Descriptive Statistics
Sample Mean
x̄ = (1/n) ∑i=1n xi
Sample Variance
s2 = (1/(n-1)) ∑i=1n (xi - x̄)2
Uses n-1 (Bessel's correction) for unbiased estimation.
Standard Deviation
s = √s2
Coefficient of Variation
CV = (s / x̄) × 100%
Skewness
g1 = (1/n) ∑ [(xi - x̄)/s]3
Kurtosis (Excess)
g2 = (1/n) ∑ [(xi - x̄)/s]4 - 3
Statistical Inference
Standard Error of Mean
SE = s / √n
Confidence Interval for Mean (t)
x̄ ± tα/2, n-1 × (s / √n)
One-Sample t-Test
t = (x̄ - μ0) / (s / √n)
df = n - 1
Two-Sample t-Test (Pooled)
t = (x̄1 - x̄2) / √[sp2(1/n1 + 1/n2)]
sp2 = [(n1-1)s12 + (n2-1)s22] / (n1 + n2 - 2)
Paired t-Test
t = d̄ / (sd / √n)
d̄ = mean of differences, sd = std dev of differences
Chi-Square Test
χ2 = ∑ (O - E)2 / E
Regression Analysis
Simple Linear Regression Slope
b1 = ∑(xi - x̄)(yi - ȳ) / ∑(xi - x̄)2
Intercept
b0 = ȳ - b1x̄
Coefficient of Determination
R2 = 1 - SSres/SStot = SSreg/SStot
Pearson Correlation
r = ∑(xi - x̄)(yi - ȳ) / √[∑(xi - x̄)2 ∑(yi - ȳ)2]
Control Chart Formulas
X-bar and R Chart
X-bar Chart Limits
UCL = X̄ + A2R̄
CL = X̄
LCL = X̄ - A2R̄
CL = X̄
LCL = X̄ - A2R̄
R Chart Limits
UCL = D4R̄
CL = R̄
LCL = D3R̄
CL = R̄
LCL = D3R̄
Individuals and Moving Range Chart
I-MR Chart Limits
UCLX = X̄ + 2.66 MR̄
LCLX = X̄ - 2.66 MR̄
UCLMR = 3.27 MR̄
LCLMR = 0
LCLX = X̄ - 2.66 MR̄
UCLMR = 3.27 MR̄
LCLMR = 0
p Chart (Proportion)
p Chart Limits
UCL = p̄ + 3√[p̄(1-p̄)/n]
CL = p̄
LCL = p̄ - 3√[p̄(1-p̄)/n]
CL = p̄
LCL = p̄ - 3√[p̄(1-p̄)/n]
c Chart (Defects)
c Chart Limits
UCL = c̄ + 3√c̄
CL = c̄
LCL = c̄ - 3√c̄
CL = c̄
LCL = c̄ - 3√c̄
Control Chart Constants
| n | A2 | A3 | D3 | D4 | d2 | B3 | B4 |
|---|---|---|---|---|---|---|---|
| 2 | 1.880 | 2.659 | 0 | 3.267 | 1.128 | 0 | 3.267 |
| 3 | 1.023 | 1.954 | 0 | 2.574 | 1.693 | 0 | 2.568 |
| 4 | 0.729 | 1.628 | 0 | 2.282 | 2.059 | 0 | 2.266 |
| 5 | 0.577 | 1.427 | 0 | 2.114 | 2.326 | 0 | 2.089 |
| 6 | 0.483 | 1.287 | 0 | 2.004 | 2.534 | 0.030 | 1.970 |
| 7 | 0.419 | 1.182 | 0.076 | 1.924 | 2.704 | 0.118 | 1.882 |
| 8 | 0.373 | 1.099 | 0.136 | 1.864 | 2.847 | 0.185 | 1.815 |
| 9 | 0.337 | 1.032 | 0.184 | 1.816 | 2.970 | 0.239 | 1.761 |
| 10 | 0.308 | 0.975 | 0.223 | 1.777 | 3.078 | 0.284 | 1.716 |
Process Capability
Cp (Potential Capability)
Cp = (USL - LSL) / 6σ
Measures spread relative to specification width. Does not account for centering.
Cpk (Actual Capability)
Cpk = min[(USL - μ)/3σ, (μ - LSL)/3σ]
Accounts for process centering. Cpk = Cp(1 - k) where k = |μ - m|/[(USL-LSL)/2]
Pp (Performance Index)
Pp = (USL - LSL) / 6s
Uses overall standard deviation s rather than within-subgroup σ.
Ppk (Performance Index)
Ppk = min[(USL - x̄)/3s, (x̄ - LSL)/3s]
Statistical Tables
t-Distribution Critical Values (Two-Tailed)
| df | α=0.10 | α=0.05 | α=0.02 | α=0.01 |
|---|---|---|---|---|
| 5 | 2.015 | 2.571 | 3.365 | 4.032 |
| 10 | 1.812 | 2.228 | 2.764 | 3.169 |
| 15 | 1.753 | 2.131 | 2.602 | 2.947 |
| 20 | 1.725 | 2.086 | 2.528 | 2.845 |
| 25 | 1.708 | 2.060 | 2.485 | 2.787 |
| 30 | 1.697 | 2.042 | 2.457 | 2.750 |
| 40 | 1.684 | 2.021 | 2.423 | 2.704 |
| 60 | 1.671 | 2.000 | 2.390 | 2.660 |
| 120 | 1.658 | 1.980 | 2.358 | 2.617 |
| ∞ | 1.645 | 1.960 | 2.326 | 2.576 |
Standard Normal (Z) Critical Values
| Confidence Level | α | zα/2 |
|---|---|---|
| 90% | 0.10 | 1.645 |
| 95% | 0.05 | 1.960 |
| 99% | 0.01 | 2.576 |
| 99.9% | 0.001 | 3.291 |
Chi-Square Critical Values (Right-Tail)
| df | α=0.10 | α=0.05 | α=0.025 | α=0.01 |
|---|---|---|---|---|
| 1 | 2.706 | 3.841 | 5.024 | 6.635 |
| 2 | 4.605 | 5.991 | 7.378 | 9.210 |
| 3 | 6.251 | 7.815 | 9.348 | 11.345 |
| 4 | 7.779 | 9.488 | 11.143 | 13.277 |
| 5 | 9.236 | 11.070 | 12.833 | 15.086 |
| 10 | 15.987 | 18.307 | 20.483 | 23.209 |
| 15 | 22.307 | 24.996 | 27.488 | 30.578 |
| 20 | 28.412 | 31.410 | 34.170 | 37.566 |