | 83.30 | 88.38 | 91.95 | 99.61 |
| 70 | 43.28 | 45.44 | 48.76 | 51.74 | 55.33 | 61.70 | 69.33 | 77.58 | 85.53 | 90.53 | 95.02 | 100.43 | 104.21 | 112.32 |
| 80 | 51.17 | 53.54 | 57.15 | 60.39 | 64.28 | 71.14 | 79.33 | 88.13 | 96.58 | 101.88 | 106.63 | 112.33 | 116.32 | 124.84 |
| 90 | 59.20 | 61.75 | 65.65 | 69.13 | 73.29 | 80.62 | 89.33 | 98.65 | 107.57 | 113.15 | 118.14 | 124.12 | 128.30 | 137.21 |
| 100 | 67.33 | 70.06 | 74.22 | 77.93 | 82.36 | 90.13 | 99.33 | 109.14 | 118.50 | 124.34 | 129.56 | 135.81 | 140.17 | 149.45 |
\(F(v_1, v_2) \text{ Distribution}\)
| \(v_2 \big\backslash v_1\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 | 24 | 30 | 40 | 60 | 120 | \(\infty\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 161.45 | 199.50 | 215.71 | 224.58 | 230.16 | 233.99 | 236.77 | 238.88 | 240.54 | 241.88 | 243.91 | 245.95 | 248.01 | 249.05 | 250.10 | 251.14 | 252.20 | 253.25 | 254.31 |
| 2 | 18.51 | 19.00 | 19.16 | 19.25 | 19.30 | 19.33 | 19.35 | 19.37 | 19.38 | 19.40 | 19.41 | 19.43 | 19.45 | 19.45 | 19.46 | 19.47 | 19.48 | 19.49 | 19.50 |
| 3 | 10.13 | 9.55 | 9.28 | 9.12 | 9.01 | 8.94 | 8.89 | 8.85 | 8.81 | 8.79 | 8.74 | 8.70 | 8.66 | 8.64 | 8.62 | 8.59 | 8.57 | 8.55 | 8.53 |
| 4 | 7.71 | 6.94 | 6.59 | 6.39 | 6.26 | 6.16 | 6.09 | 6.04 | 6.00 | 5.96 | 5.91 | 5.86 | 5.80 | 5.77 | 5.75 | 5.72 | 5.69 | 5.66 | 5.63 |
| 5 | 6.61 | 5.79 | 5.41 | 5.19 | 5.05 | 4.95 | 4.88 | 4.82 | 4.77 | 4.74 | 4.68 | 4.62 | 4.56 | 4.53 | 4.50 | 4.46 | 4.43 | 4.40 | 4.36 |
| 6 | 5.99 | 5.14 | 4.76 | 4.53 | 4.39 | 4.28 | 4.21 | 4.15 | 4.10 | 4.06 | 4.00 | 3.94 | 3.87 | 3.84 | 3.81 | 3.77 | 3.74 | 3.70 | 3.67 |
| 7 | 5.59 | 4.74 | 4.35 | 4.12 | 3.97 | 3.87 | 3.79 | 3.73 | 3.68 | 3.64 | 3.57 | 3.51 | 3.44 | 3.41 | 3.38 | 3.34 | 3.30 | 3.27 | 3.23 |
| 8 | 5.32 | 4.46 | 4.07 | 3.84 | 3.69 | 3.58 | 3.50 | 3.44 | 3.39 | 3.35 | 3.28 | 3.22 | 3.15 | 3.12 | 3.08 | 3.04 | 3.01 | 2.97 | 2.93 |
| 9 | 5.12 | 4.26 | 3.86 | 3.63 | 3.48 | 3.37 | 3.29 | 3.23 | 3.18 | 3.14 | 3.07 | 3.01 | 2.94 | 2.90 | 2.86 | 2.83 | 2.79 | 2.75 | 2.71 |
| 10 | 4.96 | 4.10 | 3.71 | 3.48 | 3.33 | 3.22 | 3.14 | 3.07 | 3.02 | 2.98 | 2.91 | 2.85 | 2.77 | 2.74 | 2.70 | 2.66 | 2.62 | 2.58 | 2.54 |
| 12 | 4.75 | 3.89 | 3.49 | 3.26 | 3.11 | 3.00 | 2.91 | 2.85 | 2.80 | 2.75 | 2.69 | 2.62 | 2.54 | 2.51 | 2.47 | 2.43 | 2.38 | 2.34 | 2.30 |
| 15 | 4.54 | 3.68 | 3.29 | 3.06 | 2.90 | 2.79 | 2.71 | 2.64 | 2.59 | 2.54 | 2.48 | 2.40 | 2.33 | 2.29 | 2.25 | 2.20 | 2.16 | 2.11 | 2.07 |
| 20 | 4.35 | 3.49 | 3.10 | 2.87 | 2.71 | 2.60 | 2.51 | 2.45 | 2.39 | 2.35 | 2.28 | 2.20 | 2.12 | 2.08 | 2.04 | 1.99 | 1.95 | 1.90 | 1.84 |
| 24 | 4.26 | 3.40 | 3.01 | 2.78 | 2.62 | 2.51 | 2.42 | 2.36 | 2.30 | 2.25 | 2.18 | 2.11 | 2.03 | 1.98 | 1.94 | 1.89 | 1.84 | 1.79 | 1.73 |
| 30 | 4.17 | 3.32 | 2.92 | 2.69 | 2.53 | 2.42 | 2.33 | 2.27 | 2.21 | 2.16 | 2.09 | 2.01 | 1.93 | 1.89 | 1.84 | 1.79 | 1.74 | 1.68 | 1.62 |
| 40 | 4.08 | 3.23 | 2.84 | 2.61 | 2.45 | 2.34 | 2.25 | 2.18 | 2.12 | 2.08 | 2.00 | 1.92 | 1.84 | 1.79 | 1.74 | 1.69 | 1.64 | 1.58 | 1.51 |
| 60 | 4.00 | 3.15 | 2.76 | 2.53 | 2.37 | 2.25 | 2.17 | 2.10 | 2.04 | 1.99 | 1.92 | 1.84 | 1.75 | 1.70 | 1.65 | 1.59 | 1.53 | 1.47 | 1.39 |
| 120 | 3.92 | 3.07 | 2.68 | 2.45 | 2.29 | 2.18 | 2.09 | 2.02 | 1.96 | 1.91 | 1.83 | 1.75 | 1.66 | 1.61 | 1.55 | 1.50 | 1.43 | 1.35 | 1.25 |
| \(\infty\) | 3.84 | 3.00 | 2.60 | 2.37 | 2.21 | 2.10 | 2.01 | 1.94 | 1.88 | 1.83 | 1.75 | 1.67 | 1.57 | 1.52 | 1.46 | 1.39 | 1.32 | 1.22 | 1.00 |
| \(v_2 \big\backslash v_1\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 | 24 | 30 | 40 | 60 | 120 | \(\infty\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 4052 | 5000 | 5403 | 5625 | 5764 | 5859 | 5928 | 5981 | 6022 | 6056 | 6106 | 6157 | 6209 | 6235 | 6261 | 6287 | 6313 | 6339 | 6366 |
| 2 | 98.50 | 99.00 | 99.17 | 99.25 | 99.30 | 99.33 | 99.36 | 99.37 | 99.39 | 99.40 | 99.42 | 99.43 | 99.45 | 99.46 | 99.47 | 99.47 | 99.48 | 99.49 | 99.50 |
| 3 | 34.12 | 30.82 | 29.46 | 28.71 | 28.24 | 27.91 | 27.67 | 27.49 | 27.35 | 27.23 | 27.05 | 26.87 | 26.69 | 26.60 | 26.50 | 26.41 | 26.32 | 26.22 | 26.13 |
| 4 | 21.20 | 18.00 | 16.69 | 15.98 | 15.52 | 15.21 | 14.98 | 14.80 | 14.66 | 14.55 | 14.37 | 14.20 | 14.02 | 13.93 | 13.84 | 13.75 | 13.65 | 13.56 | 13.46 |
| 5 | 16.26 | 13.27 | 12.06 | 11.39 | 10.97 | 10.67 | 10.46 | 10.29 | 10.16 | 10.05 | 9.89 | 9.72 | 9.55 | 9.47 | 9.38 | 9.29 | 9.20 | 9.11 | 9.02 |
| 6 | 13.75 | 10.92 | 9.78 | 9.15 | 8.75 | 8.47 | 8.26 | 8.10 | 7.98 | 7.87 | 7.72 | 7.56 | 7.40 | 7.31 | 7.23 | 7.14 | 7.06 | 6.97 | 6.88 |
| 7 | 12.25 | 9.55 | 8.45 | 7.85 | 7.46 | 7.19 | 6.99 | 6.84 | 6.72 | 6.62 | 6.47 | 6.31 | 6.16 | 6.07 | 5.99 | 5.91 | 5.82 | 5.74 | 5.65 |
| 8 | 11.26 | 8.65 | 7.59 | 7.01 | 6.63 | 6.37 | 6.18 | 6.03 | 5.91 | 5.81 | 5.67 | 5.52 | 5.36 | 5.28 | 5.20 | 5.12 | 5.03 | 4.95 | 4.86 |
| 9 | 10.56 | 8.02 | 6.99 | 6.42 | 6.06 | 5.80 | 5.61 | 5.47 | 5.35 | 5.26 | 5.11 | 4.96 | 4.81 | 4.73 | 4.65 | 4.57 | 4.48 | 4.40 | 4.31 |
| 10 | 10.04 | 7.56 | 6.55 | 5.99 | 5.64 | 5.39 | 5.20 | 5.06 | 4.94 | 4.85 | 4.71 | 4.56 | 4.41 | 4.33 | 4.25 | 4.17 | 4.08 | 4.00 | 3.91 |
| 12 | 9.33 | 6.93 | 5.95 | 5.41 | 5.06 | 4.82 | 4.64 | 4.50 | 4.39 | 4.30 | 4.16 | 4.01 | 3.86 | 3.78 | 3.70 | 3.62 | 3.54 | 3.45 | 3.36 |
| 15 | 8.68 | 6.36 | 5.42 | 4.89 | 4.56 | 4.32 | 4.14 | 4.00 | 3.89 | 3.80 | 3.67 | 3.52 | 3.37 | 3.29 | 3.21 | 3.13 | 3.05 | 2.96 | 2.87 |
| 20 | 8.10 | 5.85 | 4.94 | 4.43 | 4.10 | 3.87 | 3.70 | 3.56 | 3.46 | 3.37 | 3.23 | 3.09 | 2.94 | 2.86 | 2.78 | 2.69 | 2.61 | 2.52 | 2.42 |
| 24 | 7.82 | 5.61 | 4.72 | 4.22 | 3.90 | 3.67 | 3.50 | 3.36 | 3.26 | 3.17 | 3.03 | 2.89 | 2.74 | 2.66 | 2.58 | 2.49 | 2.40 | 2.31 | 2.21 |
| 30 | 7.56 | 5.39 | 4.51 | 4.02 | 3.70 | 3.47 | 3.30 | 3.17 | 3.07 | 2.98 | 2.84 | 2.70 | 2.55 | 2.47 | 2.39 | 2.30 | 2.21 | 2.11 | 2.01 |
| 40 | 7.31 | 5.18 | 4.31 | 3.83 | 3.51 | 3.29 | 3.12 | 2.99 | 2.89 | 2.80 | 2.66 | 2.52 | 2.37 | 2.29 | 2.20 | 2.11 | 2.02 | 1.92 | 1.80 |
| 60 | 7.08 | 4.98 | 4.13 | 3.65 | 3.34 | 3.12 | 2.95 | 2.82 | 2.72 | 2.63 | 2.50 | 2.35 | 2.20 | 2.12 | 2.03 | 1.94 | 1.84 | 1.73 | 1.60 |
| 120 | 6.85 | 4.79 | 3.95 | 3.48 | 3.17 | 2.96 | 2.79 | 2.66 | 2.56 | 2.47 | 2.34 | 2.19 | 2.03 | 1.95 | 1.86 | 1.76 | 1.66 | 1.53 | 1.38 |
| \(\infty\) | 6.63 | 4.61 | 3.78 | 3.32 | 3.02 | 2.80 | 2.64 | 2.51 | 2.41 | 2.32 | 2.18 | 2.04 | 1.88 | 1.79 | 1.70 | 1.59 | 1.47 | 1.32 | 1.00 |
< section id="references" class="level2">
References
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