Transcript Statistic

GRADING POLICY
 Quizees
 Mid-term exam
 Final exam
Grade
A
B+
B
C+
C
D+
D
E
: 25%
: 30%
: 45%
Points
> 80
75 – 79
70 - 74
60 - 69
55 - 59
50 – 54
45 - 49
< 45
=
=
=
25 point
30 points
45 points
100 points
What is statistics
The mathematics of the collection, organization, and interpretation
of numerical data, especially the analysis of population
characteristics by inference from sampling
Why statistics
• Need to make quantified statements about a phenomenon we are
interested in
• …Therefore we collect samples as proxies of the greater population of
individuals or items that make up the phenomenon we are interested in
• Anything can be expressed in statistics
Aims of the course
• Introduction to basic statistics
• Learn to use analysis tools in EXCEL
• Make you an intelligent user of data and statistics
We will bypass much of the mathematics, instead emphasizing the
understanding of underlying principles
Types of statistics
1. Descriptive statistics
Quantitative methods of organizing, summarizing, and
presenting data in an informative way (numerically, graphically)
Describe the overall characteristics of a sample
Transform raw data into more easily understood forms
2. Inferential statistics
The branch of statistics used to make inferences about a larger
population based on the data collected from a sample
Make prediction
• Parametric statistics
• Non parametric statistics
• Primary data
• Secondary data
• Quantitative data
• Qualitative data
• Discrete data
• Continuous data
Definitions
•
•
Population : all entire set of observations which we are concerned  N
Sample
: a smaller subset of obs. taken from population, should be drawn randomly
 n
sampling
population
Parameter
µ
σ2, S2
•
•
Variable
Data
sample
inference
mean
variance
statistic
x
s2
: a variety of characteristic that observed
: all the observation ,either by counting or by measuring
• Parameters : summary measure that is computed to describe a
characteristic of an population such as a mean or variance,
represented by Greek letters.
 Greek letters μ , σ , σ2
• Statistic
: is a summary measure that is computed to describe a
characteristic from a subset
 English letters
Data collection
Types of data /scale of measurement :
• Categorical /Nominal  label, identify different categories, no concept of
more or less
e.g. gender : male/female or moslem/hindu/other or fruit
• Ordinal  a set of observation ordered according to some criterion
e.g ranking, test result
• Interval  different categories, logical order, distance between category is
constant
e.g. temperature (interval data can be converted into ordinal form)
• Ratio  interval plus meaningful zero, allows ratio comparison
e.g. weight, height, etc
What do we want to know about a set of data
 DESCRIPTIVE STATISTICS
•
Shape  right/left-skewed, bell-shaped
bar graph (nominal/ordinal data)
histogram (interval/ratio data)
frequency polygon
pie-chart , pictograph
stem & leaf diagram
box & whisker plot
•
Typical value  measure of central tendency (x , μ)
other measure of location : median, modus, quartile , decile
five-number summary
•
Spread of scores  measure of variability
range
the average squared distance of each score from the
mean (s2)
standard deviation
coefficient of variation
SHAPE
bar graph
histogram
pictograph
frequency polygon
pie-chart
Histogram
•
•
Below is a grouped frequency table. It is shown (on the left) which masses went
into the count for each class. We also indicated the upper bound of each class in
red, to remind you that this value isn't counted in that class.
There is no space in between the bars
Frequency poligon
•
One way to form a frequency polygon is to connect the midpoints at the top of the
bars of a histogram with line segments (or a smooth curve). The midpoints
themselves could easily be plotted without the histogram and be joined by line
segments. Sometimes it is beneficial to show the histogram and frequency polygon
together.
A pie chart (or a circle graph) is a circular chart divided into sectors,
illustrating proportion
statisticians generally regard pie charts as a poor method of displaying
information, and they are uncommon in scientific literature. One reason is
. that it is more difficult for comparisons to be made between the size of items
in a chart when area is used instead of length.
Stem & leaf diagram  stem-plot
- Shows the spreadness of the data whether it is right-skewed, left –
skewed, or symetric (bell-shaped)
- The real data is shown
- The outlier can be seen
- We can have back-to-back stemplot to compare two data set

MEASURE OF CENTRAL TENDENCY
n
x

x


x
1
2
n
1
Mean  X 
 n  xi
n
i1
Median  The central value in an ordered set of data
Raw data
Sorted data
4
1
2
2
5
4
1
5
7
6
10
7
6
10
For an even number of values ................?
 Median
X
N
i
Mode
Frequency
• The most commonly occurring value
• For nominal data, we refer to the modal class
• Not appropriate for ordinal or (usually) interval data
Modal Class
200
180
160
140
120
100
80
60
40
20
0
25
30
35
40
45
50
55
60
Variable X
65
70
75
80
85
Box & whisker diagram/plot  Boxplot
is a convenient way of graphically depicting groups of numerical data through their fivenumber summaries: the smallest observation (sample minimum), lower quartile (Q1),
median (Q2), upper quartile (Q3), and largest observation (sample maximum). A boxplot
may also indicate which observations, if any, might be considered outliers.
Boxplots can be drawn either horizontally or vertically
Other locations
• Quartile 
If we trim away 25% of the data on either side, we are
left with the first and third quartiles
Five-number summary :
Minimum
Lower quartile – Q1
Median – Q2
Upper quartile – Q3
Maximum
Minimum
Maximum
DATA DISTRIBUTION
• Symmetric Distributions
• Mean ≈ Median (approx. equal)
• Skewed to the Left
• Mean < Median
• Mean pulled down by small values
• Skewed to the Right
• Mean > Median
• Mean pulled up by large values
May 28, 2008
Stat 111 - Lecture 3 - Numerical
Summaries
20
SPREAD OF SCORES
measure of variability (Variability refers to how "spread out" a group of
scores is.)
Range = max – min
Variance :
s2 
2
(x

x
)
 i
n 1
Standard deviation : A measure of the dispersion of a set of data
from its mean. The more spread apart the data, the higher the
deviation. Standard deviation is calculated as the square root of
variance.

s=
Coefficient of variation : CV = (std. Dev / mean) *100%
ratio of standard deviation and the mean
PROBABILITY
P(Y) non negative
0 ≤ P(Y) ≤ 1
P(A) + P(not A) = 1
DISCRETE PROB.
BINOMIAL PROBABILITY
P(H=h) =
POISSON PROBABILITY
HYPERGEOMETRIC PROB
P(X=x) =
 k  N  k

 x



 n  x
N 

n 







CONTINUOUS PROBABILITY
• Mean : µ
• Variance : σ
same mean, different std dev
• P(x1 < µ < X2) = P (z1 < Z < z2)
CONFIDENCE INTERVAL ESTIMATION
Population
Random Sample
Mean
X = 50
Mean, , is
unknown
Sample
I am 95% confident
that  is between 40 &
60.
Confidence Intervals (σ Known - this is hardly ever true)
• Assumptions
–
Population Standard Deviation Is Known
–
Population Is Normally Distributed
–
If Not Normal, use large samples
• Confidence Interval Estimate

X  Z / 2 
n
  

X  Z / 2 
n
Shortcoming of Point Estimates
^
p=
x
n , the sample proportion of x successes in a sample of size n,
is the best point estimate of the unknown value of the population proportion p
E.g
^p = 590/1000 = .59, best estimate of population proportion p
BUT
How good is this best estimate?
A confidence interval is a range (or an interval) of values used to estimate the
unknown value of a population parameter .
x
ˆ
Use p  to construct a 95% confidence interval
n
ˆ (1  pˆ )
ˆ (1  pˆ )
p
p
for p :
( pˆ  1.96
, pˆ  1.96
)
n
n
ˆ (1  pˆ )
p
written as : pˆ  1.96
n
p  Z / 2 *

n
 P  p  Z / 2 *

n
Tool for Constructing Confidence Intervals:
The Central Limit Theorem
• If a random sample of n observations is selected from a
population (any population), and x “successes” are observed,
then when n is sufficiently large, the sampling distribution of
the sample proportion p will be approximately a normal
distribution.
• n is large when np ≥ 15 and nq ≥ 15.
HYPOTHESIS TESTING
OF THE POPULATION MEAN
FUNDAMENTAL
We use samples to learn about populations
We seldom observe the populations we want to know about
Because we have to use samples, we engage in inference from
samples to populations
However, because of sampling variability, samples are not little
mirror images of the population of interest.
Given that samples are imperfect replications of populations, we
have to use techniques such as HYPOTHESIS TESTING to
determine if statements about populations are reasonable
given our observed population
INTRODUCTION
Objective : to determine whether the parameter is
significantly different with statistic
Population mean =
sample mean ?
DEFINITION
Hypothesis 
H0 : “no change” situation (hope to be disproved)
H1 : statement hoped to establish
Statistical test 
procedure in making decision : accept H0 or reject it
(use for defining the hypothesis region)
Types of error  significance level
α : 5% , 1%
Direction of research hypothesis
one-tailed test
two-tailed test
THE STEPS IN PROBLEM SOLVING
Define H0 , H1
Choose Significance level (α)
Test statistic =
samplemean  populationmean
s tan darderror
Critical point (look at the tabel)
Conclusion
Interpretation based on the conclusion
EXAMPLE : OBESITY
EXAMPLE
Main problem :
A certain type of diet for obese patients is successful if after two
months, on average, patients will lose more than 5 kg. At significant level
0f 5%, what is your conclusion if a sample of 50 patients shows an
average of weight loss of 5.5 kg with variation of 1 kg
H0 : average of weight loss = 5
H1 : average of weight loss > 5
α = 5%
Z_calc = 2.357
Critical point : 1.645
Conclusion : Z calc > 1.645  H0 is rejected
Interpretation : it is approved that ..........