What is statistics

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What is statistics

• Statistical training is necessary and important
  for many reasons. In almost any area of work, you
  must be able to read, interpret, and apply the
  results of a statistical analysis of research
  data.
•  
• What is Statistics all about? Statistics
  involves the collection, organization,
  interpretation and presentation of numerical
  information.
• Studying statistics will help you to understand
  the information and to reach correct conclusion.

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Statistics statements

• Statistics proves that cigarette smoking will
  cause lung cancer.
• According to statistics, females live longer than
  males.
• Statistically speaking, tall parents have tall
  children.

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Chapter 1 Population and Sample

• An experiment unit (or subject) is the smallest
  entity that is of interest in a statistical
  study.
• A variable is any characteristics that can be
  measured on each experiment unit in a statistical
  study.
• An observation is a value that the variable
  assumes for a single unit.
• The collection of observations assumed by the
  variables in the study is called a data set.

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Population Versus Sample

• Population the whole group
• a collection of persons, objects, or items under
  study
• Sample a portion of the whole group
• a subset of the population

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• Example1.2 Now we are interested in the heights
  of MSU students. We measured the heights (and/or
  weight, gender) of all the students and recorded
  as follows X1, X2,. Randomly select 50
  students to measure their heights. Explain the
  concepts above.
• Population
• All MSU students
• Experiment unit
• A MSU student
• variables
• height,( and/or weight, gender)
• Observations
• any one of X1,X2,.
• Data set
• X1,X2,.
• Sample
• the 50 selected students

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• Bias is a systematic tendency of the sample to
  misrepresent the population.
• A simple random sample of size n consists of n
  elements chosen from the population in such a way
  that all samples of that size have the same
  chance of being selected.

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• A Census is a sample consisting of the entire
  population.  
• Why dont we always do a census?
• Time time consuming,
• Cost cost more,
• Inaccessible population units study all the
  sunfish in Lake Michigan, say.
• Destructive testing destroy the unit

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For example, (the lottery sampling)

• 100 balls are mixed thoroughly in a bag. draw 10
  balls randomly from the bag. Try twice
  with/without replacement.

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Exercise 1.1

• A stock market investor is interested in oil
  stocks.She collects last years price/earnings
  ratios on ten randomly selected oil stocks. The
  data of ratios are X1,X2,, X10.
• What is our population?
• What is the variable? Give one observation.
• What is our sample and sample size?
• What is our data set?

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Exercise1.2

• Want to know the average height of 2nd grade
  students in East Lansing Public schools. Instead
  of measure all 2nd grade students, we sample the
  60 2nd grade students randomly, The data of
  heights are X1,X2,, X60.
• What is our population?
• What is the variable? Give one observation.
• What is our sample and sample size?
• What is our data set?

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Statistics terms

• experimental unit, subject
• variable
• population
• sample
• census
• s r s simple random sample

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Chapter 2 Univariate Data

• A univariate data set is a data set in which one
  measurement (variable) has been made on each
  experiment unit.
• A bivariate data set is a data set in which two
  measurements (variables) have been made on each
  experiment unit.
• A multivariate data set is a data set in which
  several measurements (variables) have been made
  on each experiment unit.

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• A Numerical variable (also called a quantitative
  or measure variable) is a variable whose values
  are numbers obtained by a count or measurement.
  For example weight, height.
• A discrete variable is a numerical variable that
  can assume a finite number or at most a countable
  infinite number of values (Countable means you
  can associate the values with the counting number
  1,2,3,that is, the values can be counted).
• A continuous variable is a numerical variable
  that can take any number on an interval of the
  real number line. For example, height, weight.

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Types of Variables

• A categorical variable (also called a qualitative
  variable) is a variable whose values are
  classifications or categories.
• For example, Gender Male, Female
• Occupation Student, doctor, teacher,

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Types of Data
Discrete Can only take on certain values in an
interval.
Numerical (Quantitative)
Continuous Can take on infinitely many values
in an interval.
Ordinal There is a sense of ordering.
Categorical (Qualitative)
Nominal There is no sense of ordering.
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Discrete Example

• number of people in a room This can only be 0,
  1, 2, 3, etc. It cant be 0.2, 2.2333,4.5511,
  etc.
• Usually discrete variables are those where you
  are counting.

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Continuous Examples

• age of a car Usually you will say your car is 1
  year old, 2 years old, etc. But the age of a car
  isnt just 1,2,3, etc. It would be more exact to
  say its 1.32433535 years old.
• height of an object We tend to measure in
  inches 52, 43, etc. But an object can take
  on a much more precise measurement than this.
• Other examples weights, volume, etc.

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Ordinal Examples

• military private, sergeant, lieutenant,
  general, etc. There is a sense of ordering.
• degree high school, B.S., M.S., Ph.D., etc.
• classification freshman, sophomore, junior,
  senior

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Nominal

• color of an object green, red, white
• maker of a vehicle chevy, ford, dodge

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Be cautious!

• Coding of categorical variable does not make it
  numerical.
• For example
• Gender Male -- 0 Female -- 1
• zip code, area code, telephone number, social
  security number, etc. These are all nominal.

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Exercise 2.1 Classify the following as
categorical or numerical (discrete or
continuous).

• a. Age of freshmen in MSU
• b. Faculty rank
• c. Weight of newborn babies
• d. Murder rate in a major city
• e. Number of children in a family
• f. Brand of television set

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display categorical variable

• frequency table
• pie chart
• bar chart bar graph

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Frequency table
Example2 Time/CNN telephone poll of 500 adult
Americans Has the amount of crime in your
community increased in the past 5 years?

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Pie Chart

• A circle or pie is divided into pieces
  corresponding to the categories of the variable
  so that the size of the slice is proportional to
  the relative frequency of the category.

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Pie Chart
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Bar Chart

• is a picture consisting of horizontal and
  vertical axis with rectangles that represent the
  frequency (relative frequency) of the categories
  of a variable.

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Bar chart of CNN telephone poll
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Example3 Sampling 100 students from MSU students
to get their level information, here are the
data Fr 12, So 24, Jr 32, Sr 24, Gr 8.
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display numerical data

• stem leaf
• dot plot ok for small data set
• histogram

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Example 1

• A psychologist wishes to test a new method to
  improve rote memorization by college students. A
  sample of 20 college students were taught by this
  method and then asked to memorize a list of 100
  word phrases. The following numbers of correct
  word phrases were recorded for the 20 students.
• 84 59 82 78 74 96 44 76 85 66
• 77 91 62 54 72 65 84 38 76 70

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Dotplot
84 59 82 78 74 96 44 76 85 66 77 91
62 54 72 65 84 38 76 70

• Distribution of a variable specifies the
  distinct values that the variable assumes and
  how often these values occur. The distribution
  illustrates the pattern of the variation in the
  data.

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Stem and leaf Plot.

• 84 59 82 78 74 96 44 76 85 66
  
• 77 91 62 54 72 65 84 38 76 70
• Step1. One observation plotted
• 3
• 4
• 5
• 6
• 7
• 8 4
• 9

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• 84 59 82 78 74 96 44 76 85 66
• 77 91 62 54 72 65 84 38 76 70
• Step2. Fill stem and leaf plot
• 3 8
• 4 4
• 5 9 4
• 6 6 2 5
• 7 8 4 6 7 2 6 0
• 8 4 2 5 4
• 9 6 1

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• 84 59 82 78 74 96 44 76 85 66
  
• 77 91 62 54 72 65 84 38 76 70
• Step3. Ordered stem and leaf plot
• 3 8
• 4 4
• 5 4 9
• 6 2 5 6
• 7 0 2 4 6 6 7 8
• 8 2 4 4 5
• 9 1 6

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Example 2

• The following is the concentration of mercury in
  25 lake trout caught in a major lake
• 2.2 3.4 3.0 2.6 3.8 1.8 2.8 3.2
  3.7
• 1.4 2.7 3.6 1.9 2.2 3.0 3.3 2.3
• 1.7 2.6 3.5 3.0 2.9 3.4 3.1 2.4
• Exercise create a stem leaf plot for this data.

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Stem and leaf plot

• Leaf unit 0.1
• 1 4 8 9 7
• 2 2 2 3 4 6 6 7 8 9
• 3 0 0 0 1 2 3 4 4 5 6 7 8

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double-stem stem and leaf plot

• Leaf unit 0.1
• 1 4
• 1 8 9 7
• 2 2 3 4
• 2 6 8 7 6 9
• 3 4 0 2 0 3 0 4 1
• 3 8 7 6 5

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Ordered double-stem stem and leaf plot

• Leaf unit 0.1
• 1 4
• 1 7 8 9
• 2 2 3 4
• 2 6 6 7 8 9
• 3 0 0 0 1 2 3 4 4
• 3 5 6 7 8

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Histogram

• The histogram is a graphical means of
  displaying the numerical data. If we slice up the
  entire span of values covered by the quantitative
  variable into equal-width piles called bins
  (classes), a histogram plots the bin counts
  (class counts) as the heights of bars
• It can be constructed from the stem and leaf
  plot each stem defines an interval of values as
  a class. The class limits are the smallest and
  largest possible values for the interval. Now go
  back to Example 1.

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Steps of construction

• find class limits and class boundaries
• find class frequency and construct frequency
  table
• label horizontal axis using continuous scale
• label vertical axis for (relative) frequency
• draw bars using class boundaries and (relative)
  frequency

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Histogram

• Used to represent continuous grouped data
• Does not have any gaps between bars

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Grouped frequency table
of correct word phrases in a rote memorization
study
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Constructed Histogram
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Relative frequency histogram
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Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

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Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Eg boundary between 154 and 155 is 154.5
Answer (a) find class limits and class boundaries
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Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class boundaries
Eg boundary between 154 and 155 is 154.5
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Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class boundaries
heights of 325 students
Height cm
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Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) label horizontal axis using
continuous scale
heights of 325 students
140
150
160
170
180
190
200
Height cm
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Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class widths
heights of 325 students
120
10
51
Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class widths
heights of 325 students
52
Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class widths
heights of 325 students
120
10
53
Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class widths
heights of 325 students
120
50
10
54
Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class widths
heights of 325 students
120
50
10
55
Histogram - example
Question The heights of 325 students were
measured to the nearest cm

• Draw a histogram to illustrate the above data.

Answer (a) find class widths
heights of 325 students
120
50
10
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Exercise create a histogram for this data set.

• The following is the concentration of mercury in
  30 lake trout caught in a major lake
• 2.2 3.4 3.0 2.6 3.8 1.8 2.8 3.2
  3.7 3.5
• 1.4 2.7 3.6 1.9 2.2 3.0 3.3 2.3
  3.3 3.6
• 1.7 2.6 3.5 3.0 2.9 3.4 3.1 2.4
  3.4 3.8
• Use limits 1-1.4, 1. 5-1.9, 2-2.4, 2. 5-2.9,
  3-3.4, 3.5-4.

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Frequency Polygons
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Frequency Polygons
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Population Frequency Curve

• When the number of observation is very large
  and the class limits are reduced in size, the
  distribution takes on the appearance of a
  continuous curve similar to what might be
  expected if the entire population of values are
  graphed. Usually the entire population is not
  available. However, A stem and leaf plot,
  histogram, or frequency polygon obtained from the
  representative sample should closely approximate
  the shape of population frequency curve.

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Population Frequency Curve