Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

MEASURES OF
VARIATION
(Class Lecture in Statistics)
Part 5

3
...
1 - 2

Why is it important to
understand variation?
•
•

A measure of the center by itself
can be misleading
Example:
Two nations with the same
median family income are very
different if one has extremes of
wealth and poverty and the
other has little variation
among families (see the
following table)
...
1 - 3

Example of variation

MEAN
MEDIAN

Data Set A Data Set B
50,000
10,000
60,000
20,000
70,000
70,000
80,000
120,000
90,000
130,000
70,000
70,000
70,000
70,000

Data set B has more variation
about the mean

3
...

3
...
1 - 6

Definition
The range of a set of data values
is the difference between the
maximum data value and the
minimum data value
...
1 - 7

Example of range
...
1 - 8

Range (cont
...


3
...


3
...
1 - 11

Steps to calculate the sample
standard deviation
1
...
Find the squared deviations from
the sample mean for each sample
data value:
2

(x  x)

3
...
Divide the sum in step 3 by n-1
5
...
1 - 12

Example: Standard Deviation
Given the data set:
8, 5, 12, 8, 9, 15, 21, 16, 3

Find the standard deviation

3
...
78
n
9

3
...
78) 2  7
...
78) 2  33
...
78) 2  1
...
78) 2  7
...
78) 2  3
...
78) 2  17
...
78) 2  104
...
78) 2  27
...
78) 2  60
...
1 - 15

Example: Standard Deviation
•

Add the squared deviations (last
column in the table above)

7
...
41  1
...
73  3
...
81  104
...
25  60
...
57
3
...
57 / 8  32
...
95  5
...
7
3
...

Round only the final answer, not values
in the middle of a calculation
...
1 - 18

Sample Standard Deviation
(Shortcut Formula)

n(x ) – (x)
n (n – 1)
2

s=

2

3
...
1 - 20

Example: Standard Deviation
•

We need to find each the following:

n

 (x

2

)

x

2

x

3
...
1 - 22

Example: Standard Deviation
• Thus:

n  25

(
x
)


2

x

109


x

2

 47
3
...
86  0
...
1 - 24

Standard Deviation Important Properties
 The standard deviation is a measure of variation
of all values from the mean
...

 The value of the standard deviation s can
increase dramatically with the inclusion of one
or more outliers (data values far away from all
others)
...

3
...

DIRECTIONS:
Find standard deviation

3
...
0 millibecquerels
3
...
Arrow to the right to
CALC
...
When 1-Var Stats
appears on the home screen, tell
the calculator the name of the list
containing the data

3
...
15 (mean)

x

 5966

2
x
  898586

S x  14
...
1 - 29

Population Standard
Deviation

 =

 (x – µ)

2

N

This formula is similar to the previous
formula, but instead, the population
mean and population size are used
...
1 - 30

Variance
 The variance of a set of values is a
measure of variation equal to the
square of the standard deviation
...
1 - 31

Unbiased Estimator
The sample variance s2 is an
unbiased estimator of the
population variance 2, which
means values of s2 tend to target
the value of 2 instead of
systematically tending to
overestimate or underestimate 2
...
1 - 32

Variance
 Unlike standard deviation, the
units of variance do not match the
units of the original data set, they
are the square of the units in the
original data set
...
1 - 33

Variance - Notation
s = sample standard deviation
s2 = sample variance

 = population standard deviation

 2 = population variance

3
...
1 - 35



---