Inferences From Nominal Data: the ?2 Statistic

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Chapter 22

• Inferences From Nominal Data the ?2 Statistic

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?2 test of frequencies

• Pronounced ki - square
• Can be used for nominal level data
• data with only the category property
• The ?2 test is based on a comparison of expected
  frequencies (fe) versus observed frequencies (fo)

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Example of frequencies (refresher)

• Twenty-five miners were asked about their
  political affiliation - Republican, Democrat,
  Independent, None
• Since the data, political affiliation, have only
  the category property, they are nominal level,
  and can only be counted

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Observed Frequencies
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What was expected?

• Lets say, that on average, throughout the
  country, the percentages of people who report the
  following political party affiliations are
• Democrat - 45
• Republican - 40
• Independent - 10
• None - 5

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Is the survey of miners different from the
national average?

• Computing expected frequencies is then done by
  multiplying the number of observations with the
  percent expected
• fe n( expected)

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Expected frequencies

• On average, then we would expect
• Democrats 25 (.45) 11.25
• Republicans 25 (.40) 10
• Independents 25 (.10) 2.5
• None 25 (.05) 1.25

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?2 statistic

• ?2 statistic is calculated using the following
  formula

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?2 test on miners political affiliation
Party fo fe (fo fe) (fo fe)2 (fo fe)2 fe
Dem 6 11.25 -5.25 27.5625 2.45
Rep 10 10 0 0 0
Ind 5 2.5 2.5 6.25 2.5
None 4 1.25 2.75 7.5625 6.05
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df in ?2 test

• df number of categories - 1
• In the present case,
• 4 categories - 1 3

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Interpretations of ?2 statistic

• If the differences between the observed and
  expected frequencies are small, these differences
  squared will also be small, thus making the sum
  of these squared differences small also
• Thus, small differences between observed and
  expected small ?2 statistic

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Interpretations of ?2 statistic

• However, if the differences between observed and
  expected frequencies are large, then these
  differences squared will be large also, making
  the sum of the squared differences large
• Thus, large differences large ?2statistic

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Interpretations of ?2 statistic

• How large is large?
• Fortunately, the ?2 statistic is based on a
  probability distribution of known parameters
• Table G in your text provides critical values for
  ?2 tests

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Survey of Miners

• HO Frequency of Dems 11.25
• Frequency of Reps 10
• Frequency of Ind 2.5
• Frequency of None 1.25
• HA HO is incorrect
• ?2 .05, ?2crit .05 (df 3) 7.81

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Survey of Miners

• Since the obtained ?2 11.00 is larger than the
  critical ?2 7.81, we
• Reject HO that the obtained frequencies are
  11.25, 10, 2.5, and 1.25 respectively, and
• Conclude that the obtained frequencies were
  different than expected