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Dr. McLoughlin Math 104 Statistical Formulae, page 1 of 3
MATH 104
FUNDAMENTALS OF MATHEMATICS II
DR. MCLOUGHLIN’S CLASS
STATISTICAL FORMULAE
HANDOUT 6
Let S be a well defined sample space.
Let D = {X1, X2, X3, . . . , Xn} be a finite data set meaning X1, X2, X3, . . . , Xn is a finite
random sample from S.
SOME MEASURES OF CENTRAL TENDENCY
Definition 1: The arithmetic mean of the sample is the value X where
n
X  X2   Xn
1
X=
Xj = 1
n
n j1

Definition 2: Let X be a random variable with a probability mass or density function. Let X1, X2, X3, . . . , Xn be
a finite random sample from S. Let w1, w2, w3, . . . , wn be real numbers signifying the ‘importance’ of X1, X2, X3,
. . . , Xn respectively. The weighted arithmetic mean of the sample is the value
WX where
n
 (w  X )
j
j
j1
WX =
.
n
w
j
j1
Definition 3: The geometric mean of the sample is the value G where
n
G= n
X
j
= n X1  X 2 
 Xn
j1
Definition 4: The harmonic mean of the sample is the value H where
H=
n
n

j1
1
Xj
=
n
1
1


X1 X 2

1
Xn
Definition 5: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
a finite random sample from S. The mode of the sample is the value of X which occurs most frequently such
that there is at least one value that occurs with lesser frequency.
Definition 6: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
a finite random sample from S. The median of the sample is the value of X which occurs in the centre of
ordered values of the sample if there are an odd number of values and it is the arithmetic mean of the centre two
values if there are an even number of values.
Dr. McLoughlin Math 104 Statistical Formulae, page 2 of 3
SOME MEASURES OF CENTRAL DISPERSION, ‘SPREAD,’ OR VARIABILITY
Definition 7: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
a finite random sample from S. The range is the (highest value – lowest value) .
Definition 8: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
a finite random sample from S. The Inter-Quartile range (IRQ) is a measure of dispersion found by first finding
the median of the sample. Then breaking the sample into two equal sized sub-groups, find the median each
group. The median of the first sub-group is the first quartile, Q1. The median of the sample is the second
quartile, Q2. The median of the second sub-group is the third quartile, Q3.
The IRQ = Q3 – Q1.
Definition 9: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
a finite random sample from S. The coefficient of variation of the sample is the value C where C =
s
 (100%)
X
Definition 10: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
n
2
a finite random sample from S. The variance, s , is defined as
s2 
 (X
k 1
k
 X) 2
n 1
Definition 11: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
n
a finite random sample from S. The standard deviation, s, is defined as
s=

 (X
k 1
k
 X)2
(n  1)
Definition 12: Let X be a random variable with a probability mass or density function. Let X 1, X2, X3, . . . , Xn be
n
a finite random sample from S. The Mean Absolute Deviation (MAD) is defined as
MAD 
X
k 1
Computational examples:
Suppose the X i are given by X1 = 0.1, X2 = 0.1, X3 = 0.1, and X4 = 10.
X = 2.575, S2 = 24.5025, S =
24.5025 = 4.95, mo = 0.1, md = 0.1, etc.
Suppose the X i are given by X1 = 0.1, X2 = 10, X3 = 10, and X4 = 10.
X = 7.525, S2 = 57.1725, S =
57.1725 , S  7.561249897, mo = 10, md = 10, etc.
Suppose the X i are given by X1 = 1, X2 = 2, X3 = 3, and X4 = 4.
X =
5 2 5
,S = ,S=
2
3
5
, S  1.290994449, m0 does not exist, md = 2.5, etc.
3
k
n
X
Dr. McLoughlin Math 104 Statistical Formulae, page 3 of 3
x  x
x
P(X  x) = P(Z  z) when X ~ N(x, , ) when X ~ N(x, , )
To convert from X values to Z values it is the case that: Z 
N (x, , )  1 e
2
 (xi  )2
2 2
where x

 (z)
N (z,   0,   1)  1 e 2 where z  
2
2
The approximate areas (probabilities) are as follows: