Overview of Central Tendency Measures in Statistics
ADNAN MENDERES UN
I
VERS
I
TY
Faculty of Engineering
MAT
2
54
Probability & Statistics
Week #
3
Olcay ÜZENGİ AKTÜRK, Assoc. Prof
.Dr.
Measures of central tendency
and variability
Data Types
2
•
Primary
•
Ours
•
Secondary
•
Not ours
•
Qualitative data
(words)
•
Blue, short
•
Quantitative
data
(numbers)
•
1,
2.5, 10000
•
Discret
e
(countable)
•
1 car, 206 students
•
Continuous
(
measurable
)
•
165 cm, 52.5 kg
Sampling Techniques
•
Probability Sampling
(every member of
the population has equal chance)
•
Simple Random Sampling (Lottery)
•
Systematic Sampling (every 4
th
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)
3
Presentation of Data
4
Other Graphical Methods
Box Plot (Box and Whisker)
5
Scatter Plot
How can you represent a huge amount of data (numbers) by
using only one
(or two)
number
(s)
?
6
Minimum
of them?
Maximum of them?
Average of them?
????
“CENTRAL TENDENCY”
7
Mean (average)
Median (middle)
Mode (most)
•
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
.
Sample Mean
8
2
3
7
7
= 2+3+7+7/4
= 4.75
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 (read “
x
bar”).
Sample Mean
9
10
Trimmed Mean
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.
Sample Median
11
2
3 3
7 7
7
= 3+7/2
= 5
2
2
3 3
7
7
= 3
Sample Mode
12
2
3
7
7
unimodal
2
2
3
7
7
bimodal
The most repeated value in observations
7
2
7
13
Sample #1 :
Sample #2 :
No mode !
No mode !
Measures of spread or variability ????
Sample
Range
14
The difference between the lowest and the highest
value of that sample.
2
3
7
7
The range is 7-2 = 5
Variance & Standard Deviation
15
Variance & Standard Deviation
16
1
3
5
11
= 1+3+5+11/4
= 5
Population vs. Sample
17
Commonly used Symbols
for a
Sample
and for a
Population
.
Example from textbook
18
Example from textbook
19
Example from textbook
20
Example from textbook
21
Example from textbook
22
MAT254-02 Probability & Statistics
23
END of LECTURE #3
A measure of central tendency in statistics aims to describe a dataset by identifying its central position. This includes commonly used statistics like the mean. This article covers various aspects of central tendency measures, including data types, sampling techniques, graphical methods, and more.
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ADNAN MENDERES UNIVERSITY Faculty of Engineering Measures of central tendency and variability MAT 254 Probability & Statistics Week #3 Olcay ZENG AKT RK, Assoc. Prof.Dr.
Data Types 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 Primary Ours Secondary Not ours 2
Sampling Techniques Probability Sampling (every member of the population has equal chance) Simple Random Sampling (Lottery) Systematic Sampling (every 4thsample) 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) 3
Age Frequency Presentation of Data 12 2 13 13 14 27 15 4 Graphical Method Textual Method Tabular Method Rearrangement from lowest to highest Stem-and-leaf plot Frequency distribution table (FDT) Relative FDT Cumulative FDT Contingency Table Bar Chart Histogram Frequency Polygon Pie Chart Less than, greater than Ogive Percentage of causes of child death in Egypt accident 10% congenital 10% diarrhea 50% chest infection 30% 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 4
Other Graphical Methods Box Plot (Box and Whisker) Lowest value Highest value Median Upper Quartile Lower Quartile Inter-Quartile Range Range Scatter Plot Correlation between Doppler velocimetry (RI) and baby birth weight 1 0.9 0.8 0.7 0.6 0.5 RI 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 6
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) 7
? Sample Mean 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. x The mean of a sample is denoted by (read x bar ). 2 3 7 7 ? = 2+3+7+7/4 = 4.75 8
x Sample Mean 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 = 21 2 . $ x 9
x Trimmed Mean (..) tr 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. + + + + + + + 1 1 1 2 2 2 2 100 = x 10 ( ) tr 8 = 11 1 . $ x 10 ( ) tr . 1 = 67 $ x ( 20 ) tr 10
x~ Sample Median 2 2 3 3 7 7 2 3 3 7 7 7 ? = 3+7/2 = 5 ? = 3 11
Sample Mode The most repeated value in observations 2 2 3 7 7 2 3 7 7 2 7 7 unimodal bimodal 12
Sample #1 : 1 51 101 151 201 ~ x = = 101 ; 101 x No mode ! Sample #2 : 99 100 101 102 103 ~ x = = 101 ; 101 x 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 1 3 5 11 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 ? = 1+3+5+11/4 = 5 Standard deviation ? = 18.666 = 4.32 16
Population vs. Sample Commonly used Symbols for a Sample and for a Population. 17
END of LECTURE #3 MAT254-02 Probability & Statistics 23
