Statistics

 Form Latin word ‘Statis’ means ‘Political
State’.
 Science of Uncertainty
 It Deals with what could be, what might be or
what probably is.
The basis to verify theories and laws in every
discipline.
Overall, it is a method which deals with
numerical facts and figures.

 The Indian army is going to grow by 9-10%
per annum in coming 5 yrs.
 The male female ratio in India is 972 as per
2001 census.
 Indian population is growing by 2% every
year.
 Attendance of a student should be 75% for
appearing in exams.
And many more……

 What is Science?
 Originated form Latin word “Scientia” meaning knowledge.
 Knowledge attained through study or
practice.
 Knowledge covering general truths of the
operation of generals laws (esp obtained and tested
through scientific method) and concerned with physical
world.

 Statistics is not a body of substantive
knowledge, but a body of methods for
obtaining knowledge.
 It can be accepted as scientific method than
a complete science.
Scientific Methods:-Classifies facts, sees their mutual
relation through experimentation, observation, logical
arguments from accepted postulates

 Population. Universe. The entire category under consideration. This
is the data which we have not completely examined but to which our
conclusions refer. The population size is usually indicated by a
capital N.
◦ Examples: every user of twitter; all female user of facebook.
 Sample. That portion of the population that is available, or to be
made available, for analysis. A good sample is representative of the
population. We will learn about probability samples and how they
provide assurance that a sample is indeed representative. The
sample size is shown as lower case n.
◦ If your company manufactures one million laptops, they might take a
sample of say, 500, of them to test quality. The population size is N =
1,000,000 and the sample size is n= 500.
Introduction 7

Parameter. A characteristic of a population. The
population mean, µ and the population standard
deviation, σ, are two examples of population
parameters. If you want to determine the
population parameters, you have to take a
census of the entire population. Taking a census
is very costly.
Statistic. A statistic is a measure that is derived
from the sample data. For example, the sample
mean, 𝑋, and the sample standard deviation, s,
are statistics. They are used to estimate the
population parameters.
Introduction 8

Example of statistical inference from quality control:
 GE manufactures LED bulbs and wants to know how
many are defective. Suppose one million bulbs a
year are produced in its new plant in Staten Island.
The company might sample, say, 500 bulbs to
estimate the proportion of defectives.
◦ N = 1,000,000 and n = 500
◦ If 5 out of 500 bulbs tested are defective, the sample
proportion of defectives will be 1% (5/500). This statistic
may be used to estimate the true proportion of defective
bulbs (the population proportion).
◦ In this case, the sample proportion is used to make
inferences about the population proportion.
Introduction 10

 Descriptive Statistics. Those statistics that summarize a
sample of numerical data in terms of averages and other
measures for the purpose of description, such as the mean
and standard deviation.
◦ Descriptive statistics, as opposed to inferential statistics, are not
concerned with the theory and methodology for drawing inferences
that extend beyond the particular set of data examined, in other
words from the sample to the entire population. All that we care about
are the summary measurements such as the average (mean).
◦ Thus, a teacher who gives a class, of say, 35 students, an exam is
interested in the descriptive statistics to assess the performance of
the class. What was the class average, the median grade, the standard
deviation, etc.? The teacher is not interested in making any inferences
to some larger population.
◦ This includes the presentation of data in the form of graphs, charts,
and tables.
Introduction 11

 Primary data. This is data that has been compiled
by the researcher using such techniques as
surveys, experiments, depth interviews,
observation, focus groups.
 Types of surveys. A lot of data is obtained using
surveys. Each survey type has advantages and
disadvantages.
◦ Mail: lowest rate of response; usually the lowest cost
◦ Personally administered: can “probe”; most costly;
interviewer effects (the interviewer might influence the
response)
◦ Telephone: fastest
◦ Web: fast and inexpensive
Introduction 12

 Secondary data. This is data that has been
compiled or published elsewhere, e.g., census
data.
◦ The trick is to find data that is useful. The data was
probably collected for some purpose other than
helping to solve the researcher’s problem at hand.
◦ Advantages: It can be gathered quickly and
inexpensively. It enables researchers to build on past
research.
◦ Problems: Data may be outdated. Variation in
definition of terms. Different units of measurement.
May not be accurate (e.g., census undercount).
Introduction 13

 Nonprobability Samples – based on convenience or
judgment
◦ Convenience (or chunk) sample - students in a class, mall
intercept
◦ Judgment sample - based on the researcher’s judgment as to
what constitutes “representativeness” e.g., he/she might say these
20 stores are representative of the whole chain.
◦ Quota sample - interviewers are given quotas based on
demographics for instance, they may each be told to interview
100 subjects – 50 males and 50 females. Of the 50, say, 10
nonwhite and 40 white.
 The problem with a nonprobability sample is that we do not
know how representative our sample is of the population.
Introduction 14

 Probability Sample. A sample collected in such
a way that every element in the population has
a known chance of being selected.
 One type of probability sample is a Simple
Random Sample. This is a sample collected in
such a way that every element in the
population has an equal chance of being
selected.
 How do we collect a simple random sample?
◦ Use a table of random numbers or a random number
generator.
Introduction 15

 Other kinds of probability samples (beyond the
scope of this course).
◦ systematic random sample.
 Choose the first element randomly, then every kth
observation, where k = N/n
◦ stratified random sample.
 The population is sub-divided based on a characteristic
and a simple random sample is conducted within each
stratum
◦ cluster sample
 First take a random sample of clusters from the
population of cluster. Then, a simple random sample
within each cluster. Example, election district, orchard.
Introduction 16

◦ Measures of Location
 Measures of central tendency: Mean; Median; Mode
 Measures of noncentral tendency - Quantiles
 Quartiles; Quintiles; Percentiles
◦ Measures of Dispersion
 Range
 Interquartile range
 Variance
 Standard Deviation
 Coefficient of Variation
◦ Measures of Shape
◦ Skewness
Descriptive Statistics I 17

 Measures of location place the data set on the scale
of real numbers.
 Measures of central tendency (i.e., central location)
help find the approximate center of the dataset.
 These include the mean, the median, and the
mode.

 The sample mean is the sum of all the observations
(∑Xi) divided by the number of observations (n):
𝑋 = 𝑖=1
𝑛
𝑋𝑖
𝑛
where ΣXi = X1 + X2 + X3 + X4 + … + Xn
 Example. 1, 2, 2, 4, 5, 10. Calculate the mean.
Note: n = 6 (six observations)
∑Xi = 1 + 2+ 2+ 4 + 5 + 10 = 24
𝑋= 24 / 6 = 4.0

 The median is the middle value of the ordered data
 To get the median, we must first rearrange the
data into an ordered array (in ascending or
descending order). Generally, we order the data
from the lowest value to the highest value.
 Therefore, the median is the data value such that
half of the observations are larger and half are
smaller. It is also the 50th percentile (we will be
learning about percentiles in a bit).
 If n is odd, the median is the middle observation of
the ordered array. If n is even, it is midway between
the two central observations.

 The mode is the value of the data that
occurs with the greatest frequency.
Example. 1, 1, 1, 2, 3, 4, 5
Answer. The mode is 1 since it occurs three times. The
other values each appear only once in the data set.
Example. 5, 5, 5, 6, 8, 10, 10, 10.
Answer. The mode is: 5, 10.
There are two modes. This is a bi-modal dataset.

 Quartiles split a set of ordered data into four parts.
◦ Imagine cutting a chocolate bar into four equal pieces… How
many cuts would you make? (yes, 3!)
 Q1 is the First Quartile
◦ 25% of the observations are smaller than Q1 and 75% of the
observations are larger
 Q2 is the Second Quartile
◦ 50% of the observations are smaller than Q2 and 50% of the
observations are larger. Same as the Median. It is also the 50th
percentile.
 Q3 is the Third Quartile
◦ 75% of the observations are smaller than Q3and 25% of the
observations are larger

 Dispersion is the amount of spread, or
variability, in a set of data.
 Why do we need to look at measures of
dispersion?
 Consider this example:
A company is about to buy computer chips that must
have an average life of 10 years. The company has a
choice of two suppliers. Whose chips should they buy?
They take a sample of 10 chips from each of the
suppliers and test them. See the data on the next slide.

We see that supplier B’s chips have a longer average life.
However, what if the company offers
a 3-year warranty?
Then, computers manufactured
using the chips from supplier A
will have no returns
while using supplier B will result in
4/10 or 40% returns.
Supplier A chips
(life in years)
Supplier B chips
(life in years)
11 170
11 1
10 1
10 160
11 2
11 150
11 150
11 170
10 2
12 140
𝑋A = 10.8 years 𝑋 𝐵 = 94.6 years
MedianA = 11 years MedianB = 145 years
sA = 0.63 years sB = 80.6 years
RangeA = 2 years RangeB = 169 years

 We will study these five measures of
dispersion
◦ Range
◦ Interquartile Range
◦ Standard Deviation
◦ Variance
◦ Coefficient of Variation

 Range = Largest Value – Smallest Value
Example: 1, 2, 3, 4, 5, 8, 9, 21, 25, 30
Answer: Range = 30 – 1 = 29.
 The range is simple to use and to explain to
others.
 One problem with the range is that it is
influenced by extreme values at either end.

 IQR = Q3 – Q1
 Example (n = 15):
0, 0, 2, 3, 4, 7, 9, 12, 17, 18, 20, 22, 45, 56, 98
Q1 = 3, Q3 = 22
IQR = 22 – 3 = 19 (Range = 98)
 This is basically the range of the central 50% of
the observations in the distribution.
 Problem: The interquartile range does not take
into account the variability of the total data (only
the central 50%). We are “throwing out” half of
the data.

 The standard deviation, s, measures a kind of
“average” deviation about the mean. It is not really
the “average” deviation, even though we may think
of it that way.
 Why can’t we simply compute the average deviation
about the mean, if that’s what we want?
𝑖=1
𝑛
(𝑋𝑖 − 𝑋)
𝑛
 If you take a simple mean, and then add up the
deviations about the mean, as above, this sum will
be equal to 0. Therefore, a measure of “average
deviation” will not work.

 Instead, we use:
𝑠 = 𝑖=1
𝑛
(𝑋𝑖− 𝑋)2
𝑛−1
 This is the “definitional formula” for standard deviation.
 The standard deviation has lots of nice properties,
including:
◦ By squaring the deviation, we eliminate the problem of the
deviations summing to zero.
◦ In addition, this sum is a minimum. No other value subtracted
from X and squared will result in a smaller sum of the deviation
squared. This is called the “least squares property.”
 Note we divide by (n-1), not n. This will be referred to
as a loss of one degree of freedom.

Example. Two data sets, X and Y. Which of
the two data sets has greater variability?
Calculate the standard deviation for each.
We note that both sets of data have the
same mean:
𝑋 = 3
𝑌 = 3
(continued…)
Xi Yi
1 0
2 0
3 0
4 5
5 10

SX = 10
4
= 1.58
SY = 80
4
= = 4.47
[Check these results with your calculator.]
X 𝑋 (X-𝑋) (X-𝑋)2
1 3 -2 4
2 3 -1 1
3 3 0 0
4 3 1 1
5 3 2 4
∑=0 10
Y 𝑌 (Y-𝑌) (Y- 𝑌)2
0 3 -3 9
0 3 -3 9
0 3 -3 9
5 3 2 4
10 3 7 49
∑=0 80

The variance, s2, is the standard deviation (s)
squared. Conversely, 𝑠 = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒.
Definitional formula: 𝑠2 = 𝑖=1
𝑛
𝑋 𝑖−𝑋
2
𝑛−1
Computational formula: 𝑠2
=
𝑖=1
𝑛
𝑋 𝑖
2− 𝑖=1
𝑛 𝑋𝑖
2
𝑛
𝑛−1
This is what computer software
(e.g., MS Excel or your calculator key) uses.

We see that supplier B’s chips have a longer average life.
However, what if the company offers
a 3-year warranty?
Then, computers manufactured
using the chips from supplier A
will have no returns
while using supplier B will result in
4/10 or 40% returns.
Supplier A chips
(life in years)
Supplier B chips
(life in years)
11 170
11 1
10 1
10 160
11 2
11 150
11 150
11 170
10 2
12 140
𝑋A = 10.8 years 𝑋 𝐵 = 94.6 years
MedianA = 11 years MedianB = 145 years
sA = 0.63 years sB = 80.6 years
RangeA = 2 years RangeB = 169 years

 A sample space is the set of all possible
outcomes of an experiment.
 A random variable is a rule for associating a
number with each element in a sample space.
 Suppose there are 8 balls in a bag. The
random variable X is the weight, in kg, of a
ball selected at random. Balls 1, 2 and 3
weigh 0.1kg, balls 4 and 5 weigh 0.15kg and
balls 6, 7 and 8 weigh 0.2kg
34

 There are two types of random variables:
◦ A Discrete random variable can take on only
specified, distinct values.
◦ A Continuous random variable can take on any
value within an interval.
 Thus, there are also two types of probability
distributions:
◦ Discrete probability distributions
◦ Continuous probability distributions
Probability Distributions 35

 Called a Probability density function. The
probability is interpreted as "area under the
curve."
 1) The random variable takes on an infinite # of
values within a given interval
 2) the probability that X = any particular value is
0. Consequently, we talk about intervals. The
probability is = to the area under the curve.
 3) The area under the whole curve = 1.
Normal Distribution 36

 Probabilities are obtained by getting the area
under the curve inside of a particular interval.
The area under the curve = the proportion of
times under identical (repeated) conditions that a
particular range of values will occur.
 3 Characteristics of the Normal distribution:
◦ It is symmetric about the mean μ.
◦ Mean = median = mode. [“bell-shaped” curve]
◦ f(X) decreases as X gets farther and farther away from
the mean. It approaches horizontal axis asymptotically:
- ∞ < X < + ∞. This means that there is always some
probability (area) for extreme values.

 The probability density function for the normal
distribution:
𝑓 𝑋 =
1
σ 2π
𝑒
−
1
2
𝑋−μ
σ
2
X
f(X) the height of the curve, represents the relative
frequency at which the corresponding values
occur.

Note that the normal distribution is defined by
two parameters, μ and σ . You can draw a
normal distribution for any μ and σ
combination. There is one normal distribution,
Z, that is special. It has a μ = 0 and a σ = 1.
This is the Z distribution, also called the
standard normal distribution. It is one of
trillions of normal distributions we could have
selected.

 Any normal distribution can be converted into a standard normal
distribution by transforming the normal random variable into the
standard normal random variable:
𝑍 =
𝑋 − μ
σ
 This is called standardizing the data. It will result in (transformed) data
with μ = 0 and σ = 1.
 The areas under the curve for the Standard Normal Distribution (Z) has
been computed and tabled. See, for example
http://www.statsoft.com/textbook/distribution-tables/#z
 Please note that you may find different tables for the Z-distribution. The
table we use here gives you the area under the curve from 0 to z. Some
books provide a slightly different table, one that gives you the area in
the tail. If you check the diagram that is usually shown above the table,
you can determine which table you have. In the table on the next slide,
the area from 0 to z is shaded so you know that you are getting the area
from 0 to z. Also, note that table value can never be more than .5000.
The area from 0 to infinity is .5000.

 Estimation
 Hypothesis Testing
Both activities use sample statistics (for
example, X̅) to make inferences about a
population parameter (μ).
Estimation 41

 Why don’t we just use a single number (a point
estimate) like, say, X̅ to estimate a population
parameter, μ?
 The problem with using a single point (or value) is
that it will very probably be wrong. In fact, with a
continuous random variable, the probability that the
variable is equal to a particular value is zero. So,
P(X̅=μ) = 0.
 This is why we use an interval estimator.
 We can examine the probability that the interval
includes the population parameter.
Estimation 42

 How wide should the interval be? That depends upon how
much confidence you want in the estimate.
 For instance, say you wanted a confidence interval
estimator for the mean income of a college graduate:
 The wider the interval, the greater the confidence you will
have in it as containing the true population parameter μ.
Estimation 43
You might have That the mean income is between
100% confidence $0 and $∞
95% confidence $35,000 and $41,000
90% confidence $36,000 and $40,000
80% confidence $37,500 and $38,500
… …
0% confidence $38,000 (a point estimate)

 To construct a confidence interval estimator
of μ, we use:
X̅ ± Zα σ /√n (1-α) confidence
where we get Zα from the Z table.
 When n≥30, we use s as an estimator of σ.
Estimation 44

 To be more precise, the α is split in half since
we are constructing a two-sided confidence
interval. However, for the sake of simplicity,
we call the z-value Zα rather than Za/2 .
Estimation 45
-Z/2 Z/2
/2 /2

 You work for a company that makes smart TVs,
and your boss asks you to determine with
certainty the exact life of a smart TV. She tells
you to take a random sample of 100 TVs.
 What is the exact life of a smart TV made by this
company?
Sample Evidence:
n = 100
X̅ = 11.50 years
s = 2.50 years
Estimation 46

 Since your boss has asked for 100% confidence, the
only answer you can accurately provide is: -∞ to + ∞
years.
 After you are fired, perhaps you can get your job
back by explaining to your boss that statisticians
cannot work with 100% confidence if they are working
with data from a sample. If you want 100%
confidence, you must take a census. With a sample,
you can never be absolutely certain as to the value of
the population parameter.
 This is exactly what statistical inference is: Using
sample statistics to draw conclusions (e.g., estimates)
about population parameters.
Estimation 47

n = 100
X̅ = 11.50 years
S = 2.50 years
at 95% confidence:
11.50 ± 1.96*(2.50/√100)
11.50 ± 1.96*(.25)
11.50 ± .49
The 95% CIE is: 11.01 years ---- 11.99 years
Estimation 48

 We are 95% certain that the interval from 11.01
years to 11.99 years contains the true population
parameter, μ.
 Another way to put this is, in 95 out of 100
samples, the population mean would lie in
intervals constructed by the same procedure
(same n and same α).
 Remember – the population parameter (μ ) is
fixed, it is not a random variable. Thus, it is
incorrect to say that there is a 95% chance that
the population mean will “fall” in this interval.
Estimation 49

The sample:
n = 100
X̅ = 18 years
s = 4 years
 Construct a confidence interval estimator
(CIE) of the true population mean life (µ), at
each of the following levels of confidence:
◦ (a)100% (b) 99% (c) 95% (d) 90% (e) 68%
Estimation 50

 In this problem we use s as an unbiased estimator
of σ: E(s) = σ
 σ = s =
 95% Confidence Interval Estimator:
Estimation 51

(a) 100% Confidence
[α = 0, Zα = ∞]
100% CIE: −∞ years ↔ +∞ years
(b) 99% Confidence
α = .01, Zα = 2.575 (from Z table)
18 ± 2.575 (4/√100)
18 ± 1.03
99% CIE: 16.97 years ↔ 19.03 years
(c) 95% Confidence
α = .05, Zα = 1.96 (from Z table)
18 ± 1.96 (4/√100)
18 ± 0.78
95% CIE: 17.22 years ↔ 18.78 years
Estimation 52

(d) 90% Confidence
α = .10, Zα = 1.645 (from Z table)
18 ± 1.645 (4/√100)
18 ± 0.66
90% CIE: 17.34 years ↔ 18.66 years
(e) 68% Confidence
α = .32, Zα =1.0 (from Z table)
18 ± 1.0 (4/√100)
18 ± 0.4
68% CIE: 17.60 years ↔ 18.40 years
Estimation 53

 How can we keep the same level of confidence and
still construct a narrower CIE?
 Let’s look at the formula one more time: X̅ ± Zασ/√n
 The sample mean is in the center. The more
confidence you want, the higher the value of Z, the
larger the half-width of the interval.
 The larger the sample size, the smaller the half-
width, since we divide by √n.
 So, what can we do? If you want a narrower interval,
take a larger sample.
 What about a smaller standard deviation? Of course, this
depends on the variability of the population. However, a more
efficient sampling procedure (e.g., stratification) may help. That
topic is for a more advanced statistics course.
Estimation 54

 Once you are working with a sample, not the
entire population, you cannot be 100% certain of
population parameters. If you need to know the
value of a parameter certainty, take a census.
 The more confidence you want to have in the
estimator, the larger the interval is going to be.
 Traditionally, statisticians work with 95%
confidence. However, you should be able to use
the Z-table to construct a CIE at any level of
confidence.
Estimation 55

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