Higher Engineering Mathematics, Sixth Edition

672 Higher Engineering Mathematics

Themodeis the most commonly occurring value in a set of data.

Standard deviation:

σ=

√√ √ √

[∑{ f(x− ̄x)^2

} ∑ f

] for a population

Binomial probability distribution:

Ifn=number in sample,p=probability of the occurrence of an event andq= 1 −p, then the probability of 0 , 1 , 2 , 3 ,...occurrences is given by:

qn, nqn−^1 p,

n(n− 1 ) 2!

qn−^2 p^2 ,

n(n− 1 )(n− 2 ) 3!

qn−^3 p^3 ,...

(i.e. successive terms of the(q+p)nexpansion). Normal approximation to a binomial distribution: Mean=np Standard deviationσ=

√ (npq)

Poisson distribution:

Ifλis 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 correlationr=

∑ xy √[(∑ 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 andXandYare the means of the XandYvalues respectively.

Normal probability distribution:

Partial areas under the standardized normal curve — see Table 58.1 on page 564.

Student’stdistribution:

Percentile values (tp) for Student’stdistributionwithν 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 withνdegrees of freedom—see Table 76.1, page 60, on the website. χ^2 =

∑ { (o−e)^2 e

} whereoandeare the observed and expected frequencies.

Symbols:

Population number of membersNp, meanμ, standard deviationσ.

Sample number of membersN, meanx, standard deviations.

Sampling distributions mean of sampling distribution of meansμx standard error of meansσx standard error of the standard deviationsσs.

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:

σx=

σ √ N

√( Np−N Np− 1

)

for a finite population and/or for sampling without replacement, and

σx=

σ √ N

for an infinite population and/or for sampling with replacement.

The relationship between sample mean

and population mean:

μx=μfor all possible samples of sizeNare drawn from a population of sizeNp.