Download 01/18/2008

Univariate Statistics
Demonstration Day
Topics
Questions
Calculating measures of central tendency &
dispersion by hand and in Excel
Summation Notation & Rules
Skewness
Questions
Which measure of central tendency is most
appropriate for the following distributions
and why?
–
–
–
–
Bimodal distribution?
Skewed distribution?
Dataset with an outlier (an extreme value)?
Normal distribution (unimodal)?
Symbols Review
n
: the number of observations in a sample
N
: the number of elements in the population
Σ
: this (capital sigma) is the symbol for sum
i
: the starting point of a series of numbers
X
: one element in our dataset, usually has a subscript (e.g., i, min, max)
x
: the sample mean

: the population mean
s2
: the sample variance
σ2
: the population variance
s
: the sample standard deviation
σ
: the population standard deviation
Equations Review
Sample mean
Sample standard deviation
n
x
x
i 1
n
n
i
s
 ( x  x)
i 1
i
n 1
Summation Notation
The order of operations for statistical
equations
Similar to Please Excuse My Dear Aunt
Sally from algebra
Summation Notation: Examples
Example I: All observations are included in the sum:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
10
x
i 1
i
 x1  x2  x3  x4  x5  x6  x7  x8  x9  x10
 1  2  3  4  5  6  7  8  9 10
Example II: Only observations 3 through 5 are included in the sum:
5
x
i 3
i
 x3  x4  x5  3  4  5  12
Summation Notation: Rules
Rule I: Summing a constant n times yields a result of n*b:
n
 b  b  b      b  nb
i 1
Here we are simply using the summation notation to carry
out a multiplication, e.g.:
5
4
i 1
 4  4  4  4  4  4  5  20
Summation Notation: Rules
Rule II: Constants may be taken outside of the summation
sign
n
 ax
i 1
n
 ax
i 1
i
i
n
 a  xi
i 1
 ax1  ax2      axn
n
 a( x1  x2      xn )  a  xi
i 1
• Rule II: Constants may be taken outside of the
summation sign
Example: Now let a = 3, and let the values of a set
(n = 3) of x and y values be:
x1 = 4, x2 = 5, x3 = 6
y1 = 7, y2 = 8, y3 = 9
3
3
 ax
i 2
 a  xi  3(5  6)  33
i
i 2
 ax
i
 a xi  3(4  5  6)  45
Summation Notation: Rules
Rule III: The order in which addition operations are carried
out is unimportant
n
 (x
i 1
i
n
n
i 1
i 1
 yi )   xi   yi
 ( x1  x2  x3     xn1  xn )
+
( y1  y2  y3      yn1  yn )
• Rule III: The order in which addition operations are
carried out is unimportant
Example: Now let a = 3, and let the values of a set
(n = 3) of x and y values be:
x1 = 4, x2 = 5, x3 = 6
y1 = 7, y2 = 8, y3 = 9
2
2
2
 (x  y )   x   y
i 1
i
i
i
i 1
2
i 1
i
 (4  5)  (7  8)  24
 ( x  y )  (4  7)  (5  8)  24
i 1
i
i
Summation Notation: Rules
Rule IV: Exponents are handled differently depending on
whether they are applied to the observation term or the whole
sum
n
k
k
k
k
x

x

x





x
 i 1 2
n
i 1
k


k
x

(
x

x





x
)
 i 
1
2
n
 i 1 
n
• Rule IV: Exponents are handled differently depending
on whether they are applied to the observation term or
the whole sum
Example: Now let the values of a set (n = 3) of x values be:
x1 = 4, x2 = 5, x3 = 6
2
2
2
2
x

4

5

6
 77
i =
( xi ) 2  (4  5  6) 2  225
Summation Notation: Rules
Rule V: Products are handled much like exponents
n
x y
i
i 1
i
 ( x1 y1  x2 y2      xn yn )
n
n
n
x y  x y
i 1
n
n
x y
i 1
i
i 1
i
i
i
i 1
i
i 1
i
 ( x1  x2      xn )  ( y1  y2      yn )
• Rule V: Products are handled much like exponents
Example: Now let the values of a set (n = 3) of x and y
values be:
x1 = 4, x2 = 5, x3 = 6
y1 = 7, y2 = 8, y3 = 9
x y
i
 4  7  5  8  6  9  122
i
x  y
i
i
 (4  5  6)  (7  8  9)  360
Summation Notation: Compound Sums
We frequently use tabular data (or data drawn from matrices), with which we
can construct sums of both the rows and the columns (compound sums), using
subscript i to denote the row index and the subscript j to denote the column
index:
Columns
Rows
2
3
 x
i 1 j 1
ij
x11 x12 x13
x21 x22 x23
 ( x11  x12  x13  x21  x22  x23 )
Pearson’s Skew Equation
3( x  median)

s
Excel’s Skew Equation