Measurement and Scaling Fundamentals' - PowerPoint PPT Presentation

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Measurement and Scaling Fundamentals'

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Title: Measurement and Scaling Fundamentals'


1
  • Measurement and Scaling Fundamentals.
  • Comparison Tests.

2
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.
  •  

3
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.

4
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
7
3
8
Finish
8.2
9.1
9.6
15.2
14.1
13.4
5
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.

6
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.

7
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

8
Are Women More Likely to Watch Than Men?
  • You can understand the data in crosstabs
  • But how do you read it?

9
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?

10
Are Watchers More Likely to Be Female?
  • Calculate proportions for rows
  • Compare across columns
  • How might this be different?

11
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

12
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?

13
Shop Data
  • Do a crosstab on any two categorical variables
  • Indicate frequencies and percentages
  • Chi-Square significance test

14
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

15
Crosstabs in SPSS

16
CrosstabsTable

17
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)

18
Chi-Square Dialogue Box

19
Chi-square from Crosstabs (2)

interpretation
from this
value
Chi-square value
20
  • 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).

21
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.

22
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.

23
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.
24
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 _________
25
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.

26
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.

27
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
28
Primary Scales of Measurement
29
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.

30
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

31
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!

32
Measuring Variability
33
Measuring Variability
34
Measuring Variability
35
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?

36
Measuring Variability
37
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.

38
Measuring Variability
  • Medium Variance
  • High Variance
  • Low Variance

39
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.

40
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.

41
T-Test
  • The formula for the Standard Deviation of the
    Difference is very straightforward

42
T-Test
  • The final formula for the T statistic is

43
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

44
t-tests in SPSS
.
45
Paired t-test in SPSS
.
46
t-test Printout

interpretation
from this
value
47
  • 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).

48
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

49
ANOVA Dialogue in SPSS
.
50
ANOVA Printout (1)

interpretation
from this
value
51
ANOVA Printout (2)

52
  • 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).
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