Practice
• Page 65– 2.1
1 2.0 2.0 2.01 2.0 2.0 4.01 2.0 2.0 6.03 6.0 6.0 12.04 8.0 8.0 20.06 12.0 12.0 32.010 20.0 20.0 52.07 14.0 14.0 66.07 14.0 14.0 80.03 6.0 6.0 86.02 4.0 4.0 90.02 4.0 4.0 94.01 2.0 2.0 96.01 2.0 2.0 98.01 2.0 2.0 100.050 100.0 100.050 100.0
10.0011.0012.0015.0016.0017.0018.0019.0020.0021.0022.0023.0024.0031.0040.00Total
Valid
Total
Frequency PercentValidPercent
CumulativePercent
VAR00001
• Positive Skew
0
2
4
6
8
10
12
0 5 10 15 20 25 30 35 40 45
Note
• Slides online
12 13 9 15 18 18 13 7
16 8 22 15 19 15 9 14
16 10 6 16 14 20 17 13
12 15 15 17 16 11 24 20
19 15 14 24 22 17 21 14
Neuroticism Score
f Neuroticism Score
f
24 2 14 4
23 0 13 3
22 2 12 2
21 1 11 1
20 2 10 1
19 2 9 2
18 2 8 1
17 3 7 1
16 4 6 1
15 6 N = 40
Class Interval
Midpoint f Cumulative f
25-23 24 2 40
22-20 21 5 38
19-17 18 7 33
16-14 15 14 26
13-11 12 6 12
10-8 9 4 6
7-5 6 2 2
Histogram
0
24
68
10
1214
16
6 9 12 15 18 21 24
Neuroticism
Fre
qu
ency
Frequency Polygon
0
24
68
10
1214
16
6 9 12 15 18 21 24
Neuroticism
Fre
qu
ency
Stem-and-Leaf Display
• Stem-and-leaf display with a bigger data set
• Note: The stem-and-leaf is like a histogram turned sideways!
Describing Distributions
• Bell-shaped distribution
Describing Distributions
Describing Distributions
0
2
4
6
8
10
12
0 2 4 6 8 10 12 14
Happiness Score
Fre
quen
cy
Kurtosis
• The relative concentration of scores in the center of the distribution
• Mesokurtic
Kurtosis
• The relative concentration of scores in the center of the distribution
• Platykurtic
Kurtosis
• The relative concentration of scores in the center of the distribution
• Leptokurtic
12 13 9 15 18 18 13 7
16 8 22 15 19 15 9 14
16 10 6 16 14 20 17 13
12 15 15 17 16 11 24 20
19 15 14 24 22 17 21 14
Measures of Central Tendency
• Give one value that represents an entire group of scores
• Mean
• Median
• Mode
Mean
• On your tests you get: 70%, 80%, 80%, 90%
• The mean is 80%
• You know how to do this!
Mean
X
XN
= the mean
= an instruction to add (sigma) “the sum of”
= a score
= the number of scores
Practice
• What is the mean of:
5, 8, 6, 3, 2, 2, 9
Mean = 35 / 7 = 5
10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5
Mean = 107.8 / 8 = 13.48
Characteristics of the mean
• If the mean is subtracted from each score and the differences added, the sum will equal zero
Characteristics of the mean
70, 80, 80, 90 Mean = 80
70 - 80 = -1080 - 80 = 080 - 80 = 090 - 80 = 10 = 0
Characteristics of the mean
• The mean is the point about which the sum of the squared deviations is minimized
Characteristics of the mean
70, 80, 80, 90 Mean = 80
70 - 80 = -102= 10080 - 80 = 02 = 080 - 80 = 02 = 090 - 80 = 102 = 100 = 200
Population vs. Sample
_x = The mean of a sample
= The mean of a population
*They are both calculated the same way!
Population vs. Sample
The Median
• The point that divides a distribution of scores into two parts that are equal in size
The Median
10, 5, 13, 6, 14, 17, 2, 6, 9
2, 5, 6, 6, 9, 10, 13, 14, 17
The Median
10, 5, 13, 6, 14, 17, 2, 6, 9
2, 5, 6, 6, 9 10, 13, 14, 17
The Median
5, 8, 9, 15, 20, 25, 50
The Median
5, 8, 9, 15 20, 25, 50
The Median
5, 8, 9, 12, 15, 18, 22, 30, 32, 40
The Median
5, 8, 9, 12, 15, 18, 22, 30, 32, 40
The Median
5, 8, 9, 12,15, 18,22, 30, 32, 40
16.5
Practice
• What is the median of:
5, 8, 6, 3, 2, 2, 9
2, 2, 3, 5, 6, 8, 9
(7+1) / 2 = 4
Median = 5
Practice
• What is the median of:
10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5
10.5, 10.5, 11.6, 11.9, 12.9, 14.7, 15.5, 20.2
(8+1) / 2 = 4.5
(11.9 + 12.9) / 2 = 12.4
Median = 12.4
The Mode
• The most frequently occurring score
5, 6, 8, 9, 10, 10, 10, 12, 14, 14
Mode = 10
Practice
• What is the mode of:
5, 8, 6, 3, 2, 2, 9
Mode = 2
10.5, 11.6, 12.9, 14.7, 10.5, 11.9, 20.2, 15.5
Mode = 10.5
Determining Skewness
Determining Skewness
• For example:
Mean = 4
Median = 10
Mean Median
Determining Skewness
• For example:
Mean = 10
Median = 4
Median Mean
Determining Skewness
• Mean < Median = Negative Skew
• Mean > Median = Positive Skew
• Mean = Median = No Skew
Which should you use?
Mean Median Mode
Nominal NO NO OK
Ordinal NO OK OK
Interval OK OK OK
Ratio OK OK OK
Happiness Score
f
15 9
14 12
13 10
11 2
10 3
5 1
2 1
What is the mean, median, and mode?
Happiness Score
f score(f)
15 9 135
14 12 168
13 10 130
11 2 22
10 3 30
5 1 5
2 1 2
Mean = 492 / 38 = 12.95
Happiness Score
f
15 9
14 12
13 10
11 2
10 3
5 1
2 1
Mode = 14
Happiness Score
f
15 9
14 12
13 10
11 2
10 3
5 1
2 1
(38 + 1)/ 2 = 19.5
Median = 14
The Test Scores of 3 Students
• Joe = 78, 60, 92, 80, 80
• Bob = 47, 100, 98, 45, 100
• Mary = 78, 79, 77, 78, 78
The Test Scores of 3 Students
• Joe = 78, 60, 92, 80, 80– Mean = 78
• Bob = 47, 100, 98, 45, 100– Mean = 78
• Mary = 78, 79, 77, 78, 78– Mean = 78
Variability
• Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together
Range
• The highest score minus the lowest score
• Joe = 78, 60, 92, 80, 80
• Range = 92 - 60 = 32
Range
• The highest score minus the lowest score
• Bob = 47, 100, 98, 45, 100
• Range = 100 - 45 = 55
Range
• The highest score minus the lowest score
• Mary = 78, 79, 77, 78, 78
• Range = 79 - 77 = 2
The Test Scores of 3 Students
• Joe = 78, 60, 92, 80, 80– Mean = 78 Range = 32–
• Bob = 47, 100, 98, 45, 100– Mean = 78 Range = 55
• Mary = 78, 79, 77, 78, 78– Mean = 78 Range = 2
Range
• In general - the larger the range score, the more variance
• Pro: Easy to calculate
• Con: The range only depends on two extreme scores; can be misleading
Range
20, 62, 54, 32, 28, 44, 72, 69, 50
1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99
Range
20, 62, 54, 32, 28, 44, 72, 69, 50
Range = 49
1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,5, 5, 5, 5, 5, 99
Range = 98!!
Interquartile Range
• The range of scores that make up the middle 50 percent of the distribution
• Need to find the 25th percentile score and the 75th percentile score
Interquartile Range
50%
Interquartile Range
.25 (N) = The location of the 25th percentile score counting from the bottom
.25 (N) = The location of the 75th percentile score counting from the top
N = the number of cases
*If the answer is not even simply average
*Similar to how you found the median!!
Interquartile Range
IQR = 75th percentile - 25th percentile
Interquartile Range
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Interquartile Range
2, 5, 66, 10, 14, 16, 29, 40, 56, 62, 82, 99
.25 (12) = 3
Counting 3 from the bottom the 25th percentile score = 6
Interquartile Range
2, 5, 6, 10, 14, 16, 29, 40, 56, 62,62, 82, 99
.25 (12) = 3
Counting 3 from the top the 75th percentile score = 62
Interquartile Range
2, 5, 6,6, 10, 14, 16, 29, 40, 56, 62,62, 82, 99
IQR = 75th percentile - 25th percentile
56 = 62 - 6
Score f
80 4
72 5
70 9
69 10
68 6
54 5
52 1N = 40
Practice
Score f
80 4
72 5
70 9
69 10
68 6
54 5
52 1
N = 40
70 - 68 = 240(.25) = 10
Practice
• Find the range for:
• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
• 102.58, 51.25, 58.00, 96.34, 54.43
Practice
• Find the range for:
• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22– Range =76 - 4 = 72
• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2– Range = 78.6 - 3.2 = 75.4
• 102.58, 51.25, 58.00, 96.34, 54.43– Range = 102.58 - 51.25 = 51.33
Practice
• Find the interquartile range for:
• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
• 102.58, 51.25, 58.00, 96.34, 54.43
Practice
• Find the interquartile range for:
• 8, 4, 10, 15, 25, 56, 76, 64, 43, 4, 56, 22
• 4, 4, 8, 10, 15, 22, 25, 43, 56, 56, 64, 76
• (12) .25 = 3
• 56 - 8 = 48
Practice
• Find the interquartile range for:
• 8.5, 68.2, 78.3, 59.5, 78.6, 75.2, 12.9, 3.2
• 3.2, 8.5, 12.9, 59.5, 68.2, 75.2, 78.3, 78.6• (8).25 = 2• 78.3 - 8.5 = 69.8
Practice• Find the interquartile range for:
• 102.58, 51.25, 58.00, 96.34, 54.43
• 51.25, 54.43, 58.00, 96.34, 102.58• (5).25 = 1.25• (51.25+54.43)/2 = 52.84• (96.34+102.58)/2 = 99.46• 99.46-52.84 = 46.62
Boxplots
• The boxplot graphically displays three different characteristics of the distribution– Extreme scores– Interquartile range– Median
Boxplot
100N =
NEUR
40
30
20
10
0
Boxplot
100N =
NEUR
40
30
20
10
0
Interquartile range
25th - 75th percentile
Boxplot
100N =
NEUR
40
30
20
10
0
Extreme Scores
Boxplot
100N =
NEUR
40
30
20
10
0
Median
Boxplot
100N =
NEUR
40
30
20
10
0
Skew -- Look at the “whiskers” to determine if the distribution is skewed
Create a boxplot
• Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Create a boxplot
• Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median =
25th =
75th =
Lowest =
Highest =
Create a boxplot
• Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median = 22.5
25th = 6
75th = 62
Lowest = 2
Highest = 99
12N =
VAR00004
120
100
80
60
40
20
0
-20
Neuroticism
130130N =
MNEURCNEUR
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
Extraversion
130130N =
MEXTRACEXTRA
5.0
4.5
4.0
3.5
3.0
2.5
2.0
106120
15272
Conscientiousness
128128N =
MCONCCON
5
4
3
2
1
0
99999N =
VAR00006VAR00005VAR00004VAR00003VAR00002
120
100
80
60
40
20
0
-20
A
B
C
D
E
Which distribution has a positive skew?
99999N =
VAR00006VAR00005VAR00004VAR00003VAR00002
120
100
80
60
40
20
0
-20
A
B
C
D
E
Which distribution has a negative skew?
99999N =
VAR00006VAR00005VAR00004VAR00003VAR00002
120
100
80
60
40
20
0
-20
A
B
C
D
E
Which distribution is most compact?
99999N =
VAR00006VAR00005VAR00004VAR00003VAR00002
120
100
80
60
40
20
0
-20
A
B
C
D
E
Which distribution has a median close to 25?
99999N =
VAR00006VAR00005VAR00004VAR00003VAR00002
120
100
80
60
40
20
0
-20
A
B
C
D
E
Which distribution is most symmetrical?
99999N =
VAR00006VAR00005VAR00004VAR00003VAR00002
120
100
80
60
40
20
0
-20
A
B
C
D
E
Which distribution has has the largest range?
Review
Measures of central tendency– Mean – Median– Mode
• Measures of variability– Range– IQR
Variability
• Range
• IQR
• Problem with range and IQR– variability is still measured with only two
numbers!
Deviation Score Formula
• Deviation scores
Name Score x - DeviationScore
Joe 15
Bill 12
Mary 17
Sue 20
Name Score x - DeviationScore
Joe 15
Bill 12
Mary 17
Sue 20
x = 64
Name Score x - DeviationScore
Joe 15 15 - 16
Bill 12 12 - 16
Mary 17 17 - 16
Sue 20 20 - 16
x = 64
Name Score x - DeviationScore
Joe 15 15 - 16 -1
Bill 12 12 - 16 -4
Mary 17 17 - 16 1
Sue 20 20 - 16 4
x = 64 (x- ) = 0
Name Score x - DeviationScore
Joe 26
Bill 6
Mary 1
Sue 31
Name Score x - DeviationScore
Joe 26
Bill 6
Mary 1
Sue 31
= 16
Name Score x - DeviationScore
Joe 26 26 - 16 10
Bill 6 6 - 16 -10
Mary 1 1 - 16 -15
Sue 31 31 - 16 15
= 16
Sample 1 vs. Sample 2
• Sample 1:
Raw scores: 15, 12, 17, 20
• Sample 2:
Raw scores: 26, 6, 1, 31
Sample 1 vs. Sample 2
• Sample 1:
Raw scores: 15, 12, 17, 20
Deviation scores: -1, -4, 1, 4
• Sample 2:
Raw scores: 26, 6, 1, 31
Deviation scores: 10, -10, -15, 15
Deviation Scores
• As variability increases the absolute value of the deviation scores also goes up!
• How can we use this information to create a measure of variability?
How about?
How about?
Formula
Formula
- 1
Why “ – 1” ?
• Without it the answer is biased -- its answer tends to be too small
• Page 53 – 56 explain
• Don’t worry about why -- unless you want too!!
Name x
Bart 5
Lisa 8
Homer 2
Ned 10
Maggie 5
( )
Name x
Bart 5
Lisa 8
Homer 2
Ned 10
Maggie 5
= 6
( )
Name x
Bart 5 -1
Lisa 8 2
Homer 2 -4
Ned 10 4
Maggie 5 -1
= 6
( )
Name x
Bart 5 -1 1
Lisa 8 2 4
Homer 2 -4 16
Ned 10 4 16
Maggie 5 -1 1
= 6
( )
Name x
Bart 5 -1 1
Lisa 8 2 4
Homer 2 -4 16
Ned 10 4 16
Maggie 5 -1 1= 6 = 38
( )
Formula
- 1
Formula
- 1
38
Formula
- 1
38
5
Formula
- 1
38
59.5
Formula
- 1
38
59.53.08
Practice
• For the sample data below calculate s
6, 8, 4, 3, 4, 5
Obs x
Score1 6 1 1
Score2 8 3 9
Score3 4 -1 1
Score4 3 -2 4
Score5 4 -1 1
Score6 5 0 0
( )
= 5
= 16
-1
16
61.79
-1
Practice
2.462.47