Describing Data Using
Statistics
Statistics and Risk
Management
Copyright © Texas Education Agency, 2012. All rights reserved.
1
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Tools of Knowledge
Statistics:
The study of collecting,
organizing, and analyzing data.
• For Research
• For Reporting
Copyright © Texas Education Agency, 2012. All rights reserved.
3
When Did This Come About?
 We find Statistical Information (State of Things) in 1749.
 By the 18th century, the term "statistics" designated the
systematic collection of demographic and economic data
by states.
 In the early 19th century, the meaning of "statistics"
broadened, then including the discipline concerned with
the collection, summary, and analysis of data.
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Subjects
• Population: is the over all group of
subjects to which the research will apply.
• Sample: is a smaller, more practical, group
size that represents the larger population.
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Research Statistics?
• Descriptive Statistics: Procedures used to
organize and present data in a convenient
and communicable form.
• Inferential Statistics: Procedures employed
arrive at broader conclusions or inferences
about populations on the basis of samples.
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Data Types
 Qualitative Data
 Quantitative Data
 Discrete data
 Continuous data
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Sample Measurements
• Count: Number of Scores
• Low/High: The Extreme Values
• Range: From the Low to the High
Copyright © Texas Education Agency, 2012. All rights reserved.
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The Count = 24
•
•
•
•
•
•
•
•
•
•
•
•
9:00 pm
9:00 am
1:00 pm
6:00 pm
10:00 am
12:00 am
2:00 pm
10:00 pm
8:00 am
3:00 pm
11:00 am
8:00 pm
210
205
200
180
170
170
170
161
160
152
150
143
•
•
•
•
•
•
•
•
•
•
•
•
4:00 pm
5:00 pm
6:00 am
7:00 am
7:00 pm
11:00 pm
12:00 am
2:00 am
1:00 am
3:00 am
4:00 am
5:00 am
Copyright © Texas Education Agency, 2012. All rights reserved.
140
130
120
110
100
80
40
40
30
20
10
31
9
The High = 210
•
•
•
•
•
•
•
•
•
•
•
•
9:00 pm
9:00 am
1:00 pm
6:00 pm
10:00 am
12:00 am
2:00 pm
10:00 pm
8:00 am
3:00 pm
11:00 am
8:00 pm
210
205
200
180
170
170
170
161
160
152
150
143
•
•
•
•
•
•
•
•
•
•
•
•
4:00 pm
5:00 pm
6:00 am
7:00 am
7:00 pm
11:00 pm
12:00 am
2:00 am
1:00 am
3:00 am
4:00 am
5:00 am
Copyright © Texas Education Agency, 2012. All rights reserved.
140
130
120
110
100
80
40
40
30
20
10
31
10
The Low = 31
•
•
•
•
•
•
•
•
•
•
•
•
9:00 pm
9:00 am
1:00 pm
6:00 pm
10:00 am
12:00 am
2:00 pm
10:00 pm
8:00 am
3:00 pm
11:00 am
8:00 pm
210
205
200
180
170
170
170
161
160
152
150
143
•
•
•
•
•
•
•
•
•
•
•
•
4:00 pm
5:00 pm
6:00 am
7:00 am
7:00 pm
11:00 pm
12:00 am
2:00 am
1:00 am
3:00 am
4:00 am
5:00 am
Copyright © Texas Education Agency, 2012. All rights reserved.
140
130
120
110
100
80
40
40
30
20
10
31
11
The Range = 169
• 9:00 pm
210
• 4:00 pm
140
• 9:00 am
205
• 5:00 pm
130
• 1:00 pm
200
• 6:00 am
120
• 6:00 pm
180
• 7:00 am
110
• 10:00 am 170
• 7:00 pm
100
• 12:00 am 170
• 11:00 pm 80
• 2:00 pm
170
• 12:00 am 40
• 10:00 pm 161
• 2:00 am
40
• 8:00 am
160
• 1:00 am
30
• 3:00 pm
152
• 3:00 am
20
• 11:00 am 150
• 4:00 am
10
• 8:00 pm
143
• 5:00 am
31
What about a sample with an odd number of scores? 12
Copyright © Texas Education Agency, 2012. All rights reserved.
Measures of Central Tendencies
• Mode: The most repeated score
• Median: The score(s) in the middle
• Mean: The average of all scores (A Building
Block of Statistics)
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13
The Median: (143+140)/2=141.5
•
•
•
•
•
•
•
•
•
•
•
•
9:00 pm
9:00 am
1:00 pm
6:00 pm
10:00 am
12:00 am
2:00 pm
10:00 pm
8:00 am
3:00 pm
11:00 am
8:00 pm
210
205
200
180
170
170
170
161
160
152
150
143
•
•
•
•
•
•
•
•
•
•
•
•
4:00 pm
5:00 pm
6:00 am
7:00 am
7:00 pm
11:00 pm
12:00 am
2:00 am
1:00 am
3:00 am
4:00 am
5:00 am
140
130
120
110
100
80
40
40
30
20
10
31
What about a sample with an odd number of scores? 14
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The Mode
•
•
•
•
•
•
•
•
•
•
•
•
6:00 am
7:00 am
8:00 am
9:00 am
10:00 am
11:00 am
12:00 am
1:00 pm
2:00 pm
3:00 pm
4:00 pm
5:00 pm
120
110
160
205
170
150
170
200
170
152
140
130
•
•
•
•
•
•
•
•
•
•
•
•
6:00 pm
7:00 pm
8:00 pm
9:00 pm
10:00 pm
11:00 pm
12:00 am
1:00 am
2:00 am
3:00 am
4:00 am
5:00 am
Copyright © Texas Education Agency, 2012. All rights reserved.
180
100
143
210
161
80
40
30
40
20
10
31
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The Mean (Average)
Sum
Each Score
Mean (Average)
Notation: Sample data uses
CAPITAL Roman letters
Number of
Scores
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Spreadsheet Formulas
=Sum(B2:B11)/Count(B2:B11)
Or
=Average(B2:B11)
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Not Enough!! Why?
Sample One
Sample Two
5.0
6.4
5.0
4.5
6.0
8.0
1.4
6.0
1.0
10.5
Total: 26.9
Total: 26.9
Mean: 5.38
Mean: 5.38
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Deviation from the Mean
Sample One
Sample Two
5.0-5.38 = -.38
6.4-5.38 = 1.02
5.0-5.38 = -.38
4.5-5.38 = -.88
6.0-5.38 = .62
8.0-5.38 = 2.62
1.4-5.38 = -3.98
6.0-5.38 = .62
1.0-5.38 = -4.38
10.5-5.38 = 5.12
Tot. Deviation: 0.0
Tot. Deviation: 0.0
Mean: 5.38
Mean: 5.38
That does not work!
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Let’s Square Them
Sample One
Sample Two
(5.0-5.38)2 = .14
(6.4-5.38)2 = 1.04
(5.0-5.38)2 = .14
(8.0-5.38)2 = 6.86
(1.4-5.38)2 = 15.84
(6.0-5.38)2 = .38
(4.5-5.38)2 = .77
(6.0-5.38)2 = .38
(1.0-5.38)2 = 19.18
(10.5-5.38)2 = 26.21
Tot.Variance: 2.47
Tot.Variance: 68.47
Mean: 5.38
Mean: 5.38
Works, but values are exaggerated!
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Standard Deviation
Take the total variances and divide them by
N-1. Then take the square root of that.
Sample One
(2.47/(5-1))^.5 = .79
Standard
Deviation of
the Sample
Sample Two
(68.47/(5-1))^.5 = 4.14
Score
𝑺𝑿 =
Sample Mean
𝑿−𝑿
𝑵−𝟏
𝟐
The X subscript only
Number of
shows the S is for the
Scores
sample
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Do you sum the
differences then
square them or
square the
differences then
sum them?
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Degrees of Freedom
Why do we divide by N-1 instead of just N?
N-1 is referred to as Degrees of Freedom. It was
decided that df is used because it makes the
resulting amount larger by using a consistent
methodology….that’s all. You will see it again.
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Spreadsheet Formula
=STDEV.P(B2:B11)
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