Download Population and Sample Means

Population and Sample Means
X


i
N
Slide 12.1A
Population and Sample Means
X


i
N
X

X
i
n
Slide 12.1
Mean Advantages and Disadvantages
Advantages
 commonly understood
 all data have one descriptive mean
Disadvantages
 extreme scores distort mean
 tedious if computed by hand
Slide 12.2
SAMPLE DATA and the Mean
1
3
X

X
i
n
5
5
7
9
Slide 12.3A
SAMPLE DATA and the Mean
1
3
5
X

X
i
n
1 3  5  5  7  9
X
6
5
7
9
Slide 12.3B
SAMPLE DATA and the Mean
1
3
5
X

X
i
n
1 3  5  5  7  9
X
6
5
7
9
Slide 12.3C
SAMPLE DATA and the Mean
1
3
5
5
7
X

X
i
n
1 3  5  5  7  9
X
6
30
X
6
9
Slide 12.3D
SAMPLE DATA and the Mean
1
3
5
5
7
X

X
i
n
1 3  5  5  7  9
X
6
30
X
6
5
9
Slide 12.3
Median Advantages and
Disadvantages
Advantages
 not distorted by extreme scores
 useful to detect deviations from
normal distributions
Disadvantages
 may be tedious to find by hand
Slide 12.4
Mode Advantages and Disadvantages
Advantages
 not distorted by extreme scores
 useful for both qualitative and
quantitative data
Disadvantages
 data may not have a true mode
 useless if many modes
Slide 12.5
Assessing Dispersion by
Looking at Spread
Data
2
Mean = 5
5
8
Slide 12.6A
Assessing Dispersion by
Looking at Spread
Data
2
5
8
Mean = 5
How far from the
mean are the data?
Slide 12.6
Starting to Assess the Variance
s
2
X


i
X

n 1
2 -5 =-3
5 -5 = 0
8 -5 = 3
Slide 12.7
2
A Formula to Assess the
Variance
s
2
X


i
X

2
n 1
2 -5 =-3
9
5 -5 = 0
0
8 -5 = 3
9
Slide 12.8A
A Formula to Assess the
Variance
s
2
X


9
5 -5 = 0
0
8 -5 = 3
X  X
9
18
2
s2 
n 1
X

n 1
2 -5 =-3
i
i
2
=
Slide 12.8B
A Formula to Assess the
Variance
s
X


2
9
5 -5 = 0
0
8 -5 = 3
X  X
9
18
2
s2 
n 1
X

n 1
2 -5 =-3
i
i
2
=
9
2 18
Slide 12.8C
A Formula to Assess the
Variance
s
2
X


9
5 -5 = 0
0
8 -5 = 3
X  X
2
s2 
n 1
X

n 1
2 -5 =-3
i
i
2
=
9
18
9
2 18
THE VARIANCE
Slide 12.8
Sample and Population
Standard Deviations
s 
X
i
 X
n 1
Slide 12.9A

2
Sample and Population
Standard Deviations
s 
 
X
 X
i

2
n 1
X
 
i
N
Slide 12.9
2
SAMPLE AND POPULATION TERMS
Sample
Population
Slide 12.10A

SAMPLE AND POPULATION TERMS



n
Sample
Mean
Population
X

XX  
i
i
N
Slide 12.10B
2

SAMPLE AND POPULATION TERMS



Sample
Mean
Variance
Population
X
2

XX i  


X 
X  X
n


2
s 
s 
i
2
2
N n
1
i
2
Slide 12.10C
i

n 1

SAMPLE AND POPULATION TERMS



Sample
Mean
Variance
Standard
Deviation
Population
X
2

XX i   2



X

X

X
n


i
i
2
2
2
s 
s 
i


N 
n X
1
n 1 Xi 
s
 

n 1
Slide 12.10
Standard Normal Curve


X



2
-3
i
N
=0
= 1 
Slide 12.11
+3

z Scores when Data Do Not Already Have a
Mean of 0 and a Standard Deviation of 1
z
X 

Slide 12.12A
z Scores when Data Do Not Already Have a
Mean of 0 and a Standard Deviation of 1
z
or
X 

XX
z
s
Slide 12.12
Areas under the Standard
Normal Curve
z = -1.67
z=1
0
Slide 12.13
Areas under the
Standard Normal Curve
z = -1.75
z = 1.75
0
Slide 12.14
Areas under the
Standard Normal Curve
z=1
0
Slide 12.15
Correlation Example
Speaking Skill
Writing Skill
X
Y
1
3
2
4
3
7
4
5
5
6
Slide 12.16A
Correlation Example
Speaking Skill
X
XX
Writing Skill
Y
Y Y
1
-2
3 -2
2
-1
4 -1
3
0
7
2
4
1
5
0
5
2
6
1
Slide 12.16B
Correlation Example
Speaking Skill
X
XX
Writing Skill
Y
Y Y
XX
*
1
-2
3 -2
4
2
-1
4 -1
1
3
0
7
2
0
4
1
5
0
0
5
2
6
1
2
Slide 12.16C
Y Y
Correlation Example
Speaking Skill
X
XX
Writing Skill
Y
Y Y
XX
*
1
-2
3 -2
4
2
-1
4 -1
1
3
0
7
2
0
4
1
5
0
0
5
2
6
1
2
Slide 12.16D
  X  X  Y  Y 
Y Y
=7
Correlation Example
Speaking Skill
X
XX
Writing Skill
Y
Y Y
XX
*
1
-2
3 -2
4
2
-1
4 -1
1
3
0
7
2
0
4
1
5
0
0
5
2
6
1
2
Slide 12.16
  X  X  Y  Y 
n-1
Y Y
=7
=4
Correlation Computation
7
 1. 75
4
r
s X * sY
Slide 12.17A
Correlation Computation
7
 1. 75
4
r
s X * sY
Slide 12.17B
Correlation Computation
7
 1. 75
4
r
s X * sY
1. 75
r
1.58 *1.58
1. 75
r
. 70
2.5
Slide 12.17C
Correlation Computation
7
 1. 75
4
r
s X * sY
1. 75
r
1.58 *1.58
1. 75
r
. 70
2.5
Slide 12.17D
Correlation Computation
7
 1. 75
4
r
s X * sY
1. 75
r
1.58 *1.58
1. 75
r
. 70
2.5
Slide 12.17