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1_INTRODUCTION TO STATISTICS AND STATISTICAL INFERENCE.ppt
1. Note: Most of the Slides were taken from
Elementary Statistics: A Handbook of Slide
Presentation prepared by Z.V.J. Albacea, C.E.
Reano, R.V. Collado, L.N. Comia and N.A.
Tandang in 2005 for the Institute of Statistics,
CAS, UP Los Banos
Training on Teaching
Basic Statistics for
Tertiary Level Teachers
Summer 2008
INTRODUCTION TO
STATISTICS AND
STATISTICAL
INFERENCE
2. Session 1.2
TEACHING BASIC STATISTICS ….
Florence Nightingale on Statistics
“...the most important science in the whole
world: for upon it depends the practical
application of every other science and of every
art: the one science essential to all political
and social administration, all education, all
organization based on experience, for it only
gives results of our experience.”
“To understand God's thoughts, we must study
statistics, for these are the measures of His
purpose.”
3. Session 1.3
TEACHING BASIC STATISTICS ….
Realities about Statistics
The man in the street distrusts statistics and
despises [his image of] statisticians, those who
diligently collect irrelevant facts and figures and
use them to manipulate society.
“There are three kinds of lies: lies, damned lies, and
statistics” – Mark Twaine
One can not go about without statistics.
“Statistics are like bikinis. What they reveal is suggestive,
but what they conceal is vital.” – Aaron Levenstein
4. Session 1.4
TEACHING BASIC STATISTICS ….
Definition of Statistics
plural sense: numerical facts, e.g. CPI,
peso-dollar exchange rate
singular sense: scientific discipline
consisting of theory and methods for
processing numerical information
that one can use when making
decisions in the face of uncertainty.
5. Session 1.5
TEACHING BASIC STATISTICS ….
History of Statistics
The term statistics came from the Latin phrase
“ratio status” which means study of practical
politics or the statesman’s art.
In the middle of 18th century, the term statistik
(a term due to Achenwall) was used, a German
term defined as “the political science of several
countries”
From statistik it became statistics defined as a
statement in figures and facts of the present
condition of a state.
6. Session 1.6
TEACHING BASIC STATISTICS ….
Application of Statistics
Diverse applications
“During the 20th Century statistical thinking
and methodology have become the
scientific framework for literally dozens of
fields including education, agriculture,
economics, biology, and medicine, and with
increasing influence recently on the hard
sciences such as astronomy, geology, and
physics. In other words, we have grown
from a small obscure field into a big
obscure field.” – Brad Efron
7. Session 1.7
TEACHING BASIC STATISTICS ….
Application of Statistics
Comparing the effects of five kinds of
fertilizers on the yield of a particular
variety of corn
Determining the income distribution of
Filipino families
Comparing the effectiveness of two diet
programs
Prediction of daily temperatures
Evaluation of student performance
8. Session 1.8
TEACHING BASIC STATISTICS ….
Two Aims of Statistics
Statistics aims to uncover
structure in data, to explain
variation…
Descriptive
Inferential
9. Session 1.9
TEACHING BASIC STATISTICS ….
Areas of Statistics
Descriptive statistics
methods concerned w/
collecting, describing, and
analyzing a set of data
without drawing
conclusions (or inferences)
about a large group
Inferential statistics
methods concerned
with the analysis of a
subset of data leading
to predictions or
inferences about the
entire set of data
10. Session 1.10
TEACHING BASIC STATISTICS ….
Example of Descriptive Statistics
Present the Philippine population by constructing a
graph indicating the total number of Filipinos counted
during the last census by age group and sex
11. Session 1.11
TEACHING BASIC STATISTICS ….
Example of Inferential Statistics
A new milk formulation designed to improve the psychomotor
development of infants was tested on randomly selected infants.
Based on the results, it was concluded that the new milk formulation is
effective in improving the psychomotor development of infants.
12. Session 1.12
TEACHING BASIC STATISTICS ….
Inferential Statistics
Larger Set
(N units/observations) Smaller Set
(n units/observations)
Inferences and
Generalizations
13. Session 1.13
TEACHING BASIC STATISTICS ….
Key Definitions
The universe/physical population is the collection of
things or observational units under consideration.
A variable is a characteristic observed or measured on
every unit of the universe.
The statistical population is the set of all possible values
of the variable.
Measurement is the process of determining the value or
label of the variable based on what has been observed.
An observation is the realized value of the variable.
Data is the collection of all observations.
14. Session 1.14
TEACHING BASIC STATISTICS ….
Key Definitions
Parameters are numerical measures
that describe the population or universe
of interest. Usually donated by Greek
letters; (mu), (sigma), (rho),
(lambda), (tau), (theta), (alpha) and
(beta).
Statistics are numerical measures of a
sample
15. Session 1.15
TEACHING BASIC STATISTICS ….
VARIABLES
Qualitative Quantitative
Continuous
Discrete
Types of Variables
Qualitative variable
Describes the quality or
character of something
Quantitative variable
Describes the amount or
number of something
a. Discrete
countable
b. Continuous
Measurable (measured
using a continuous scale
such as kilos, cms, grams)
c. Constant
16. Session 1.16
TEACHING BASIC STATISTICS ….
Levels of Measurement
1. Nominal
Numbers or symbols used to classify units
into distinct categories
2. Ordinal scale
Accounts for order; no indication of distance
between positions
3. Interval scale
Equal intervals (fixed unit of measurement);
no absolute zero
4. Ratio scale
Has absolute zero
17. Session 1.17
TEACHING BASIC STATISTICS ….
Methods of Collecting Data
Objective Method
Subjective Method
Use of Existing Records
19. Session 1.19
TEACHING BASIC STATISTICS ….
Mean Median Mode
Summary Measures
Variation
Variance
Standard Deviation
Coefficient of
Variation
Range
Location
Maximum
Minimum
Percentile
Quartile
Decile
Median
Interquartile
Range
Skewness
Kurtosis
Central
Tendency
20. Session 1.20
TEACHING BASIC STATISTICS ….
A single value that is used to identify
the “center” of the data
it is thought of as a typical value of
the distribution
precise yet simple
most representative value of the
data
Measures of Central Tendency
21. Session 1.21
TEACHING BASIC STATISTICS ….
Mean
Most common measure of the center
Also known as arithmetic average
1 1 2
N
i
i N
X
X X X
N N
1 2
1
n
i
n
i
x
x x x
x
n n
Population Mean:
Sample Mean:
22. Session 1.22
TEACHING BASIC STATISTICS ….
Properties of the Mean
may not be an actual
observation in the data set
can be applied in at least
interval level
easy to compute
every observation contributes
to the value of the mean
23. Session 1.23
TEACHING BASIC STATISTICS ….
Properties of the Mean
subgroup means can be combined to come up
with a group mean (use weighted mean)
easily affected by extreme values
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5
Mean = 6
24. Session 1.24
TEACHING BASIC STATISTICS ….
Median
Divides the observations into two equal
parts
If the number of observations is odd, the
median is the middle number.
If the number of observations is even, the
median is the average of the 2 middle
numbers.
Sample median denoted as
while population median is denoted as
x
~
~
25. Session 1.25
TEACHING BASIC STATISTICS ….
Properties of a Median
may not be an actual observation in
the data set
can be applied in at least ordinal level
a positional measure; not affected by
extreme values
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
26. Session 1.26
TEACHING BASIC STATISTICS ….
Mode
occurs most frequently
nominal average
may or may not exist
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
27. Session 1.27
TEACHING BASIC STATISTICS ….
Properties of a Mode
can be used for qualitative as
well as quantitative data
may not be unique
not affected by extreme values
can be computed for
ungrouped and grouped data
28. Session 1.28
TEACHING BASIC STATISTICS ….
Mean, Median & Mode
Use the mean when:
sampling stability is desired
other measures are to be
computed
29. Session 1.29
TEACHING BASIC STATISTICS ….
Mean, Median & Mode
Use the median when:
the exact midpoint of the
distribution is desired
there are extreme
observations
30. Session 1.30
TEACHING BASIC STATISTICS ….
Mean, Median & Mode
Use the mode when:
when the "typical" value is
desired
when the dataset is measured
on a nominal scale
31. Session 1.31
TEACHING BASIC STATISTICS ….
Measures of Location
A Measure of Location summarizes a
data set by giving a value within the
range of the data values that describes
its location relative to the entire data set
arranged according to magnitude
(called an array).
Some Common Measures:
Minimum, Maximum
Percentiles, Deciles, Quartiles
32. Session 1.32
TEACHING BASIC STATISTICS ….
Maximum and Minimum
Minimum is the smallest value in the
data set, denoted as MIN.
Maximum is the largest value in the
data set, denoted as MAX.
33. Session 1.33
TEACHING BASIC STATISTICS ….
Percentiles
Numerical measures that give the
relative position of a data value
relative to the entire data set.
Divide an array (raw data arranged
in increasing or decreasing order
of magnitude) into 100 equal parts.
The jth percentile, denoted as Pj, is
the data value in the the data set
that separates the bottom j% of the
data from the top (100-j)%.
34. Session 1.34
TEACHING BASIC STATISTICS ….
EXAMPLE
Suppose LJ was told that relative
to the other scores on a certain
test, his score was the 95th
percentile.
This means that (at least) 95%
of those who took the test had
scores less than or equal to LJ’s
score, while (at least) 5% had
scores higher than LJ’s.
35. Session 1.35
TEACHING BASIC STATISTICS ….
Deciles
Divide an array into ten equal
parts, each part having ten
percent of the distribution of
the data values, denoted by Dj.
The 1st decile is the 10th
percentile; the 2nd decile is the
20th percentile…..
36. Session 1.36
TEACHING BASIC STATISTICS ….
Quartiles
Divide an array into four equal
parts, each part having 25% of
the distribution of the data
values, denoted by Qj.
The 1st quartile is the 25th
percentile; the 2nd quartile is
the 50th percentile, also the
median and the 3rd quartile is
the 75th percentile.
37. Session 1.37
TEACHING BASIC STATISTICS ….
Measures of Variation
A measure of variation is a
single value that is used to
describe the spread of the
distribution
A measure of central tendency
alone does not uniquely
describe a distribution
38. Session 1.38
TEACHING BASIC STATISTICS ….
Mean = 15.5
s = 3.338
11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5
s = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Data C
A look at dispersion…
39. Session 1.39
TEACHING BASIC STATISTICS ….
Two Types of Measures of
Dispersion
Absolute Measures of Dispersion:
Range
Inter-quartile Range
Variance
Standard Deviation
Relative Measure of Dispersion:
Coefficient of Variation
40. Session 1.40
TEACHING BASIC STATISTICS ….
Range (R)
The difference between the maximum and
minimum value in a data set, i.e.
R = MAX – MIN
Example: Pulse rates of 15 male residents of a
certain village
54 58 58 60 62 65 66 71
74 75 77 78 80 82 85
R = 85 - 54 = 31
41. Session 1.41
TEACHING BASIC STATISTICS ….
Some Properties of the Range
The larger the value of the
range, the more dispersed
the observations are.
It is quick and easy to
understand.
A rough measure of
dispersion.
42. Session 1.42
TEACHING BASIC STATISTICS ….
Inter-Quartile Range (IQR)
The difference between the third quartile and
first quartile, i.e.
IQR = Q3 – Q1
Example: Pulse rates of 15 residents of a
certain village
54 58 58 60 62 65 66 71
74 75 77 78 80 82 85
IQR = 78 - 60 = 18
43. Session 1.43
TEACHING BASIC STATISTICS ….
Some Properties of IQR
Reduces the influence of
extreme values.
Not as easy to calculate
as the Range.
44. Session 1.44
TEACHING BASIC STATISTICS ….
Variance
important measure of variation
shows variation about the mean
Population variance
Sample variance
N
X
N
i
i
1
2
2
)
(
1
)
(
1
2
2
n
x
x
s
n
i
i
45. Session 1.45
TEACHING BASIC STATISTICS ….
Standard Deviation (SD)
most important measure of variation
square root of Variance
has the same units as the original data
Population SD
Sample SD
N
X
N
i
i
1
2
)
(
1
)
(
1
2
n
x
x
s
n
i
i
47. Session 1.47
TEACHING BASIC STATISTICS ….
Remarks on Standard Deviation
If there is a large amount of variation,
then on average, the data values will be
far from the mean. Hence, the SD will be
large.
If there is only a small amount of
variation, then on average, the data
values will be close to the mean. Hence,
the SD will be small.
48. Session 1.48
TEACHING BASIC STATISTICS ….
Mean = 15.5
s = 3.338
11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5
s = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Data C
Comparing Standard Deviations
(comparable only when units of measure are the same and
the means are not too different from each other)
49. Session 1.49
TEACHING BASIC STATISTICS ….
Example: Team A - Heights of five marathon players in inches
5”
65 “ 65 “ 65 “ 65 “ 65 “
Mean = 65
S = 0
Comparing Standard Deviations
50. Session 1.50
TEACHING BASIC STATISTICS ….
Example: Team B - Heights of five marathon players in inches
62 “ 67 “ 66 “ 70 “ 60 “
Mean = 65”
s = 4.0”
Comparing Standard Deviation
51. Session 1.51
TEACHING BASIC STATISTICS ….
Properties of Standard Deviation
It is the most widely used measure of
dispersion. (Chebychev’s Inequality)
It is based on all the items and is rigidly
defined.
It is used to test the reliability of measures
calculated from samples.
The standard deviation is sensitive to the
presence of extreme values.
It is not easy to calculate by hand (unlike the
range).
52. Session 1.52
TEACHING BASIC STATISTICS ….
Chebyshev’s Rule
It permits us to make statements about
the percentage of observations that
must be within a specified number of
standard deviation from the mean
The proportion of any distribution that
lies within k standard deviations of the
mean is at least 1-(1/k2) where k is
any positive number larger than 1.
This rule applies to any distribution.
53. Session 1.53
TEACHING BASIC STATISTICS ….
For any data set with mean () and
standard deviation (SD), the following
statements apply:
At least 75% of the observations are
within 2SD of its mean.
At least 88.9% of the observations are
within 3SD of its mean.
Chebyshev’s Rule
54. Session 1.54
TEACHING BASIC STATISTICS ….
Illustration
At least 75%
At least 75% of the observations
are within 2SD of its mean.
55. Session 1.55
TEACHING BASIC STATISTICS ….
Example
The midterm exam scores of 100 STAT 1 students
last semester had a mean of 65 and a standard
deviation of 8 points.
Applying the Chebyshev’s Rule, we can say that:
1. At least 75% of the students had scores
between 49 and 81.
2. At least 88.9% of the students had scores
between 41 and 89.
56. Session 1.56
TEACHING BASIC STATISTICS ….
Coefficient of Variation (CV)
measure of relative variation
usually expressed in percent
shows variation relative to mean
used to compare 2 or more groups
Formula :
100%
Mean
SD
CV
58. Session 1.58
TEACHING BASIC STATISTICS ….
Measure of Skewness
Describes the degree of departures of the
distribution of the data from symmetry.
The degree of skewness is measured by
the coefficient of skewness, denoted as SK
and computed as,
SD
Median
Mean
K
3
S
59. Session 1.59
TEACHING BASIC STATISTICS ….
What is Symmetry?
A distribution is said to be
symmetric about the mean,
if the distribution to the left
of mean is the “mirror
image” of the distribution to
the right of the mean.
Likewise, a symmetric
distribution has SK=0 since
its mean is equal to its
median and its mode.
60. Session 1.60
TEACHING BASIC STATISTICS ….
SK > 0
positively
skewed
Measure of Skewness
SK < 0
negatively skewed
61. Session 1.61
TEACHING BASIC STATISTICS ….
Measure of Kurtosis
Describes the extent of peakedness or
flatness of the distribution of the data.
Measured by coefficient of kurtosis (K)
computed as,
4
1
4
3
N
i
i
X
K
N
62. Session 1.62
TEACHING BASIC STATISTICS ….
K = 0
mesokurtic
K > 0
leptokurtic
K < 0
platykurtic
Measure of Kurtosis
63. Session 1.63
TEACHING BASIC STATISTICS ….
Box-and-Whiskers Plot
Concerned with the symmetry of the
distribution and incorporates
measures of location in order to study
the variability of the observations.
Also called as box plot or 5-number
summary (represented by Min, Max,
Q1, Q2, and Q3).
Suitable for identifying outliers.
64. Session 1.64
TEACHING BASIC STATISTICS ….
The diagram is made up of a box which
lies between the first and third
quartiles.
The whiskers are the straight lines
extending from the ends of the box to
the smallest and largest values that
are not outliers.
Box-and-Whiskers Plot
65. Session 1.65
TEACHING BASIC STATISTICS ….
Steps to Construct a Box-and-Whiskers plot
Step 1: Draw a rectangular box whose left edge is at the
Q1 and whose right edge is at the Q3 so the box width
is the IQR. Then draw a vertical line segment inside
the box where the median is found.
Q1 Q3
Md
75 78 85
66. Session 1.66
TEACHING BASIC STATISTICS ….
Step 2: Place marks at distances 1.5 IQR from
either end of the box. (1.5 IQR =15)
100
Q1 Q3
Md
75 78 85
60
1.5 IQR 1.5 IQR
Steps to Construct a Box-and-Whiskers plot
67. Session 1.67
TEACHING BASIC STATISTICS ….
Step 3:Draw the horizontal line
segments known as the “whiskers”
from each of the end box to the
largest and smallest values in the data
set that are not outliers.
(An observation beyond 1.5 IQR is
an outlier.)
Steps to Construct a Box-and-Whiskers plot
68. Session 1.68
TEACHING BASIC STATISTICS ….
Step 4: For every outlier, draw a dot. If two or more dots
have the same values, draw the dots side by side.
Q1 Q3
Md
75 78 85
60 100
1.5 IQR 1.5 IQR
98
55
.
.
Steps to Construct a Box-and-Whiskers plot