Learning Objectives

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Learning Objectives

• Calculate the mean, median, and mode for small
  sets of data and also frequency distribution
• Understand the advantage and disadvantage of each
  type of average
• Understand the need for measure of spread and to
  be able to use the cumulative frequency ogive to
  find the interquartile range
• Calculate the standard deviation for small sets
  of data and for frequency distribution

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Summary Measures

• Average measures location or central
  tendency, tells at what general level the data
  are.
• Range measures scatter or dispersion,
  indicates how widely spread the data are.
• The symmetry of the data, measures the "shape" of
  the data, tells us how equally the data are
  dispersed.

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Measures of Location - The Arithmetic Mean

• The arithmetic mean
• For example, the mean of 3, 4, 5, 5, 5, 6, 7, 8,
  11 is 54/9 6
• Or the mean

• X f fx
• 0 364 0
• 1 362 362
• 2 226 452
• 3 44 132
• 4 4 16
• Total 1000 962
• .962

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The Arithmetic Mean

• The arithmetic mean for grouped data
• where
• fi frequency of ith class interval
• xi mid-point of ith class interval
• j number of
  class intervals

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Suppose that a survey of the prices of 60items
sold in a shop gives the results below(five
class intervals for prices)
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Measures of Location - Median

• This is the middle value of a set of numbers
• Median for ungrouped data the middle item or the
  arithmetic mean of the middle two values.
• E.g. the median of 3, 4, 5, 5, 5, 6, 7, 8, 11
  is 5,
• whereas the median of 3, 4, 5, 5, 5, 6, 7, 8,
  11, 12 is (56)/2 5.5

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Measures of Location - Mode

• The mode is the most frequently occurring value
  of the variable
• The mode of ungrouped data
• 5, 9, 7, 14, 8, 7, 3 is 7
• With a frequency distribution, the mode is the
  value with the greatest frequency

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The mode of a grouped frequency distribution
30 35 40 45 50 55
60
Modal value
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The mode of a grouped frequency distribution
Modal value
40 45 50 55 60 65 70 75 80
13
scored by each of twenty participants in a
driving competition
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by a sample of 20 views in a 19 part serial
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from work in a 1-year period for a sample of 20
employees
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Answers for calculations

• Mean is the first-choice as long as the data are
• symmetrical
• Median should be considered when there are large
  outliers
• Mode is good measure when the data have two or
  more clusters

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Measures of Dispersion (or Scatter)

• Range largest reading - smallest reading
• E.g., 1, 3, 4, 5, 5, 5, 6, 7, 8, 13,
  Range 13 - 1 12
• Interquartile Range Range of middle 50 of
  readings
• E.g., 1, 3, 4, 5, 5, 5, 6, 7, 8, 13
• remove
• Interquartile Range 7 - 4 3

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75
Q1 the lower quartile 25
50
Q3 the upper quartile 25
IQR (Q3-Q1)
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Semi-IQR IQR/2
Q1
Q2
Q3
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Measures of Dispersion (or Scatter)

• Mean absolute deviation (MAD) - the average
  distance of the readings from their arithmetic
  mean
• where is the
  arithmetic mean
• modulus
• E.g., the MAD of 3, 4, 5, 5, 5, 6, 7, 8, 11,
  where the mean 6 and n 9. x - -3, -2,
  -1, -1, -1, 0, 1, 2, 5. Thus,
• MAD 3 2 1 1 1 0 1 2 5 / 9
  1.8

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Measures of Dispersion (or Scatter)

• MAD grouped data MAD
• 4.1

MAD 71.2/60 1.19
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Measures of Dispersion (or Scatter)

• Population variance
• Sample variance

S2 792 / (7-1) 132
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Measures of Dispersion (or Scatter)

• Standard deviation
• sample standard deviation
• population standard deviation
• Grouped data

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Summary

• Measures of location the arithmetic mean is
  chosen unless there is good reason to do
  otherwise
• Measures of dispersion
• range is easily understood, distorted by outliers
• imterqurtile range is easily understood but not
  well known
• MAD is sensible but unfamiliar and difficult to
  handle mathematically
• variance is mathematically tractable but with
  wrong units, difficult to understand
• standard deviation is mathematically tractable
  and well-known

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