Transcript Q 1 + Q 3 - TonyReiter

Statistics
Two Meanings
 Specific numbers
 Method of analysis
1
Statistics
 Specific number
numerical measurement determined by a
set of data
Example: Twenty-three percent of people
polled believed that there are
too many polls.
2
Statistics
 Method of analysis
a collection of methods for planning
experiments, obtaining data, and then
then organizing, summarizing, presenting,
analyzing, interpreting, and drawing
conclusions based on the data
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Statistics
 Descriptive Statistics
•
Handles the collection and presentation of
data
• Inferential Statistics
•
Analyzing and drawing conclusions from
numerical information
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Definitions
Population
the complete collection of all
elements (scores, people,
measurements, and so on) to be
studied. The collection is complete
in the sense that it includes all
subjects to be studied.
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Definitions
Census
the collection of data from every
element in a population
Sample
a subcollection of elements drawn
from a population
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Definitions
 Parameter
a numerical measurement describing
some characteristic of a population
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Definitions
 Parameter
a numerical measurement describing
some characteristic of a population
population
parameter
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Definitions
Statistic
a numerical measurement describing
some characteristic of a sample
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Definitions
Statistic
a numerical measurement describing
some characteristic of a sample
sample
statistic
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Definitions
Quantitative data
numbers representing counts or
measurements
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Definitions
Quantitative data
numbers representing counts or
measurements
 Qualitative (or categorical or
attribute) data
can be separated into different categories
that are distinguished by some nonnumeric
characteristics
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Definitions
Quantitative data
the incomes of college graduates
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Definitions
Quantitative data
the incomes of college graduates
 Qualitative (or categorical or
attribute) data
the genders (male/female) of college
graduates
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Definitions
Discrete
data result when the number of possible values is
either a finite number or a ‘countable’ number of
possible values
0, 1, 2, 3, . . .
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Definitions
Discrete
data result when the number of possible values is
either a finite number or a ‘countable’ number of
possible values
0, 1, 2, 3, . . .
 Continuous
(numerical) data result from infinitely many possible
values that correspond to some continuous scale
that covers a range of values without gaps,
interruptions, or jumps
2
3
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Definitions
Discrete
The number of eggs that hens lay; for
example, 3 eggs a day.
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Definitions
Discrete
The number of eggs that hens lay; for
example, 3 eggs a day.
 Continuous
The amounts of milk that cows produce;
for example, 2.343115 gallons a day.
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Definitions
 nominal level of measurement
characterized by data that consist of names,
labels, or categories only. The data cannot be
arranged in an ordering scheme (such as low
to high)
Example: survey responses yes, no, undecided
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Definitions
 ordinal level of measurement
involves data that may be arranged in some
order, but differences between data values
either cannot be determined or are meaningless
Example: Course grades A, B, C, D, or F
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Definitions
 interval level of measurement
like the ordinal level, with the additional property
that the difference between any two data values is
meaningful. However, there is no natural zero
starting point (where none of the quantity is
present)
Example: Years 1000, 2000, 1776, and 1492
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Definitions
 ratio level of measurement
the interval level modified to include the natural
zero starting point (where zero indicates that
none of the quantity is present). For values at
this level, differences and ratios are meaningful.
Example: Prices of college textbooks
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Levels of Measurement
 Nominal - categories only
 Ordinal - categories with some order
 Interval - differences but no natural
starting point
 Ratio - differences and a natural starting
point
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Levels of Measurement
 Nominal - categories only
 Ordinal - categories with some order
 Interval - differences but no natural
starting point
 Ratio - differences and a natural starting
point
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Stem-and Leaf Plot
Stem
Raw Data (Test Grades)
67 72
89
85
88 90
75
89
99 100
6
7
8
9
10
Leaves
7
25
5899
09
0
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Example: Create a Stem and Leaf Plot for the
following data which represents ages of CEO's:
53
45
41
36
55
50
37
48
52
62
43
46
39
50
52
61
48
37
48
55
59
52
39
41
50
The TI-83 will not create the Stem and Leaf Plot for
you completely, but it will allow you to sort the data
which makes creating the chart by hand easy. Here
is what to do:
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1. Enter the data into a free list (use L1 if it is available). Recall
that you do this by hitting STAT, then 1 for Edit and clear L1
if necessary. After you have entered the data into L1 the screen
should look like this:
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2. Now hit 2nd MODE for quit to get to the homescreen.
Now hit STAT to get this screen:
3. Now select 2 to get SortA( which stands for Sort Ascending).
Your screen will look like this:
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4. Enter the list you wish to sort in this case L1 (hit 2nd 1).
Your screen looks like this
5. Now hit enter, the screen will say done.
Hit Stat then edit to get back to the editor.
Your data should be sorted.
Here is what the screen should look like:
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Stem Leaf
3
3
67799
4
1135
4
6888
5
00222355
5
9
6
12
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Definitions
 Median
•
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
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Definitions
 Median
•
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
 often denoted by x~
(pronounced ‘x-tilde’)
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Definitions
 Median
•
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
 often denoted by x~
(pronounced ‘x-tilde’)
 is not affected by an extreme value
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6.72
3.46
3.60
6.44
3.46
3.60
6.44
6.72
(even number of values)
no exact middle -- shared by two numbers
3.60 + 6.44
2
MEDIAN is 5.02
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6.72
3.46
3.60
6.44
3.46
3.60
6.44
6.72
(even number of values)
no exact middle -- shared by two numbers
3.60 + 6.44
MEDIAN is 5.02
2
6.72
3.46
3.60
6.44
26.70
3.46
3.60
6.44
6.72
26.70
(in order -
exact middle
odd number of values)
MEDIAN is 6.44
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Qualitative vs Quantitative
•
Number of students who turn a paper in late.
•
Sex of the next baby born in a hospital.
•
Amount of fluid in a machine to fill bottles of soda pop.
•
Brand of a personal computer.
•
Zip Codes.
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Discrete vs Continuous
•
Price of a textbook.
•
The length of a new born baby.
•
The number of bad checks received by a store.
•
Concentration of a contaminant in a solution.
•
Actual weight of a 1-lb can of coffee.
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Measures of Position
Quartiles, Deciles,
Percentiles
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Quartiles
Q1, Q2, Q3
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Quartiles
Q1, Q2, Q3
divides ranked scores into four equal parts
25%
25%
25% 25%
Q1 Q2 Q3
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Deciles
D1, D2, D3, D4, D5, D6, D7, D8, D9
divides ranked data into ten equal parts
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Deciles
D1, D2, D3, D4, D5, D6, D7, D8, D9
divides ranked data into ten equal parts
10% 10% 10%
D1
D2
D3
10% 10% 10%
D4
D5
10% 10% 10% 10%
D6
D7
D8
D9
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Quartiles
Q1 = P25
Q2 = P50
Q3 = P75
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• Range The difference between the
highest and lowest score
• Interquartile Range (or IQR): Q3 - Q1
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• Interquartile Range (or IQR): Q3 - Q1
• Semi-interquartile Range: (Q3- Q1)/2
• Midquartile: (Q1+ Q3)/2
• 10 - 90 Percentile Range: P90 - P10
• Midrange: (smallest + largest)/2
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Finding the Percentile of a
Given Score
Percentile of score x =
number of scores less than x
• 100
total number of scores
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Finding the Score
Given a Percentile
R=
k
100
•n
n
k
R
Pk
total number of values in the data set
percentile being used
locator that gives the position of a value
kth percentile
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Start
Finding the Value
of the
kth Percentile
Sort the data.
(Arrange the data in
order of lowest to
highest.)
Compute
L= k
n
100
(
)
where
n = number of values
k = percentile in question
Is
L a whole
number
?
No
Yes
The value of the kth percentile
is midway between the Lth value
and the next value in the
sorted set of data. Find Pk by
adding the L th value and the
next value and dividing the
total by 2.
Change L by rounding
it up to the next
larger whole number.
The value of Pk is the
Lth value, counting
from the lowest
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Stem-and Leaf Plot
Stem
Raw Data (Test Grades)
67 72
89
88 90
Find P50
P50=
85
75
6
7
8
9
10
89
99 100
 k 
R 
n
 100 
 50 
R 
  10  5
100


P50= the mean of the 5th and 6th
score or 88.5
Find P33
P33=
Leaves
7
25
5899
09
0
 k 
R 
n
 100 
 33 
R 
  10  3.3
100


P33= round up to 4, the fourth score is 85
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Stem-and Leaf Plot
Stem
Raw Data
16 19 22 23 24 25 26 27
28 28 29 30 31 31 34
Find P50
P50=
 k 
R 
n
 100 
 50 
R 
  15  7.5
100


P50= round up to 8, the eight
score is 27
Find P30
P30=
1
2
3
Leaves
69
234567889
0114
 k 
R 
n
 100 
 30 
R 
  15  4.5
100


P30= round up to 5, the fifth score is 24
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Stem-and Leaf Plot
Stem
Raw Data
16 19 22 23 24 25 26
27 28 28 29 30 31 31
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What percentile is 30?
11/15 = 73rd percentile
1
2
3
Leaves
69
234567889
0114
What percentile is 22?
2/15 = 13th percentile
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