Summary Statistics: Mean, Median, Standard Deviation, and More

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Summary Statistics Mean, Median, Standard
Deviation, and More

• Seek simplicity and then distrust it.
• (Dr. Monticino)

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Assignment Sheet

• Read Chapter 4
• Homework 3 Due Wednesday Feb. 9th
• Chapter 4
• exercise set A 1 -6, 8, 9
• exercise set C 1, 2, 3
• exercise set D 1 - 4, 8,
• exercise set E 4, 5, 7, 8, 11, 12
• Quiz 2 will be over Chapter 2
• Quiz 3 on basic summary statistic calculations
  mean, median, standard deviation, IQR, SD units
• If youd like a copy of notes - email me

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Overview

• Measures of central tendency
• Mean (average)
• Median
• Outliers
• Measures of dispersion
• Standard deviation
• Standard deviation units
• Range
• IQR
• Review and applications

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Central Tendency

• Measures of central tendency - mean and median -
  are useful in obtaining a single number summary
  of a data set
• Mean is the arithmetic average
• Median is a value such that at least 50 of the
  data is less and at least 50 is greater

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Example

• Calculate mean and median for following data sets

37 44 55 78 100 111 125 151 161 37 44 55 69 90
120 125 152 157 161
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Outliers and Robustness

• Mean can be sensitive to outliers in data set
• Not robust to data collection errors or a single
  unusual measurement
• Blind calculation can give misleading results

mean 170.35
median 151
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Outliers and Robustness

• Always a good idea to plot data in the order that
  it was collected
• Spot outliers
• Identify possible data collection errors

mean without outliers 150.14
median without outliers 149
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Outliers and Robustness

• Median can be a more robust measure of central
  tendency than mean
• Life expectancy
• U.S. males mean 80.1, median 83
• U.S. females mean 84.3, median 87
• Household income
• Mean 51,855, median 38,885
• .3 account for 12 of income
• Net worth
• Mean 282,500, median 71,600

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Which Central Tendency Measure?

• Calculate mean, median and mode
• Plot data
• Create histogram to inspect mode(s)
• Do not delete data points
• If analyze data without outliers, report and
  explain outliers
• Many statistical studies involve studying the
  difference between population means
• Reporting the mean may be dictated by objective
  of study

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Which Central Tendency Measure?

• If data is
• Unimodal
• Fairly symmetric
• Mean is approximately equal to median
• Then mean is a reasonable measure of central
  tendency

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Which Central Tendency Measure?

• If data is
• Unimodal
• Asymmetric
• Then report both median and mean
• Difference between mean and median indicates
  asymmetry
• Median will usually be the more reasonable
  summary of central tendency

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Which Central Tendency Measure?

• If data is
• Not unimodal
• Then report modes and cautiously mean and median
• Analyze data for differences in groups around the
  modes

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Limitations of Central Tendency

• Any single number summary may not adequately
  represent data and may hide differences between
  data sets
• Example

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Measures of Dispersion

• Including an additional statistic - a measure of
  dispersion - can help distinguish between data
  sets which have similar central tendencies
• Range max - min
• Standard deviation root mean square difference
  from the mean

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Measures of Dispersion

• Examples
• Range

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Measures of Dispersion

• Examples
• Standard deviation

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Measures of Dispersion

• Both range and standard deviation can be
  sensitive to outliers
• However, many data sets can be characterized by
  mean and SD
• If the values of the data set are distributed in
  an approximately bell shape, the
• 68 of the data will be within 1 SD unit of
  mean, 95 will be within 2 SD units and nearly
  all will be within 3 SD units

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Measures of Dispersion

• Example
• Suppose data set has mean 35 and SD 7
• How many SD units away from the mean is 42?
• How many SD units away from the mean is 38?
• How many SD units away from the mean is 30?
• Assuming bell shape distribution, 95 are
  between what two values?

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Measures of Dispersion

• A robust measure of dispersion is the
  interquartile range
• Q1 value such that 25 of data less than, and
  75 greater than
• Q3 value such that 75 less than, and 25
  greater than
• IQR Q3 - Q1

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Example

• Calculate range, standard deviation and
  interquartile range for the following data sets

1 98 99 100 100 100 102 102 104 107 95 98 99
100 100 100 102 102 104 107
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Assignment, Discussion, Evaluation

• Read Chapter 4
• Discussion problems
• Chapter 4
• exercise set A 1 -6, 8, 9
• exercise set C 1, 2, 3
• exercise set D 1 - 4, 8,
• exercise set E 4, 5, 7, 8, 11, 12
• Quiz 3 on basic summary statistic calculations
  mean, median, standard deviation, IQR, SD units

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Review of Definitions

• Measures of central tendency
• Mean (average)
• Median
• If odd number of data points, middle value
• If even number of data points, average of two
  middle values

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Question and Examples

• Can mean be larger than median? Can median be
  larger than mean?
• Give examples
• Can mean be a negative number? Can the median?
• The average height of three men is 69 inches.
  Two other men enter the room of heights 73 and 70
  inches. What is the average height of all five
  men?

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Questions and Examples

• The average of a data set is 30.
• A value of 8 is added to each element in the data
  set. What is the new average?
• Each element of the data set is increased by 5.
  What is the new average?
• Suppose that data consists of only 1s and 0s
• What does the average represent?
• Application an experiment is performed and only
  two outcomes can occur
• Label one type of outcome 1 and the other 0
• For the data set 31, 45, 72, 86, 62, 78, 50, find
  the median, Q1 (25th percentile) and Q3 (75th
  percentile)

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Review of Definitions

• Measures of dispersion
• Standard deviation
• Range max - min
• IQR Q3 - Q1

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Questions and Examples

• Can the SD be negative? Can the range? Can the
  IQR?
• Can the SD equal 0?
• For the data set 3,1,5,2,1,6 find the SD, range
  and IQR
• The average weight for U.S. men is 175 lbs and
  the standard deviation is 20 lbs
• If a man weighs 190 lbs., how many standard
  deviation units away from the mean weight is he?
• Assuming a normal (bell-shaped) distribution for
  weight, ninety-five percent of U.S. men weigh
  between what two values?

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Questions and Examples

• The average of a data set is 23 and the standard
  deviation is 5
• A value of 8 is added to each element in the data
  set. What is the new standard deviation?
• Each element of the data set is increased by 5.
  What is the new standard deviation?  
• (Dr. Monticino)