Measurement and Scaling Fundamentals'

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• Measurement and Scaling Fundamentals.
• Comparison Tests.

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Measurement and Scaling

• Measurement means assigning numbers or other
  symbols to characteristics of objects according
  to certain prespecified rules.
• One-to-one correspondence between the numbers and
  the characteristics being measured.
• The rules for assigning numbers should be
  standardized and applied uniformly.
• Rules must not change over objects or time.
•  

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Measurement and Scaling

• Scaling involves creating a continuum upon which
  measured objects are located.
• Consider an attitude scale from 1 to 100. Each
  respondent is assigned a number from 1 to 100,
  with 1 Extremely Unfavorable, and 100
  Extremely Favorable. Measurement is the actual
  assignment of a number from 1 to 100 to each
  respondent. Scaling is the process of placing
  the respondents on a continuum with respect to
  their attitude toward department stores.

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Primary Scales of Measurement
Scale Nominal Numbers Assigned to
Runners Ordinal Rank Order of
Winners Interval Performance Rating on a
0 to 10 Scale Ratio Time to Finish, in
Seconds
Finish
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3
8
Finish
8.2
9.1
9.6
15.2
14.1
13.4
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Primary Scales of Measurement Nominal Scale

• The numbers serve only as labels or tags for
  identifying and classifying objects.
• When used for identification, there is a strict
  one-to-one correspondence between the numbers and
  the objects.
• The numbers do not reflect the amount of the
  characteristic possessed by the objects.
• The only permissible operation on the numbers in
  a nominal scale is counting.
• Only a limited number of statistics, all of which
  are based on frequency counts, are permissible,
  e.g., percentages, and mode.

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Review

• What is a crosstab?
• A table based on two variables, where the cell
  entries are the counts or percentages of cases
  that fall in that row or column category.

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A Quick Example

• You claim that women are more likely to watch
  Sex and the City than men.
• Your friend, Dingbat Moronson, tells you that he
  has a male friend who watches the show, so you
  cannot generalize.
• Resolved to prove your point by doing just that,
  you collect the following randomly sampled data

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Are Women More Likely to Watch Than Men?

• You can understand the data in crosstabs
• But how do you read it?

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Are Women More likely to Watch than Men?

• Compare values of the dv across values of iv.
  That is
• Calculate proportions for columns
• Compare across rows
• Take that, Dingbat!
• Is this the same thing as saying watchers are
  more likely to be female?

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Are Watchers More Likely to Be Female?

• Calculate proportions for rows
• Compare across columns
• How might this be different?

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Class Survey

• Hyp Vegetarians are more likely to oppose the
  death penalty than meat eaters.
• Rationale Vegis uniformly oppose killing.
• SPSS (row) Vegi, (column) Dpenalty, (row)
  Percentages

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Why are people Vegetarians?

• Rationale might it be concern for relatives?
• Hyp People who were mammals in their previous
  life are more likely to be vegetarians.
• NB Multiple categories. Might recode help?
  Other Probs?

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Shop Data

• Do a crosstab on any two categorical variables
• Indicate frequencies and percentages
• Chi-Square significance test

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Chi-Square Analysis on SPSS

• The Chi-square statistic is a measure of
  association or test of independence between two
  variables consisting of nominal data. For example
  Gender and Type of Sentence. As you may recall
  from previous studies a table of observations
  concerning two sets of variables is constructed
  when undertaking this type of analysis. This is
  known as a Crosstabs and can be facilitated in
  SPSS using the Statistics - Summarise -
  Crosstabs menu items, then you can select the
  appropriate nominal data variables

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Crosstabs in SPSS

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CrosstabsTable

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Chi-Square from Crosstabs (1)

• The crosstabs facility within SPSS has a
  dialogue box from which a range of analyses and
  extra functions can be selected. These include
  chi-square and inclusion of percentages in the
  table.
• Interpretation of the chi-square test is quick
  and can be made by observation of the
  significance value on the top line of the
  analysis section. It is usual to compare this to
  0.05 (5 significance)

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Chi-Square Dialogue Box

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Chi-square from Crosstabs (2)

interpretation
from this
value
Chi-square value
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• Chi-square Interpretation
• In a chi-square test the null hypothesis should
  state that there is no association between two
  variables. The alternative hypothesis should
  state that the two variables are associated
• Ho - No association between Gender and Type of
  Sentence
• H1 - Is association between Gender and Type of
  Sentence
• as the significance on the printout is below
  0.05 (i.e. .00000) you should reject the null
  hypothesis (at the 5 level) and accept the
  alternative (If it would have been greater than
  0.05 the reverse would be the case).

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Primary Scales of Measurement Ordinal Scale

• A ranking scale in which numbers are assigned to
  objects to indicate the relative extent to which
  the objects possess some characteristic.
• Can determine whether an object has more or less
  of a characteristic than some other object, but
  not how much more or less.
• Any series of numbers can be assigned that
  preserves the ordered relationships between the
  objects.
• In addition to the counting operation allowable
  for nominal scale data, ordinal scales permit the
  use of statistics based on centiles, e.g.,
  percentile, quartile, median.

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Comparative Scaling TechniquesRank Order Scaling

• Respondents are presented with several objects
  simultaneously and asked to order or rank them
  according to some criterion.
• It is possible that the respondent may dislike
  the brand ranked 1 in an absolute sense.
• Furthermore, rank order scaling also results in
  ordinal data.
• Only (n - 1) scaling decisions need be made in
  rank order scaling.

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Preference for Toothpaste Brands Using Rank
Order Scaling

Instructions Rank the various brands of
toothpaste in order of preference. Begin by
picking out the one brand that you like most and
assign it a number 1. Then find the second most
preferred brand and assign it a number 2.
Continue this procedure until you have ranked all
the brands of toothpaste in order of preference.
The least preferred brand should be assigned a
rank of 10. No two brands should receive the
same rank number. The criterion of preference is
entirely up to you. There is no right or wrong
answer. Just try to be consistent.
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Preference for Toothpaste Brands Using Rank
Order Scaling

Form
Brand Rank Order 1. Crest _________
2. Colgate _________ 3.
Aim _________ 4. Gleem
_________ 5. Macleans
_________
6. Ultra Brite _________ 7. Close Up
_________ 8. Pepsodent _________
9. Plus White _________ 10.
Stripe _________
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Primary Scales of Measurement Interval Scale

• Numerically equal distances on the scale
  represent equal values in the characteristic
  being measured.
• It permits comparison of the differences between
  objects.
• The location of the zero point is not fixed.
  Both the zero point and the units of measurement
  are arbitrary.
• Statistical techniques that may be used include
  all of those that can be applied to nominal and
  ordinal data, and in addition the arithmetic
  mean, standard deviation, and other statistics
  commonly used in marketing research.

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Primary Scales of Measurement Ratio Scale

• Possesses all the properties of the nominal,
  ordinal, and interval scales.
• It has an absolute zero point.
• It is meaningful to compute ratios of scale
  values.
• All statistical techniques can be applied to
  ratio data.

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Illustration of Primary Scales of Measurement
Nominal Ordinal
Ratio Scale
Scale
Scale Preference

spent last No. Store
Rankings
3 months 1. Lord
Taylor 2. Macys 3. Kmart 4. Richs 5. J.C.
Penney 6. Neiman Marcus 7.
Target 8. Saks Fifth Avenue 9. Sears
10.Wal-Mart
IntervalScale Preference Ratings 1-7 11-17
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Primary Scales of Measurement
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Variability

• In order to know whether a difference between two
  means is important, we need to know how much the
  scores vary around the means.

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Variability

• Holding the difference between the means constant
• With High Variability the two groups nearly
  overlap
• With Low Variability the two groups show very
  little overlap

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Measuring Variability

• Since we need a measure of variability why not
  just compute the average difference between each
  score and the group mean?
• A high variability group would have a large
  average difference.
• A low variability group would have a small
  average difference.
• Lets try it with a 5 member Group!

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Measuring Variability
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Measuring Variability
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Measuring Variability
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Measuring Variability

• It does not work!
• The positive and negative difference cancel each
  other out.
• The sum of the differences will always be zero.
• What if we squared the differences so that they
  would always be positive?

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Measuring Variability
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Measuring Variability

• Now were getting somewhere.
• In our example, the average squared difference is
  equal to 8.
• The average squared difference is called the
  Variance.
• High Variability Groups will have a high
  Variance.
• Low Variability Groups will have a low Variance.

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Measuring Variability

• Medium Variance
• High Variance
• Low Variance

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Measuring Variability

• Usually its easier to work with the square root
  of the variance.
• This statistic is called the Standard Deviation.
• Most statistical tests have a short cut formula
  for computing the Standard Deviation.

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The T-Test

• The logic of the T-Test is simple
• The T Statistic The Difference Between the Two
  Groups Means Divided by Standard Deviation of
  the Difference.

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T-Test

• The formula for the Standard Deviation of the
  Difference is very straightforward

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T-Test

• The final formula for the T statistic is

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t-tests on SPSS

• A t-test is a measure of the difference between
  the means of two variables. The test is usually
  applied to interval and ratio data types. For
  example differences between two factors (1 and
  2). This test can be undertaken with different
  samples, using the Statistics - Compare Means
  menu items where either Independent Samples or
  Paired Samples can be selected for appropriate
  variables. You will observe the Paired Samples
  t-test for factor 1 and factor 2

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t-tests in SPSS
.
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Paired t-test in SPSS
.
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t-test Printout

interpretation
from this
value
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• t-test Interpretation
• In a t-test the null hypothesis should state
  that there is no difference between means. The
  alternative hypothesis should state that the two
  means are different
• Ho - There is no difference between the means
• H1 - The two means are different
• as the significance on the printout is below
  0.05 (i.e. .00000) you should reject the null
  hypothesis (at 5 level) and accept the
  alternative (If it would have been greater than
  0.05 the reverse would be the case).

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One-WayANOVA on SPSS

• A One-Way Analysis of Variance (ANOVA) is
  similar to a t-test, in that it is concerned with
  differences in means, but the test can be applied
  on two or more means. The test is usually applied
  to interval and ratio data types. For example
  differences between two factors (1 and 2). The
  test can be undertaken using the Statistics -
  Compare Means - One-Way ANOVA menu items, then
  select for appropriate variables. You will
  observe the One-Way ANOVA for factor 1 and factor
  2

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ANOVA Dialogue in SPSS
.
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ANOVA Printout (1)

interpretation
from this
value
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ANOVA Printout (2)

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• ANOVA Interpretation
• In ANOVA the null hypothesis should state that
  there is no difference between means. The
  alternative hypothesis should state that at least
  one of the means is different
• Ho - There is no difference between the means
• H1 - At least one of the means is different
• as the significance on the printout is below
  0.05 (i.e. .00000) you should reject the null
  hypothesis (at the 5 level) and accept the
  alternative (If it would have been greater than
  0.05 the reverse would be the case).