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DescriptiveStatistics3slidesperpage.pdf
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4320 Descriptive Statistics
Central Tendency and Variation
Normal Distribution
99.9
INSERT new Figure from 4th edition here
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68%
95%
99%
Symbols
f = Frequency
N = Number of measures
Σ= Summation
X = Score / measurement
X= Mean
Measures of Central Tendency
“Average”
Describe the middle characteristics of a set of scores.
Allow a score to be compared with the group of scores.
Three measures of central tendency.
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Measures of Central Tendency
Mode: The most frequent score of the distribution
Median: The point below which 50% of scores fall (most appropriate central tendency to use)
Mean: The arithmetic average of the scores in a distribution; The most commonly used measure of central tendency
Mode
Case 1
X f
5 2
4 3
3 2
2 5
1 7
Mode =1
5,4,3,2,1,1,1,2,2,3,4,5, 4,2,1,2,1,1,1
Mode
Case 1
X f
5 2
4 3
3 2
2 5
1 7
Case 2
X f
8 7
7 3
6 2
5 5
1 7
Mode =1 Mode = 8 , 1
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Median
Point at which half of the scores are below and half are above.
Not a score.
Represents the 50th percentile
Median
X
55
39
26
24
20
19
18
X
88
87
86
85
84
83
Median Median = 85.5
X
150
120
86
85
50
40
Median = 85.5
Mean X = Σ X / N
X
88 X=431
87 6
86 X=71.83
85
84
1
X
88 X= 510
87 6
86 X= 85
85
84
80
Σ = 510 N = 6Σ =431 N = 6
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Summation Notation
S = N =
Mean =
Median =
Mode =
20, 20, 21, 21, 22, 19, 20, 19, 22, 18, 18, 20, 21, 18, 19, 18, 21, 26, 30
Summation Notation
S = 393 N = 19
Mean = 20.68
Median = 20
Mode = 18, 20, 21
20, 20, 21, 21, 22, 19, 20, 19, 22, 18, 18, 20, 21, 18, 19, 18, 21, 26, 30
Measures of Variability
Definition: spread of scores.
Allows for further analysis of the scores.
Range: From minimum to maximum
score
Standard Deviation (s, σ, or SD): The
square root of the variance
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Standard Deviation
Most commonly used measure of variance
X X-X
7 +2 4
6 +1 1
5 0 0
4 -1 1
3 -2 4
Σ 25 Σ = 10
N 5 σ = 1.58
X 5
X X-X
10 +5 25
8 +3 9
5 0 0
2 -3 9
0 -5 25
Σ 25 Σ = 68
N 5 σ = 4.12
X 5
Standard Deviation
Example
20, 20, 21, 21, 22, 19, 20, 19, 22, 18, 18, 20, 21, 18, 19, 18, 21, 26, 30
SD = ?
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Example
20, 20, 21, 21, 22, 19, 20, 19, 22, 18, 18, 20, 21, 18, 19, 18, 21, 26, 30
SD = 2.96
Standard Scores
Z-score: a standardized score
T-score: a standardized ratio T = 50 + 10 * (Z)
Z Score
Sit-ups (1 min)
45, 37, 36, 35, 33, 33, 32, 21, 13, 12
Push ups (1 min)
30, 25, 24, 22, 10, 9, 8, 8, 6, 5
On which test did John Doe score better on?
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Sit Ups
Sit-ups (1 min)
45, 37, 36, 35, 33, 33, 32, 21, 13, 12
X= 29.7
s=10.2
z=(35-29.7) / 10.2 = .52
Push Ups
Push ups (1 min)
30, 25, 24, 22, 10, 9, 8, 8, 6, 5,
X= 15.1
s=8.22
z= (22-15.1)/8.22=.84
Z Score Sit ups
Push ups
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68%
95%
99%
Graphing Frequency Distributions
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Graphing Frequency Distributions
Normal or Mesokurtic Mean = Median = Mode
Leptokurtic (Positive Kurtosis)
Platykurtic (Negative Kurtosis)
Left or Negatively Skewed
Right or Positively
Skewed
Mean, Median, Mode
Examples
25 45
Examples
Jan Feb March April May June July Aug Sept Nov Dec
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Examples
D e c W
e e k 1
D e c W
e e k 2
D e c W
e e k 3
D e c W
e e k 4
Ja n W
e e k 1
Ja n W
e e k 2
M a
y W e e k 1
M a
y W e e k 2
M a
y W e e k 3
M a
y W e e k 4
