Introduction to Statistics and Statistical Inference

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

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.”

Session 1.3
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

Session 1.4
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.

Session 1.5
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.

Session 1.6
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

Session 1.7
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

Session 1.8
Two Aims of Statistics
Statistics aims to uncover
structure in data, to explain
variation…
 Descriptive
 Inferential

Session 1.9
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

Session 1.10
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

Session 1.11
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.

Session 1.12
Inferential Statistics
Larger Set
(N units/observations) Smaller Set
(n units/observations)
Inferences and
Generalizations

Session 1.13
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.

Session 1.14
Key Definitions
 Parameters are numerical measures
that describe the population or universe
of interest. Usually donated by Greek
letters; µ (mu), σ (sigma), ρ (rho), λ
(lambda), τ (tau), θ (theta), α (alpha) and
β (beta).
 Statistics are numerical measures of a
sample

Session 1.15
VARIABLES
Qualitative Quantitative
ContinuousDiscrete
Types of Variables
Qualitative variable
 Describes the quality or
character of something
Quantitative variable
 Describes the amount or
number of something
a. Discrete
 countable
a. Continuous
 Measurable (measured
using a continuous scale
such as kilos, cms, grams)
a. Constant

Session 1.16
Levels of Measurement
1. Nominal
 Numbers or symbols used to classify units
into distinct categories
1. Ordinal scale
 Accounts for order; no indication of distance
between positions
1. Interval scale
 Equal intervals (fixed unit of measurement);
no absolute zero
1. Ratio scale
 Has absolute zero

Session 1.17
Methods of Collecting Data
 Objective Method
 Subjective Method
 Use of Existing Records

Session 1.18
Methods of Presenting Data
Textual
Tabular
Graphical

Session 1.19
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

Session 1.20
 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

Session 1.21
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
µ = + + +
= =
∑ K
1 21
n
i
ni
x
x x x
x
n n
= + + +
= =
∑ K
Population Mean:
Sample Mean:

Session 1.22
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

Session 1.23
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

Session 1.24
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~
µ~

Session 1.25
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

Session 1.26
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

Session 1.27
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

Session 1.28
Mean, Median & Mode
Use the mean when:
 sampling stability is desired
 other measures are to be
computed

Session 1.29
Mean, Median & Mode
Use the median when:
 the exact midpoint of the
distribution is desired
 there are extreme
observations

Session 1.30
Mean, Median & Mode
Use the mode when:
 when the "typical" value is
desired
 when the dataset is measured
on a nominal scale

Session 1.31
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

Session 1.32
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.

Session 1.33
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)%.

Session 1.34
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.

Session 1.35
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…..

Session 1.36
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.

Session 1.37
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

Session 1.38
Mean = 15.5
s = 3.33811 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…

Session 1.39
Two Types of Measures of
Dispersion
Absolute Measures of Dispersion:
 Range
 Inter-quartile Range
 Variance
 Standard Deviation
Relative Measure of Dispersion:
 Coefficient of Variation

Session 1.40
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

Session 1.41
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.

Session 1.42
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

Session 1.43
Some Properties of IQR
 Reduces the influence of
extreme values.
 Not as easy to calculate
as the Range.

Session 1.44
Variance
 important measure of variation
 shows variation about the mean
Population variance
Sample variance
N
X
N
i
i∑=
−
= 1
2
2
)( µ
σ
1
)(
1
2
2
−
−
=
∑=
n
xx
s
n
i
i

Session 1.45
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
)( µ
σ
1
)(
1
2
−
−
=
∑=
n
xx
s
n
i
i

Session 1.46
(Sample) Data: 10 12 14 15 17 18 18
24
n = 8 Mean =16
309.4
7
2)1624(2)1618(2)1617(2)1615(2)1614(2)1612(2)1610(
=
−+−+−+−+−+−+−
=s
Computation of Standard Deviation

Session 1.47
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.

Session 1.48
Mean = 15.5
s = 3.33811 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)

Session 1.49
Example: Team A - Heights of five marathon players in inches
5”
65 “ 65 “ 65 “ 65 “ 65 “
Mean = 65
S = 0
Comparing Standard Deviations

Session 1.50
Example: Team B - Heights of five marathon players in inches
62 “ 67 “ 66 “ 70 “ 60 “
Mean = 65”
s = 4.0”
Comparing Standard Deviation

Session 1.51
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).

Session 1.52
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.

Session 1.53
For any data set with mean (µ) 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

Session 1.54
Illustration
At least 75%
At least 75% of the observations
are within 2SD of its mean.

Session 1.55
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.

Session 1.56
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

Session 1.57
Comparing CVs
 Stock A: Average Price = P50
SD = P5
CV = 10%
 Stock B: Average Price = P100
SD = P5
CV = 5%

Session 1.58
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
MedianMean
K
−
=
3
S

Session 1.59
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.

Session 1.60
SK > 0
positively
skewed
Measure of Skewness
SK < 0
negatively skewed

Session 1.61
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
µ
σ
=
−
= −
∑

Session 1.62
K = 0
mesokurtic
K > 0
leptokurtic
K < 0
platykurtic
Measure of Kurtosis

Session 1.63
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.

Session 1.64
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

Session 1.65
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 Q3Md
75 78 85

Session 1.66
Step 2: Place marks at distances 1.5 IQR from
either end of the box. (1.5 IQR =15)
100
Q1 Q3Md
75 78 8560
1.5 IQR 1.5 IQR
Steps to Construct a Box-and-Whiskers plot

Session 1.67
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

Session 1.68
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 8560 100
1.5 IQR 1.5 IQR
9855
.
.
Steps to Construct a Box-and-Whiskers
plot

Introduction to Statistics and Statistical Inference

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