672 Higher Engineering Mathematics
The mode is the most commonly occurring value in a
set of data.
Standard deviation:
σ =
f (x − ¯
x) 2
f
for a population
Binomial probability distribution:
If n =number in sample, p =probability of the occurrence of an event and q = 1 − p, then the probability of
0, 1, 2, 3, ... occurrences is given by:
q
n
, nq
n−1 p,
n(n − 1)
2!
q
n−2 p
2
,
n(n − 1)(n − 2)
3!
q
n−3 p
3
, ...
(i.e. successive terms of the (q + p)
n expansion).
Normal approximation to a binomial distribution:
Mean = np Standard deviation σ =
√
(npq)
Poisson distribution:
If λ is the expectation of the occurrence of an event then
the probability of 0, 1, 2, 3, ... occurrences is given by:
e
−λ
, λe
−λ
, λ
2 e −λ
2!
, λ
3 e −λ
3!
, ...
Product-moment formula for the linear
correlation coefficient:
Coefficient of correlation r =
x y
x 2
y 2
where x = X − X and y = Y − Y and (X 1 , Y 1 ),
(X 2 , Y 2 ), ... denote a random sample from a bivariate
normal distribution and X and Y are the means of the
X and Y values respectively.
Normal probability distribution:
Partial areas under the standardized normal curve — see
Table 58.1 on page 564.
Student’s t distribution:
Percentile values (t p ) for Student’s t distribution with ν
degrees of freedom — see Table 74.2, page 38, on the
website.
Chi-square distribution:
Percentile values (χ 2
p ) for the Chi-square distribution
with ν degrees of freedom—see Table 76.1, page 60, on
the website.
χ 2 =
(o − e) 2
e
where o and e are the observed and
expected frequencies.
Symbols:
Population
number of members N p , mean μ, standard deviation σ .
Sample
number of members N , mean x, standard deviation s.
Sampling distributions
mean of sampling distribution of means μ x
standard error of means σ x
standard error of the standard deviations σ s .
Standard error of the means:
Standard error of the means of a sample distribution, i.e.
the standard deviation of the means of samples, is:
σ x =
σ
√
N
N p − N
N p − 1
for a finite population and/or for sampling without
replacement, and
σ x =
σ
√
N
for an infinite population and/or for sampling with
replacement.
The relationship between sample mean
and population mean:
μ x = μ for all possible samples of size N are drawn
from a population of size N p .
The mode is the most commonly occurring value in a
set of data.
Standard deviation:
σ =
f (x − ¯
x) 2
f
for a population
Binomial probability distribution:
If n =number in sample, p =probability of the occurrence of an event and q = 1 − p, then the probability of
0, 1, 2, 3, ... occurrences is given by:
q
n
, nq
n−1 p,
n(n − 1)
2!
q
n−2 p
2
,
n(n − 1)(n − 2)
3!
q
n−3 p
3
, ...
(i.e. successive terms of the (q + p)
n expansion).
Normal approximation to a binomial distribution:
Mean = np Standard deviation σ =
√
(npq)
Poisson distribution:
If λ is the expectation of the occurrence of an event then
the probability of 0, 1, 2, 3, ... occurrences is given by:
e
−λ
, λe
−λ
, λ
2 e −λ
2!
, λ
3 e −λ
3!
, ...
Product-moment formula for the linear
correlation coefficient:
Coefficient of correlation r =
x y
x 2
y 2
where x = X − X and y = Y − Y and (X 1 , Y 1 ),
(X 2 , Y 2 ), ... denote a random sample from a bivariate
normal distribution and X and Y are the means of the
X and Y values respectively.
Normal probability distribution:
Partial areas under the standardized normal curve — see
Table 58.1 on page 564.
Student’s t distribution:
Percentile values (t p ) for Student’s t distribution with ν
degrees of freedom — see Table 74.2, page 38, on the
website.
Chi-square distribution:
Percentile values (χ 2
p ) for the Chi-square distribution
with ν degrees of freedom—see Table 76.1, page 60, on
the website.
χ 2 =
(o − e) 2
e
where o and e are the observed and
expected frequencies.
Symbols:
Population
number of members N p , mean μ, standard deviation σ .
Sample
number of members N , mean x, standard deviation s.
Sampling distributions
mean of sampling distribution of means μ x
standard error of means σ x
standard error of the standard deviations σ s .
Standard error of the means:
Standard error of the means of a sample distribution, i.e.
the standard deviation of the means of samples, is:
σ x =
σ
√
N
N p − N
N p − 1
for a finite population and/or for sampling without
replacement, and
σ x =
σ
√
N
for an infinite population and/or for sampling with
replacement.
The relationship between sample mean
and population mean:
μ x = μ for all possible samples of size N are drawn
from a population of size N p .