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ADNAN MENDERES UNIVERSITY
Faculty of Engineering
Measures of central tendency and variability
MAT 254 Probability & Statistics
Week #3
Olcay ÜZENGİ AKTÜRK, Assoc. Prof.Dr.
Data Types
• Primary
• Ours
• Secondary
• Not ours
• Qualitative data (words)
• Blue, short
• Quantitative data (numbers)
• 1, 2.5, 10000
• Discrete (countable)
• 1 car, 206 students
• Continuous (measurable)
• 165 cm, 52.5 kg
2
Sampling Techniques
• Probability Sampling (every member of
the population has equal chance)
•
•
•
•
•
Simple Random Sampling (Lottery)
Systematic Sampling (every 4th sample)
Stratified Sampling (from each area)
Cluster or Area Sampling (form clusters)
Multi-stage Sampling (multi-stage)
• Non-probability Sampling (samples are
selected based on an inclusion rule)
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Age
Frequency
12
2
13
13
14
27
15
4
Presentation of Data
Percentage of causes of child death in Egypt
congenital
10%
accident
10%
diarrhea
50%
Stem
Leaves
1
7,8
2
0,3,3,4,5,6,7,8,9
3
4,4,5,5,7,8,8,8,8,9,9,9
4
2,3,4,4,5,6,6,8,9
5
0,0,0
chest infection
30%
4
Other Graphical Methods
Box Plot (Box and Whisker)
Lowest value
Highest value
Median
Upper Quartile
Lower Quartile
Inter-Quartile Range
Range
Scatter Plot
RI
Correlation between Doppler velocimetry (RI) and
baby birth weight
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.5
2
2.5
3
3.5
4
4.5
baby weight in kg
5
How can you represent a huge amount of data (numbers) by
using only one (or two) number(s)?
Minimum of them?
Maximum of them?
Average of them?
????
“CENTRAL TENDENCY”
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• In statistics, a measure of CENTRAL TENDENCY is a single value
that attempts to describe a set of data by identifying the central
position within that set of data. As such, measures of central
tendency are sometimes called measures of central location.
• The most commonly used statistics for measuring the center of
a set of data, arranged in order of magnitude, are the mean,
median, and mode.
 Mean (average)
 Median (middle)
 Mode (most)
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Sample Mean
𝑥
 The mean (arithmetic mean or average) of a set of data is found by adding up
all the items and then dividing by the sum of the number of items.
 The mean of a sample is denoted by
x (read “x bar”).
2
3
7
7
𝑥 = 2+3+7+7/4
= 4.75
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Sample Mean
x
Staff
1
2
3
4
5
6
7
8
9
10
Salary
1$
1$
1$
1$
2$
2$
2$
2$
100$
100$
1  1  1  1  2  2  2  2  100  100
x
10
x  21.2$
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Trimmed Mean
xtr (..)
A trimmed mean is computed by “trimming away” a certain
percent of both the largest and the smallest set of values.
For example, the 10% trimmed mean is found by eliminating the
largest 10% and smallest 10% and computing the average of the
remaining values.
xtr (10 )
xtr (10 )
1  1  1  2  2  2  2  100

8
 11.1$
xtr ( 20 )  1.67$
10
~
Sample Median x
2
3
7
7
𝑥 = 3+7/2
=5
3
7
2
2
3
7
7
3
𝑥=3
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Sample Mode
The most repeated value in observations
2
3
7
7
7
2
2
3
7
7
2
7
unimodal
bimodal
12
Sample #1 :
1
51
101
x  101 ; ~
x  101
Sample #2 :
99
100
101
x  101 ; ~
x  101
151
201
No mode !
102
103
No mode !
Measures of spread or variability ????
13
Sample Range
The difference between the lowest and the highest
value of that sample.
2
3
7
7
The range is 7-2 = 5
14
Variance & Standard Deviation
15
Variance & Standard Deviation
1
3
5
11
𝑥 = 1+3+5+11/4
=5
Variance
s2= [(1-5)2 + (3-5)2 + (5-5)2 + (11-5)2] / (4-1)
s2= [(-4)2 + (-2)2 + (0)2 + (6)2] / 3
s2= (16 + 4 + 0 + 36) / 3
s2= 56/3 = 18.666
Standard deviation
𝑠 = 18.666 = 4.32
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Population vs. Sample
Commonly used Symbols
for a Sample and for a Population.
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Example from textbook
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Example from textbook
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Example from textbook
20
Example from textbook
21
Example from textbook
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END of LECTURE #3
MAT254-02 Probability & Statistics
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