Area A(z) under the normal PDF, f(z)
1  z 
x xx
A  z   erf 

where z 


S
2  2
Example, at z = 1.06, A(z) = 0.35543.
f(z)
A(z)
A(z) is the colored
area in the sketch.
0
z
z
Critical Values for the Student’s t Distribution
Critical Values for the χ2 PDF
Critical Values of the Linear Correlation Coefficient
Modified Thompson Tau – Used for Determination of Outliers
In this table, τ is obtained from the expression τ =
•
•
t ⋅ ( n − 1)
n n − 2 + t2
, where
n is the number of data points
t is the student’s t value, based on α = 0.05 and df = n-2 (note that here df = n-2 instead of n-1). In Excel,
we calculate t as TINV(α, df), i.e., here t = TINV(α, n-2)
p-Values for the t Distribution – one tail, df = 9
f(t,df)
Area = p-value
For two tails, multiply the value by 2, since the t PDF is symmetric.
The p-value is the colored area under the t PDF in the sketch.
Example: 1-tailed p at t = 1.06: p-value = TDIST(t,df,1) = 0.15838.
2-tailed p at t = 1.06: p-value = TDIST(t,df,2) = 0.31676.
0
t-statistic
t
p-Values for the t Distribution – one tail, df = 19
f(t,df)
Area = p-value
For two tails, multiply the value by 2, since the t PDF is symmetric.
The p-value is the colored area under the t PDF in the sketch.
Example: 1-tailed p at t = 1.06: p-value = TDIST(t,df,1) = 0.15122.
2-tailed p at t = 1.06: p-value = TDIST(t,df,2) = 0.30243.
0
t-statistic
t
p-Values for the t Distribution – one tail, df = 29
f(t,df)
Area = p-value
For two tails, multiply the value by 2, since the t PDF is symmetric.
The p-value is the colored area under the t PDF in the sketch.
Example: 1-tailed p at t = 1.06: p-value = TDIST(t,df,1) = 0.14895.
2-tailed p at t = 1.06: p-value = TDIST(t,df,2) = 0.29789.
0
t-statistic
t
Folding Diagram for Aliasing Calculations
Instructions for using the folding diagram:
•
Calculate the folding frequency, ffolding = fs /2.
•
Locate f / ffolding on the folding diagram, as plotted below. Note: For values of f / ffolding greater than
5.0, the folding diagram can easily be extended, following the obvious pattern.
•
Read straight down from the value of f / ffolding to obtain the value of fa / ffolding on the bottom
(horizontal) axis.
⎛ f ⎞
•
Finally, calculate the aliasing frequency, f a = ⎜⎜ a ⎟⎟ f folding .
⎝ f folding ⎠
Etc. … continue up as needed
6.0
6.2
6.4
6.6
5.6
5.8
4.0
4.2
4.4
4.6
3.6
3.8
f /ffolding
2.0
2.2
2.4
1.6
1.8
0
0.2
5.4
0.4
3.4
6.8
7.0
5.2
4.8
3.2
5.0
3.0
2.6
2.8
1.4
0.6
1.2
0.8
f /ffolding
1.0
fa/ffolding
Alternative – an equation instead of the folding diagram:
• General equation to determine the perceived frequency of any signal frequency f when sampled at
⎛ f ⎞
any sampling frequency fs, whether there is aliasing or not: f perceived = f − f s ⋅ NINT ⎜ ⎟ , where
⎝ fs ⎠
o
o
NINT is the “nearest integer” function.
In Excel, use ROUND(x,0) to round real number x to the nearest integer.
Thermocouple Voltage Data – Table 9.2 of Wheeler, A. J. and Ganji, A. R., Introduction to
Engineering Experimentation, Ed. 2, Pearson Education Inc. (Prentice Hall), Upper Saddle
River, NJ, 2004.
Platinum 100-Ω RTD Table
Table taken from R. E. Fraser, Process Measurement and Control – Introduction to Sensors,
Communication, Adjustment, and Control, Prentice-Hall, Inc., Upper Saddle River, NJ, 2001.
Resistance values for two standard thermistors