Begin2.DVI

quartiles. Note that Q 2 will be the same as the median. If the data set is divided

into ten equal parts by numbers D 1 , D 2 , D 3 , D 4 , D 5 , D 6 , D 7 , D 8 , D 9 , then these numbers

are called deciles. If the data set is divided into one hundred equal parts by numbers

P 1 , P 2 ,... , P 99 , then these numbers are called percentiles. In general, if the data is

divided up into quartiles, deciles, percentiles or some other equal subdivision, then

the subdivisions created are called quantiles. The mode of the data set is that value

which occurs with greatest frequency. Note that the mode may not exist or even if

it does exist, it might not be a unique value. A unimodal data set is one which has

a unique single mode.

The Geometric and Harmonic Mean

The geometric mean Gassociated with the data set {X 1 , X 2 ,... , X N}is the Nth

root of the product of the numbers in the set. The geometric mean is denoted

G=N

√

X 1 X 2 ···XN (11.5)

The harmonic mean H associated with the above data set is obtain by first taking

the arithmetic mean of the reciprocals and then taking the reciprocal of the result.

The harmonic mean can be expressed using either of the relations

H=^1

1

N

∑N

i=1

1

Xi

or^1

H

=^1

N

∑N

i=1

1

Xi

(11.6)

The arithmetic mean X, geometric mean Gand harmonic mean H, satisfy the

inequalities

H≤G≤X (11.8)

The equality sign being used when all the numbers in the data set are equal to one

another.

The Root Mean Square (RMS)

The root mean square (RMS), sometimes referred to as the quadratic mean , of

the data set {X 1 , X 2 ,... , X n}is defined for a discrete set of values as

RMS =

√∑

N

j=1 Xj^2

N (11.8)