and for a continuous set of values f(x)over an interval (a, b)it is defined
RMS =
√
1
b−a
∫b
a
[f(x)]^2 dx (11.9)
The root mean square is used to measure the average magnitude of a quantity that
varies.
Mean Deviation and Sample Variance
The mean deviation (MD), sometimes called the average deviation , of a set of
numbers X 1 , X 2 ,... , X N represents a measure of the data spread from the mean and
is defined
MD =
∑N
j=1 |Xj−X|
N
=^1
N
[
|X 1 −X|+|X 2 −X|+···+|Xn−X|
]
(11 .10)
where X is the arithmetic mean of the data. The mean deviation associated with
numbers X 1 , X 2 ,... , X koccurring with frequencies f ̃ 1 ,f ̃ 2 ,... ,f ̃kcan be calculated
MD =
∑k
j=1 f ̃j|Xj−X|
N
=^1
N
[
f ̃ 1 |X 1 −X|+f ̃ 2 |X 2 −X|+···+f ̃k|Xk−X|
]
(11 .11)
The sample variance of the data set {X 1 , X 2 ,... , X n}is denoted s^2 and is calculated
using the relation
s^2 =^1
n− 1
∑n
j=1
(Xj−X)^2 =^1
n− 1
[
(x 1 −X)^2 + (x 2 −X)^2 +···+ (xn−X)^2
]
(11 .12)
where Xis the sample mean or arithmetic mean. The sample variance is a measure
of how much dispersion or spread there is in the data. The positive square root
of the sample variance is denoted s, which is called the standard deviation of the
sample.
Note that some textbooks define the sample variance as
S^2 =^1 n
∑n
j=1
(Xj−X)^2 (11 .13)
where nis used as a divisor instead of n− 1. This is confusing to beginning students
who often ask, “Why the different definitions for sample variance in different text-
books?” The reason is that for small values for n, say n < 30 , then equation (11.12)
will produce a better estimate of the true standard deviation σassociated with the
total population from which the sample is taken. For sample sizes larger than n= 30
