Chapter 1: Looking at Data Distributions

1
Chapter 1 Looking at Data Distributions
2
1. Introduction

• 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, etc.

3
Types of Variables

• Categorical Variable An individual is placed
  into one of several groups or categories. These
  groups or categories are not usually numerical.

4
Types of Variables

• Quantitative Variable (numeric) values can be
  added, subtracted, averaged, etc.
• Discrete 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.
• Continuous 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.

5
Types of Variables

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

6
Types of Variables

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

7
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.

8
2. Describing Categorical Variable Distributions

• Suppose we poll 50 students on an issue
  (Statistics is interesting!). How can we exhibit
  their responses?
• Frequency Tables
• gives counts (31 agree), proportions (31/50
  .62 agree), and percents (62 agree)

9
2. Describing Categorical Variable Distributions

• Suppose we poll 50 students on an issue
  (Statistics is interesting!). How can we exhibit
  their response?
• Bar Chart
• can have counts,
• percents or
• proportions on
• vertical axis

10
2. Describing Categorical Variable Distributions

• Suppose we poll 50 students on an issue
  (Statistics is interesting!). How can we exhibit
  their response?
• Pie Chart

11
3. Describing Numeric Variable Distributions

• To describe a distribution we need 3 items
• Shape modes, symmetric, skewed, outliers
• Center mean, median, mode
• Spread range, standard deviation, IQR

12
3. Describing 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

13
3. Describing Numeric Variable Distributions

• Center
• Mean The arithmetic average. Add up the
  numbers and divide by the number of observations.
  If the n observations are , their sample mean is
• 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.

14
3. Describing Numeric Variable Distributions

• Spread
• Range The difference between the largest and
  smallest value.
• Standard Deviation Measures spread by looking
  at how far observations are from their mean. The
  computational formula for the sample standard
  deviation is
• Variance Square of Standard Deviation

15
3. Describing Numeric Variable Distributions

• Spread
• Interquartile Range (IQR) IQRQ3-Q1
• Distance between the first quartile (Q1) and
  the third quartile (Q3).
• Q1 25 of the observations are less than
  Q1 and 75 are greater than Q1.
• or the median of the observations whose
  position in the ordered
• list is to the left of the
  location of the overall median.
• Q3 75 of the observations are less than
  Q3 and 25 are greater than Q3.
• or the median of the
  observations whose position in the ordered
  
• list is to the right of the
  location of the overall median.
• Note To compute Q1/Q3 for odd number of
  observations, the center value is excluded.

16
3. Describing Numeric Variable Distributions

• Example Suppose the age of five students are
  20, 18, 22, 20, 23.

Mean ? Median ? Q1 ? Q3 ?
Range ? Std. dev. ? Var. ? IQR ?
17
3. Describing Numeric Variable Distributions

• Examples for Q1 and Q3
• Even number of observations
• The highway mileages of the 18 cars, arranged in
  increasing order, are
• 13 13 16 19 21 21 23 23 24 26 26 27
  27 27 28 28 30 30
• Find Q1 and Q3.
• Odd number of observations
• The highway mileages of the 11 trucks arranged in
  increasing order, are
• 22 22 23 24 24 25 27 28 28 30 31
• Find Q1 and Q3.

18
3. Describing Numeric Variable Distributions

• Another example in the book shows how much 50
  consecutive shoppers spent in a store. The data
  appear as follows

19
3. Describing Numeric Variable Distributions

• How can we describe the distribution of these 50
  numbers?

20
3. Describing Numeric Variable Distributions

• How can we describe the distribution of these 50
  numbers?
• 50th percentile is also called the median the
  middle data value if ordered smallest to largest
• 25th and 75th percentiles are also called Q1 and
  Q3 respectively the middle data value of each
  half

21
3. Describing Numeric Variable Distributions

• How can we describe the distribution of these 50
  numbers?
• Stemplot (discard decimals)

• 1. Separate each observation into a stem and a
  leaf. Stems may have as many digits as needed,
  but each leaf contains only a single digit.
• Write the stems in a vertical column
• Write each leaf in the row to the right of its
  stem,
• in increasing order out from the stem.

22
3. Describing Numeric Variable Distributions

• How can we describe the distribution of these 50
  numbers?
• Histogram
• Breaks the range of values
• of a variable into intervals
• and displays only the count
• or percent of the
• observations in each interval.

23
3. Describing Numeric Variable Distributions

• How can we describe the distribution of these 50
  numbers?
• Box Plot (made up of min., Q1, median, Q3, and
  max.)
• these five numbers
• are called the
• five number summary

24
3. Describing Numeric Variable Distributions

• Outlier 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.

25
3. Describing Numeric Variable Distributions

• Now, we examine the appearance of other data
• this example is bimodal has two
  modes

26
3. Describing Numeric Variable Distributions

• Now, we examine the appearance of other data
• this only has one mode - unimodal

27
3. Describing Numeric Variable Distributions

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

28
3. Describing Numeric Variable Distributions

• Now, we examine the appearance of other data
• This is an example of
• a boxplot that is
• skewed to the right.

29
3. Describing Numeric Variable Distributions

• Symmetry versus Skewness

__________
__________
___________
30
3. Describing Numeric Variable Distributions

• Symmetry versus Skewness

__________
__________
____________
31
3. Describing Numeric Variable Distributions

• Mean versus Median for Different Distributions
• meanltmedian meanmedian
  meangtmedian

Right Skewed
Symmetric
Left Skewed
32
3. Describing Numeric Variable Distributions

• Example
• Calculate mean, median, std. dev. and IQR of
  these two observations
• 3, 3, 5, 6, 7
• 3, 3, 5, 6, 7, 80
• Conclusion
• Mean and std. dev. cant resist the influence of
  outliers. That is, mean and std. dev. are not
  resistant.
• Median and IQR are better than mean and std. dev.
  for describing a skewed distribution or a
  distribution with strong outliers. Use mean and
  std. dev. only for reasonably symmetric
  distributions that are free of outliers.

33
3. Describing Numeric Variable Distributions

• Measures of Center and Spread
• To describe distributions use
• Median and IQR Mean and
  standard deviation Median and IQR

Right Skewed
Left Skewed
Symmetric
34
3. Describing Numeric Variable Distributions

• Summary
• Shape (usually Histogram. Boxplot and Stemplot)
• Mode unimodal, bimodal?
• Symmetric? Left skewed? Right skewed?
• Center (Descriptives)
• Spread (Descriptives)
• Outlier (Boxplot)

35
4. The Normal Distribution

• Sometimes the overall pattern of a large number
  of observations is so regular that we can
  describe it by a smooth curve.
• A density curve is a curve that is always on or
  above the horizontal axis and has area exactly 1
  underneath it.
• A density curve describes the overall pattern of
  a distribution. The area under the curve and
  above any range of values is the relative
  frequency of all observations that fall in that
  range.

36
4. The Normal Distribution

• Normal curves are density curves that are
  symmetric, unimodal and bell-shaped. They
  describe normal distributions.
• A normal curve is specified by giving its mean µ
  and its standard deviation s.
• We often write that a variable (call it X) has
  normal distribution with mean m and variance s2
  in the following way

37
4. The Normal Distribution

• Here are some examples of normal distributions

m 0
m 3
m -2
s 1
s 2
s 0.5
0
-2
3
N(0,12)
N(-2,0.52)
38
4. The Normal Distribution

• 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 deviation of the mean
• Approximately 99.7 of the data falls within
  three standard deviation of the mean

39
4. The Normal Distribution Empirical Rule for
N(0,12)
40
4. The Normal Distribution

• If x is an observation from a distribution that
  has mean m and standard deviation s, the
  standardized value of x is

This is known as a Z-score. A z-score is
literally how many sds an observation is from
its mean. They are measures of relative standing.

• The standard normal distribution is the normal
  distribution with mean 0 and standard
  deviation 1. Z N(0,12) .

41
4. The Normal Distribution

• If a variable X has any normal distribution
  N(m,s2 ) , then the standardized variable
  
• has the standard normal distribution.

42
4. The Normal Distribution

• For N(0,12) we can find approximate probabilities
  associated with different values of Z using
  Empirical Rule.

43
4. The Normal Distribution

• We can find the approximate probability that Z is
  to the left of any integers using the Empirical
  Rule.

P( Z lt -4.00) ? 0 P( Z lt -3.00) ? 0.15 P( Z lt
-2.00) ? 2.5 P( Z lt -1.00) ? 16
P( Z lt 0.00) ? 50 P( Z lt 1.00) ? 84 P( Z lt
2.00) ? 97.5 P( Z lt 3.00) ? 99.85 P( Z lt
4.00) ? 100
44
4. The Normal Distribution

• We can find the approximate probability that Z is
  to the right of any integers using the symmetry.
• P( Z gt z) P( Z lt -z)
• For example, P( Z gt 3.00) P( Z lt -3.00)

• We can find the approximate probability that Z is
  between any two integers using the Empirical
  Rule.
• P( a lt Z lt b) P( Z lt b) P(Z lt a)
• Examples
• P( Z gt 3.00) ? 0.15 P( -2.00 lt Z lt 1.00) ?
  81.5
• P( Z gt -1.00) ? 84 P( 2.00 lt Z lt 3.00) ?
  2.35

45
4. The Normal Distributionexample

• P(Z lt 1.25) ? P(Z gt 0.25) ?
• A. 0.4840 A. 0.7040
• B. 0.8944 B. 0.0217
• C. 0.9989 C. 0.8485
• D. 0.1736 D. 0.4013

46
4. The Normal Distribution

• Now suppose we know X N (m, s2) and we want to
  know P(X lt x), P(X gt x) and P(x1 lt X lt x2).
• We can first convert the X to Z and then use the
  probabilities from the Empirical Rule.
• Recall that if X N (m, s2) , then
• we have

47
4. The Normal Distributionexamples

• Suppose X N ( 3, 22). Find the probability
  that X is less than 5.

• Suppose X N (-1, 52). Find the probability
  that X is greater than 11.

48
4. The Normal Distribution

• We will look at some more difficult examples
• Suppose X N (2, 32),
• Given a value z, find the corresponding x that it
  came from.
• say z 5, x ?
• How many standard deviations is x from m?
• say x 10
• Find Pr (X lt -4 or X gt 8).
• Find Pr ( -4 lt X lt 8 ).
• Find the x such that Pr ( X lt x ) ? .84
• Find the x such that Pr ( X gt x ) ? .84

49
Go back to3. Describing Numeric Variable
Distributions

• How can we describe the distribution of these 50
  numbers?
• Normal Quantile Plot (This compares the
  distribution of the sample to the Normal
  Distribution)
• the straight line
• is normal,
• compare dots
• to the line

50
Go back to3. Describing Numeric Variable
Distributions

• Summary
• Shape (Histogram, Boxplot and Stemplot Normal
  Quantile Plot)
• Mode unimodal, bimodal?
• Symmetric? Left skewed? Right skewed?
• Are the data normally distributed?
• Center (Descriptives)
• Spread (Descriptives)
• Outlier (Boxplot)

51
5. Distribution Properties

• Shift Changes adding or subtracting a number
  from the each of the values. If c gt 0, then

mean
mean c
mean - c
52
5. Distribution Properties

• The mean, median, Q1, Q3, maximum, and minimum
  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.

53
5. Distribution Properties

• Scale Changes multiplying or dividing each of
  the values by a number. If c gt 1, then

mean
meanc
mean/c
54
5. Distribution Properties

• The mean, median, Q1, Q3, maximum, and minimum
  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.

55
5. Distribution Properties

• 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.

56
5. Distribution Properties

• 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________
57
5. Distribution Properties

• 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 _________
58
5. Distribution Properties

• 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 _____________
59
5. Distribution Properties

• 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 _________________
60
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.