Transcript 1.1 Basic Equations

Introduction
Population – the entire group of concern
Sample – only a part of the whole
Based on sample, we’ll make a prediction about
the population.
Bad sampling: convenience, bias, voluntary
Good sampling: simple random sample(SRS).
Inferential Stats: making predictions or
inferences about a population based on a sample
Experiments
Observation – no attempt to influence
Experiment– deliberately imposes some treatment
Basic design principles:
Control the effects of lurking variables
Randomize which subject gets which treatment
Use large sample size to reduce chance variation
Statistical Significance:
An observed effect so big that it would rarely
occur just by chance.
Picturing Distributions with Graphs
What makes up any set of data?
• Individuals
– objects described by data
– can be
• Variables
– characteristic of individuals of particular interest
– different values possible for different people
Two kinds of variables
Categorical (Qualitative)
– describes an individual by category or quality.
– examples like
Numerical (Quantitative)
– describes an individual by number or quantity.
– discrete for variables that are
– continuous for variables that are
– examples like
Describing Categorical Variables
Tables summarize the data set by
– listing possible categories.
– giving the number of objects in each category.
– or show the count as a percentage.
Picture the distribution of a cat. var. with
– Pie charts
– Bar graphs
Pie Charts
whole is split into appropriate pieces.
Bar Graph
Horizontal line keeps track of categorical values.
Vertical bars at each value keeps track of # or %.
% #
25 20
15 12
5
4
A
B
C
D
E
F
Example 1
80 AASU students in an Elem. Stats class come from
one of four colleges (S & T, Edu, Health, Lib. Arts).
The breakdown of these 80 students is given below.
College
Count
Liberal Arts
17
Education
4
32
Health
Professions
Science &
Technology
Undeclared
23
4
80
Percent
Ex1 - Pie Chart
College
Count
Percent
Lib Arts
17
21.25%
Edu
4
5%
Health
32
40%
S&T
23
28.75%
Undeclared
4
5%
80
100%
Ex1 – Bar Graph
College
Count
Percent
Lib Arts
17
21.25%
Edu
4
5%
Health
32
40%
30
S& T
23
28.75%
20
Undeclared
4
5%
10
80
100%
%
LA
E
H
ST
U
Describing Quantitative Variables
Tables summarize the data set by
– listing possible intervals (ranges, classes).
– giving the number of individuals in each class
– or showing the number as a percentage.
Picture the distribution of a quant. var. with
– Histogram (similar to bar graph but now vertical
bars of neighboring classes touch)
Where one class ends, the next begins.
Example 2
Consider the ages of the full-time faculty in the math
dept. The breakdown of these 19 individuals is given
in the table.
Age
Class
%
Count Percent
30
20-30
5
26.3%
30-40
3
15.8%
40-50
5
26.3%
50-60
4
21.1%
60-70
2
10.5%
19
100%
20
10
10
30
50
70
Info from histograms
Helps to describe a distribution with
– pattern (shape, center, spread)
– deviations (outliers) from the rest of the data
• Could result from unusual observation or typo
– For shape, look at symmetric vs. skewed
Examples 3 and 4
%
2
4
6
8
10
12
%
v
20
40
60
80
100
Example 4 without outliers
%
30
10
5
v
20
40
60
80
100
%
20
10
5
v v
20
40
v
60
80
100
Describing Distributions with Numbers
There are better ways to describe a quantitative
data set than by an estimation from a graph.
Center: mean, median, mode
Spread: quartiles, standard deviation
Center: Mean
The mean of a data set is the arithmetic average of
all the observations.
Given a data set:
x1 , x2 ,, xn
Mean – Example 1
Your test scores in a Stats Class are: 60, 75, 92, 80
Your mean score is:
Mean – Example 2
Compare high temperatures in Savannah for July
2010 and July 2011.
July 2010 high temps: 83, 87, 84, …, 97, 100, 92
July 2011 high temps: 94, 91, 93, …, 97, 99, 99
x2010
x2011
83  87    92


31
94  91    99


31
Center: Median
The median of a data set is the middle value of
all the (ordered) observations.
Given a data set:
x1 , x2 ,, xn
Median – Examples 3/4
11 tests: 60, 77, 92, 80, 84, 93, 80, 95, 65, 66, 75
Ordered data set: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95
10 dice rolls: 2, 4, 5, 5, 6, 7, 7, 8, 9, 10
Center: Mode
The mode of a data set is the value that appears the most.
Tests data set: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95
Dice rolls: 2, 4, 5, 5, 6, 7, 7, 8, 9, 10
2010 July High Temps mode:
2011 July High Temps mode:
Spread: Quartiles
A measure of center is not useful by itself
– Are other observations close or far from center?
Take an ordered data set and find:
– M,
– Q1,
– Q3,
– IQR =
Summary of data in the “Five-Number Summary”:
Quartiles – Example 5
11 tests: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95
5-num-sum:
Visualize 5-num-sum with a boxplot.
• Draw rectangle with ends at Q1 and Q3.
• Draw line in the box for the median.
• Draw lines to the last observations within 1.5IQR of
the quartiles.
• Observations outside 1.5IQR of the quartiles are
suspected outliers.
Boxplot – Example 6
5-Num-Sum: 60, ____, 80, ____, 95
50
•
•
•
•
60
70
80
90
100
Draw rectangle with ends at Q1 and Q3
Draw line in the box for the median
Draw lines to last observations within 1.5IQR of the quartiles
Observations outside 1.5IQR of the quartiles are suspected outliers
Boxplot – Example 7
July 2010 5-Num-Sum: 83, 92, 94, 97, 102
July 2011 5-Num-Sum: 84, 91, 95, 98, 99
2010
IQR = 97-92=5
2010
2011
2011
IQR = 98-91=7
80
85
90
95
100 105
Spread: Standard Deviation
More common measure of spread (in conjunction
with the mean) is the standard deviation.
A single deviation from the mean looks like
For every value in a data set, deviations are either
positive, negative or zero.
Finding an average of those will be trouble, since
when you add the deviations together, you’ll get 0.
Example 1 data: 60, 75, 92, 80
x  76.75
To deal with this “adding to zero”, we get rid of
any negative terms by squaring each deviation.
A single squared deviation from the mean looks like:
The average of the squared deviations is called the
variance:
n-1 is called the degrees of freedom, since knowledge of the
first (n-1) deviations will automatically set the last one.
The standard deviation is the square root of the variance.
 x  x 
2
s
Observations
Deviations
i
n 1
Squared Dev
s 
2
60
75
92
80
mean=76.75
s


When to use what?
For skewed data:
For (nearly) symmetric data:
Outliers have a big impact on mean and std. dev.
Consider two data sets:
Set 1: 1, 1, 3, 5, 10
Set 2: 1, 1, 3, 5, 70