STATISTICS for the Utterly Confused, 2nd ed. SLIDES PREPARED

1
STATISTICS for the Utterly Confused, 2nd ed.

• SLIDES PREPARED
• By
• Lloyd R. Jaisingh Ph.D.
• Morehead State University
• Morehead KY

2
Chapter 3

• Data Description Numerical Measures of
  Variability for Ungrouped Univariate Data

3
Outline

• Do I Need to Read This Chapter?
• 3-1 The Range
• 3-2 The Interquartile Range
• 3-3 The Mean Absolute Deviation
• 3-4 The Variance and Standard Deviation
• 3-5 The Coefficient of Variation
• 3-6 The Empirical Rule
• Its a Wrap

4
Objectives

• Introduction of some basic statistical
  measurements of spread or variability.
• How to compute these measures and investigate
  some of their properties.

5
Introduction

• A measure of variability for a collection of data
  values is a number that is meant to convey the
  idea of spread for the data set.
• The most commonly used measures of variability
  for sample data are the
• range
• interquartile range
• mean absolute deviation
• variance or standard deviation
• coefficient of variation.

6
3-1 The Range

• Explanation of the term range The range is the
  difference between the largest and smallest
  values in the data set.
• NOTE The explanation is true for a sample as
  well as a finite population of values.

7
3-1 The Range

• Example What is the range for the following
  sample values?
• 3 8 6 14 0 4 0 12 7 0 -10
• Solution First we should arrange from smallest
  to largest.
• -10 -7 0 0 0 3 6 8 12 14
• Range 14 (-10) 24

8
3-1 The Range

• Question Why does subtracting the smallest value
  from the largest value is a measure of spread?
• The next slide shows the plot of the data set.
• Observe that the range measures the distance
  between the smallest and largest values.
• This distance gives a measurement of the spread
  of the data.

9
3-1 The Range
Range gives a measurement of spread.
10
Quick Tip

• The range does not use the concept of of
  deviations.
• It is affected by outliers (large or small values
  relative to the rest of the data set).
• The range does not utilize all the information in
  the data set only the largest and smallest
  values.
• Thus it is not a very useful measure of spread or
  variation.

11
The Range -- Example

• Example What is the range for the following
  sample values?
• 9 995 1000 1002 1014
• Solution Range 1014 9 1005
• Here the range is significantly affected by the
  outlying value of 9.

12
3-2 The Interquartile Range

• Explanation of the term interquartile range
  The interquartile range measures the spread of
  the middle 50 of an ordered data set.
• NOTE The interquartile range is obtained using
  the following steps-
• Step1 Order the data set from smallest to
  largest.
• Step 2 Find the median for the ordered set.
  Denote by Q2.

13
3-2 The Interquartile Range

• Step 3 Find the median for the first 50 of the
  ordered set. The median found in Step 2 is not
  included in this portion of the data. Denote by
  Q1.
• Step 4 Find the median for the second 50 of the
  ordered set. The median found in Step 2 is not
  included in this portion of the data. Denote by
  Q3.

14
3-2 The Interquartile Range

• Step 5 The interquartile range is computed from
  the following

15
3-2 The Interquartile Range

• The following depicts the idea of the
  interquartile range.

16
3-2 The Interquartile Range

• Example The following scores for a statistics
  10-point quiz were reported. What is the value
  of the interquartile range?
• 7 8 9 6 8 0 9 9 9
• 0 0 7 10 9 8 5 7 9

17
3-2 The Interquartile Range

• Solution With the availability of technology, it
  makes it easy to compute the interquartile range.
  
• We will present information from the MINITAB
  software and the TI-83 calculator to help compute
  the interquartile range.

18
3-2 The Interquartile Range

• MINITAB Solution The following shows the
  descriptive statistics output.

Interquartile range Q3 Q1 9 5.75 3.25.
19
3-2 The Interquartile Range

• TI-83 Solution The following shows the
  descriptive statistics output.

Interquartile range Q3 Q1 9 6 3.
Note A slight difference in the answers. TI-83
rounded Q1 to 6.
20
3-3 The Mean Absolute Deviation

• The mean absolute deviation utilizes deviations
  of the data values from the mean.
• Explanation of the term - Mean Absolute Deviation
  (MAD) The mean absolute deviation is the average
  of the absolute deviations from the mean of the
  data set.

21
3-3 The Mean Absolute Deviation

• The MAD is computed using the following formula.
  

• The formula says that you
• subtract the sample mean from each data
• value
• take the absolute values of the results
• add the absolute values together
• divide by the sample size

22
3-3 The Mean Absolute Deviation

• Example What is the MAD for the following sample
  values?
• 3 8 6 12 0 -4 10
• Solution First of all, the sample mean 5
  (Verify).
• The table on the next slide shows the
  computations

23
3-3 The Mean Absolute Deviation
MAD 32/7 4.57
24
3-3 The Mean Absolute Deviation
Question What does the MAD measure? The MAD
measures the average (absolute) distance of the
sample values from the mean of the data values.
25
3-3 The Mean Absolute Deviation
The deviations contribute to the total in
proportion to the size of the deviation.
The average distance of the sample values from
the mean is 4.57.
26
Quick Tip

• If data set A has a larger MAD than data set B,
  then it is reasonable to believe that the values
  in data set A are more spread out (variable) than
  the values in data set B.
• The MAD is sensitive to values that are very
  small or very large relative to the rest of the
  data set.

27
3-4 The Variance and Standard Deviation

• The variance and standard deviation are the
  most common and useful measures of variability.
• These two measures provide information about how
  the data vary about the mean.

28
3-4 The Variance and Standard Deviation
When the data are clustered about the mean, the
variance and standard deviation will be somewhat
small.
29
3-4 The Variance and Standard Deviation
When the data are widely scattered about the
mean, the variance and standard deviation will be
somewhat large.
30

3-4 The Variance and Standard Deviation

• Explanation of the term sample variance
  the sample variance is an approximate average of
  the squared deviations of the data values from
  the sample mean.
• The sample variance is computed from the
  following formula and is denoted by s2

31

3-4 The Variance and Standard Deviation

• Example What is the variance for the following
  sample values?
• 3 8 6 14 0 11
• NOTE Do not let the formula intimidate you. We
  will build a table to help with the computations.

32

3-4 The Variance and Standard Deviation

• We will build a table to help in the
  computations. NOTE The mean 7.

S2 132/(6 1) 132/5 26.4
33

3-4 The Variance and Standard Deviation

• In the previous example, observe that the
  variance is large relative to the size of the
  data values.
• This can be observed from the plot which shows
  that the data values are very much spread out
  about the mean value of 7.

34

3-4 The Variance and Standard Deviation

• Explanation of the term sample standard
  deviation the sample standard deviation is the
  positive square root of the variance.
• NOTE the standard deviation has the same unit
  as the variable.
• Example The sample standard deviation for the
  previous example is

35
Quick Tips

• If all of the observations have the same value,
  the sample variance (standard deviation) will be
  zero. That is, there is no variability in the
  data set.
• The variance (standard deviation) is influenced
  by outliers in the data set.
• The unit for the standard deviation is the same
  as that for the raw data.
• Thus it is preferred to use the standard
  deviation rather than the variance as the measure
  of variability.

36

3-4 The Variance and Standard Deviation

• Explanation of the term population variance
  the population variance is the average of the
  squared deviations of the data values from the
  population mean.
• The population variance is computed from the
  following formula and is denoted by s2

37

3-4 The Variance and Standard Deviation

• Explanation of the term population standard
  deviation the population standard deviation is
  the positive square root of the population
  variance.
• The population standard deviation is computed
  from the following formula and is denoted by s

38

3-5 The Coefficient of Variation

• The coefficient of variation (CV) allows us to
  compare the variation of two (or more) different
  variables.
• Explanation of the term sample coefficient of
  variation the sample coefficient of variation is
  defined as the sample standard deviation divided
  by the sample mean of the data set.
• Usually, the result is expressed as a
  percentage.

39

3-5 The Coefficient of Variation
NOTE The sample coefficient of variation
standardizes the variation by dividing it by
the sample mean.
40

3-5 The Coefficient of Variation

• The coefficient of variation has no units since
  the standard deviation and the mean have the same
  units, and thus cancel out each other.
• Because of this property, we can use this
  measure to compare the variations for different
  variables with different units.

41

3-5 The Coefficient of Variation

• Example The mean number of parking tickets
  issued in a neighborhood over a four-month period
  was 90, and the standard deviation was 5. The
  average revenue generated from the tickets was
  5,400, and the standard deviation was 775.
  Compare the variations of the two variables.
• Solution is on the next slide.

42

3-5 The Coefficient of Variation

• Solution

Since the CV is larger for the revenues, there is
more variability in the recorded revenues than
in the number of tickets issued.
43

3-5 The Coefficient of Variation

• Explanation of the term population coefficient
  of variation the population coefficient of
  variation is defined as the population standard
  deviation divided by the population mean of the
  data set.
• NOTE The population CV has the same properties
  as the sample CV.

44

3-6 The Empirical Rule

• Knowing the value of the mean and the value of
  the standard deviation for a data set can provide
  a great deal of information about the data set.
• In particular, if the data set has a single
  mound and is symmetrical (bell-shaped), then
  one can generalize some propertied of the
  distribution.
• One such generalization is called the Empirical
  Rule.

45

3-6 The Empirical Rule

• The Empirical Rule gives some general statements
  relating the mean and the standard deviation of a
  bell-shaped distribution.
• It relates the mean to one, two, and three
  standard deviations.

46

3-6 Empirical Rule

• One Sigma Rule Approximately 68 of the data
  values will lie within one standard deviation
  from the mean.
• That is, one can expect a deviation of more than
  one sigma from the mean to occur once in every
  three observations.
• This true because approximately 33
  (approximately 1/3) of the values are outside one
  standard deviation from the mean

47

3-6 Empirical Rule - One Sigma Rule
Graphical Display of the One Sigma Rule
48

3-6 Empirical Rule

• Two Sigma Rule Approximately 95 of the data
  values will lie within two standard deviations
  from the mean.
• That is, one can expect a deviation of more than
  two sigma from the mean to occur once in every
  twenty observations.
• This true because approximately 5 (1/20) of the
  values are outside two standard deviations from
  the mean

49

3-6 Empirical Rule - Two Sigma Rule
Graphical Display of the Two Sigma Rule
50

3-6 Empirical Rule

• Three Sigma Rule Approximately 99.7 of the
  data values will lie within three standard
  deviations from the mean.
• That is, one can expect a deviation of more than
  three sigma from the mean to occur once in every
  333 observations.
• This true because approximately 0.3 (1/333) of
  the values are outside three standard deviations
  from the mean

51

3-6 Empirical Rule - Three Sigma Rule
Graphical Display of the Three Sigma Rule