Chapter 1 Introduction

1
Chapter 1Introduction

• Individual objects described by a set of data
  (people, animals, or things)
• Variable Characteristic of an individual. It
  can take on different values for different
  individuals.
• Examples age, height, gender, favorite class,
  speed, moisture, etc.

2
Types of Variables

• Quantitative numerical values, can be added,
  subtracted, averaged, etc.
• ________ takes on values which are spaced. That
  is, for two values of a discrete variable that
  are adjacent, there is no value that goes between
  them.
• ________ values are all numbers in a given
  interval. That is, for two values of a
  continuous variable that are adjacent, there is
  another value that can go between the two.
• Categorical An individual is placed into one of
  several groups or categories. These groups or
  categories are not usually numerical.

3
Types of Variables

• Examples
• Numeric
• Variable Discrete Continuous Categorical
• Length
• Hours Enrolled
• Major
• Zip Code

4
Distribution of a Variable

• The distribution of a variable tells us the
  possible values for the variable and the
  probability that the variable takes these values.
• Two ways to describe a distribution
• Numerically
• Graphically

5
Categorical Variables

• Suppose we poll 46 people on an issue. How can
  we exhibit their response?
• Numerically
• Counts
• Proportions
• Percentages
• Graphically
• Frequency Tables
• Bar Charts
• Pie Charts

6
Categorical Variables

• Suppose we poll 46 people on an issue. How can
  we exhibit their response?
• Frequency Tables
• counts (14 agree)
• proportions (14/46 .304 agree)
• percents (30.4 agree)

7
Categorical Variables

• Suppose we poll 46 people on an issue. How can
  we exhibit their response?
• Bar Chart
• can have counts,
• percents or
• proportions on
• vertical axis

8
Categorical Variables

• Suppose we poll 46 people on an issue. How can
  we exhibit their response?
• Pie Chart

9
Examining a Distribution

• To describe a distribution we need 3 items
• Shape modes, symmetric, skewed
• Center mean, median
• Spread range, standard deviation, IQR
• Look for the overall pattern and for striking
  deviations
• Outlier-individual value that falls outside the
  overall pattern

10
Numeric Variable Distributions

• Shape
• Modes Major peaks in the distribution
• Symmetric The values smaller and larger than
  the midpoint are mirror images of
  each other
• Skewed to the right Right tail is much longer
  than the left tail
• Skewed to the left Left tail is much longer
  than the right tail
• Center
• Mean The arithmetic average. Add up the
  numbers and divide by the number of
  observations.
• Median List the data from smallest to
  largest. If there are an odd number of data
  values, the median is the middle one in the list.
  If there are an even number of data values,
  average the middle two in the list

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Numeric Variable Distributions

• Spread
• Range The difference in the largest and
  smallest value. (Max Min)
• Standard Deviation Measures spread by looking
  at how far observations are from their mean.
• The computational formula for the standard
  deviation is
• Interquartile Range (IQR) Distance between the
  first quartile (Q1) and the third quartile (Q3).
  IQR Q3 Q1
• Q1 25 of the observations are less than Q1
  and 75 are greater than Q1.
• Q3 75 of the observations are less than Q3
  and 25 are greater than Q3.

12
Numeric Variable Distributions

• Example 1.5 on page 11 of the book shows how much
  50 consecutive shoppers spent in a store. The
  data appear as follows

3.11 18.30 24.50 36.30 50.30
8.88 18.40 25.10 38.60 52.70
9.26 19.20 26.20 39.10 54.80
10.80 19.50 26.20 41.00 59.00
12.60 19.50 27.60 42.90 61.20
13.70 20.10 28.00 44.00 70.30
15.20 20.50 28.00 44.60 82.70
15.60 22.20 28.30 45.40 85.70
17.00 23.00 32.00 46.60 86.30
17.30 24.40 34.90 48.60 93.30
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Numerical Variables

• How can we describe the distribution of these 50
  numbers?
• Numerically
• Center Mean or Median
• Spread Quartiles, Range, IQR, or Standard
  deviation
• Graphically
• Frequency Table
• Histogram
• Boxplot
• Stem and Leaf
• Normal Quantile Plot

14
Descriptive statistics

• The descriptives box from SPSS gives the mean,
  median, variance, standard deviation, minimum,
  maximum, range, and IQR.

15
Percentiles

• 50th percentile is also called the median the
  middle data value if ordered smallest to largest
• 25th and 75th percentiles are also called the
  quartiles Q1 and Q3 respectively the middle
  data value of each half

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Frequency Table
Category Count or Frequency Percent
0 - 10 3 6.00
10 - 20 12 24.00
20 - 30 13 26.00
30 - 40 5 10.00
40 - 50 7 14.00
50 - 60 4 8.00
60 - 70 1 2.00
70 - 80 1 2.00
80 - 90 3 6.00
90 - 100 1 2.00

• Notice the amount
• spent is broken into
• categories or groups
• Recall, frequency
• tables can be used for
• categorical variables
• as well

17
Histogram

• Breaks the range of values
• of a variable into intervals
• Displays only the count
• of percent of the observations
• that fall into each interval

18
Box Plot

• Minimum, Q1, Median, Q3, and Maximum
• These five numbers
• are called the
• ____________________
• What are these points?

19
Stem and Leaf Plot

• Works best for smaller data sets
• Example 1.4 on pg 10
• Here are the numbers of homeruns that Babe Ruth
  hit in each of his 15 years with the New York
  Yankees from 1920-1934
• 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46,
  41, 34, 22

20
Normal Quantile Plot

• Normal Quantile Plot (This compares the
  distribution of the sample to the Normal
  Distribution)
• the straight line
• is normal,
• compare dots
• to the line
• If dots fall close to the normal
• line then the data comes
• from a normal distribution.

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Describing Numeric Variable Distributions

• Now, we examine the appearance of other data
• Modes are major peaks in the distribution
• The histogram below The histogram below has one
• has two modes-bimodal mode-unimodal

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Describing Numeric Variable Distributions

• Now, we examine the appearance of other data
• This example is called right This is an example
  of a boxplot skewed since the distribution
  has that is skewed to the _______.
• a long right tail.

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Describing Numeric Variable Distributions

• ________ observations that are unusually far
  from the bulk of the data.
• What are some possible explanations for outliers?
• The data point was recorded wrong.
• The data point wasnt actually a member of the
  population we were trying to sample.
• We just happened to get an extreme value in our
  sample.
• The 1.5 x IQR Criterion for Outliers Designate
  an observation a suspected outlier if it falls
  more than 1.5 x IQR below the first quartile or
  above the third quartile.

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1.5IQR Criterion Example

• Suppose you had the following data set
• -2, 15, 3, 7, 10, 21, 1, 5, 12, 8, 1, 35, 10
• List data from smallest to largest
• Find Q1, Median, Q3, Min, and Max
• IQR Q3 Q1 ______
• 1.5IQR _______
• Q1 1.5IQR ________If less than this number,
  then outlier
• Q3 1.5IQR ________If more than this number,
  then outlier
• Are there any outliers in this data set?

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Describing Numeric Variable Distributions

• Symmetry versus Skewness

__________
_________
___________
26
Mean versus Median

• For a skewed distribution, the mean is farther
  out in the longer tail than is the median.

Symmetric
Right Skewed
Left Skewed
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Strategy for Exploring Data on a Single
Quantitative Variable

1. Always plot your data make a graph usually a
   stem and leaf or histogram
2. Look for overall pattern and for outliers
3. Calculate an appropriate numerical summary to
   briefly describe center and spread
4. Sometimes the overall pattern of a large number
   of observations is so regular that it can be
   described by a smooth curve

28
Introducing the Normal Distribution

• It is customary to describe a normal distribution
  in the following way
• Properties of the Normal Distribution
• Symmetric, bell-shaped
• Mean, µ and standard deviation, s
• Area under the curve is 1

s
m
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The Normal Distribution

• Normal distributions can take on many different
  means and standard deviations. Only the general
  bell shape must remain the same.
• Here are some examples of normal distributions

m -2
m 0
m 3
s 0.5
s 1
s 2
0
-2
3
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Distribution Properties

• Introducing

The Standard Normal Distribution
Properties
1. _________________
2. _________________
3. _________________
31
Distribution Properties

• Empirical Rule (The 68-95-99.7 Rule) If the
  distribution is normal, then
• Approximately 68 of the data falls within one
  standard deviation of the mean
• Approximately 95 of the data falls within two
  standard deviations of the mean
• Approximately 99.7 of the data falls within
  three standard deviations of the mean

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Distribution PropertiesEmpirical Rule
33
Percentiles of a Standard Normal Curve
34
Empirical Rule Example

• If the grades on an exam are normally distributed
  with a mean of 68 and a variance of 16, what
  grade do you have to make to be in the top 15 of
  the class?

35
Distribution Properties

• Shift Changes adding or subtracting a number
  from the each of the values.

mean
mean c
mean - c
36
Distribution Properties

• The mean, median, Q1, Q3, minimum, and maximum
  all shift when there is a shift change. The
  shift change, say c, is added or subtracted to
  each of the statistics accordingly.
• The measures of spread (standard deviation,
  variance, IQR, and range) do not change when
  there is a shift change.

37
Distribution Properties

• Scale Changes multiplying or dividing each of
  the values by a number.

mean
38
Distribution Properties

• Scale Changes multiplying or dividing each of
  the values by a number.

meanc
39
Distribution Properties

• Scale Changes multiplying or dividing each of
  the values by a number.

mean/c
40
Distribution Properties

• The mean, median, Q1, Q3, minimum, and maximum
  all change when there is a scale change unless
  they are zero. Each is multiplied or divided by
  the scale change c.
• The measures of spread (standard deviation,
  variance, IQR, and range) always change when
  there is a scale change. The standard deviation,
  IQR, and range are multiplied or divided by the
  scale change c. The variance is multiplied or
  divided by c2.

41
Shift Change Example

• Suppose we measure the weight of everyone on a
  football team and obtain the following statistics
  for a team report
• Mean 230 lbs. Median 240 lbs.
• Std. Dev. 50 lbs. Q1 200 lbs., Q3 280 lbs.
• Variance 250 lbs. IQR 80 lbs
• Min. 170 lbs. Range 180 lbs.
• Max. 350 lbs.

42
Shift Change Example

• Now suppose we found out the scale was 10 lbs.
  under so we need to add 10 lbs. to every weight.
  What would happen to each of the following
  statistics?

Original
After Shift Change
Mean 230 lbs. Mean________
Median 240 lbs. Median_________
s 50 lbs. s_______
Q1 200 lbs. Q1________
Q3 280 lbs. Q3________
43
Shift Change Example

• Now suppose we found out the scale was 10 lbs.
  under so we need to add 10 lbs. to every weight.
  What would happen to each of the following
  statistics?

Original
After Shift Change
Variance 250 lbs.
Variance ________
IQR 80 lbs.
IQR _________
Min 170 lbs.
Min _________
Max 350 lbs.
Max _________
Range 180 lbs.
Range _________
44
Shift and Scale Change Example

• Further, suppose we found out that we are
  supposed to report the weights and statistics in
  kilograms, not lbs (Remember, 1 lb 0.6
  kilograms). What would happen to each of the
  following statistics?

After Shift Change
After Shift and Scale Change
Mean 240 lbs.
Mean ______________
Median 250 lbs.
Median ______________
s 50 lbs.
s _____________
Q1 210 lbs.
Q1 _____________
Q3 290 lbs.
Q3 _____________
45
Shift and Scale Change Example

• Further, suppose we found out that we are
  supposed to report the weights and statistics in
  kilograms, not lbs (Remember, 1 lb 0.6
  kilograms). What would happen to each of the
  following statistics?

After Shift Change
After Shift and Scale Change
Variance 250 lbs.
Variance _______________
IQR 80 lbs.
IQR _______________
Min 180 lbs.
Min _______________
Max 360 lbs.
Max ________________
Range 180 lbs.
Range _________________
46
Linear Transformations

• If you are given a mean, (or ?), and a
  standard deviation, s (or ?), and want to convert
  your data so you have a new mean, (or
  ?new), and new standard deviation, snew (or
  ?new), all you need is to remember what shift and
  scales changes affect.
• In our linear transformation formula
• a is the shift change
• b is the scale change
• Standard deviation are only affected by scale
  changes, but means are affected by both shift and
  scales changes.

47
Linear Transformation Example

• For example 12 and s 7 but we want
  25 and 10.
• snew scales
• 10 scale7
• scale 10/7
• scale 1.43
• substituting in shift scale
  
• 25 shift 1.4312
• shift 25 ? 1.4312
• shift 7.84
• So our linear transformation equation is x new
  7.84 1.43x