1. Chapter 8 A Statistics Primer
Winston Jackson and Norine Verberg
Methods: Doing Social Research, 4e

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Level of Measurement
Measures can be designed to have a higher, more complex level or a more basic, rudimentary level
Influenced by how the variable is conceptualized
Gender: can have only two categories (males and females)
Age: can be age (year of birth) or age groups (age categories, e.g., adolescent, young adult, etc.)
Influences choice of statistical analysis
Shown on Table 8.11 on page 222
Related to measurement error (Chapter 13)
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3. Three Levels of Measurement
Nominal: involves no underlying continuum; assignment of numeric values arbitrary
Examples: religious affiliation, gender, etc
Ordinal: implies an underlying continuum; values are ordered but intervals are not equal.
Examples: community size, Likert items, etc.
Ratio: involves an underlying continuum; numeric values assigned reflect equal intervals; zero point aligned with true zero.
Examples: weight, age in years, % minority
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4. Examples of Nominal Level Measures
Do you have a valid driver’s licence? [ ] Yes
[ ] No
Your sex (Circle number of your answer)
1 Male
2 Female
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5. Example of Ordinal Level Measure
The population of the place I considered my hometown when growing up was:
Rural area 1
town under 5, 
5,000 to 19, 
20,000 to 99, 
100,000 to 999, 
1,000,000 or over 6
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6. Examples of Ratio Level Measures
In the following items, circle a number to indicate the extent
to which you agree or disagree with each statement.
I would quit my present job if I won $1,000,000 through a lottery.
Strongly Disagree Strongly Agree
I would be satisfied if my child followed the same type of career as I have.
Strongly Somewhat Neither Agree Somewhat Strongly
Disagree Disagree nor Disagree Agree Agree
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7. Describing an Individual Variable
Statistics provide ways to describe and compare sets of observations (e.g., income levels, infant mortality, morbidity, crime, etc.)
Two common ways of describing a distribution (a set of scores in a data set)
Measures of central tendency
Measures of dispersion
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8. Measures of Central Tendency
A number that typifies the central scores of a set of values
Mean
Median
Mode
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Mean
The arithmetic average or average
Calculated by summing values and dividing by number of cases
Used to describe central tendency of ratio level data
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Median
The midpoint
Used to describe central tendency of ordinal level data
Calculated by ordering a set of values and then using the middlemost value (in cases of two middle values, calculate the mean of the two values).
Often used when a data set has extreme cases
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11. Table 8.7 Median for Extreme Values
CASE #
$ INCOME 1.
5,400 2.
6,600 3.
7,700 4.
10,200 5.
13,400 6.
16,400 7.
16,700 8.
18,300
← $18,300 median value 9.
19,000
10.
20,000
11.
20,500
12.
22,900
13.
24,600
14.
31,500
$54,213 mean value
15.
580,000
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Mode
The most frequently occurring value
Used to describe central tendency of nominal level data (gender, religion, nationality)
TABLE 8.8 DISTRIBUTION OF RESPONDENTS BY COUNTRY
COUNTRY NUMBER PERCENT
Canada ← mode
New Zealand
Australia
TOTAL
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13. Measures of Dispersion
Indicates dispersion or variability of values
Are scores close together or spread out?
Three common measures of dispersion:
Range
Standard deviation
Variance
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14. Table 8.9 Two Grade Distributions
SUBJECT
MARY
BETH
Sociology 78 66
Psychology 80 72
Political Science 82 88
Anthropology 90
Philosophy 94
Mean1
Range2 10 28
Standard Deviation3
3.74
12.25
Variance4
14.0
150.00
1Mean = sum of values divided by number of cases
2Range = highest value – lowest value
3See computation in Table 8.10 on p. 221.
4Variance = sd2
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Range
Gap between the lowest and highest value
Computed by subtracting the lowest from the highest
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16. Standard Deviation and Variance
The standard deviation measures the average amount of deviation from the mean value of the variable
The variance is the standard deviation squared
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17. Table 8.10 Computation of Standard Deviation, Beth’s Grades
SUBJECT
GRADE
Sociology 66
66 – 82 = –16
256
Psychology 72
72 – 82 = –10
100
Political science 88
88 – 82 = 6 36
Anthropology 90
90 – 82 = 8 64
Philosophy 94
94 – 82 = 12
144
MEAN
82.0
TOTAL
Note: The “N – 1” term is used when sampling procedures have been used. When population values are used the denominator is “N.” SPSS uses N – 1” in calculating the standard deviation in the DESCRIPTIVES procedure.
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Standardizing Data
Standardizing data facilitates making comparisons between units of different size
Also, can standardize data to create variables that have similar variability (Z scores)
Standardization of data is commonly done
Several methods of standardizing data: proportions, percentages, percentage change, rates, ratios
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Proportions
A proportion represents the part of 1 that some element represents.
Proportion female = Number female
Total persons
Proportion female = 31,216
58,520
Proportion female = .53
The females represent .53 of the population
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Percentage
A percentage represents how often something happens per 100 times
A proportion may be converted to a percentage by multiplying by 100
Females constitute 53% of the population
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Percentage Change
Percentage change is a measure of how much something has changed over a given time period.
Percentage change is:
Time 2 – Time 1 x 100
Time 1
Example: percentage change in number of women in selected occupations (Table 8.13, p. 223)
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Rates
Rates represent the frequency of an event for a standard-sized unit. Divorce rates, suicide rates, crime rates are examples.
So if we had 104 suicides in a population of 757,465 the suicide rate per 100,000 would be calculated as follows:
SR = x 100,000 =
757,465
There are suicides per 100,000
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Ratios
A ratio represents a comparison of one thing to another.
So if there are 200 burglaries per 100,000 in the U.S. and 57 per 100,000 in Canada, the U.S./Canadian burglary ratio is:
US Burglary Rate = = 3.51
Canadian Burglary Rate
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Normal Distribution
Much data in the social and physical world are “normally distributed”; this means that there will be a few low values, many more clustered toward the middle, and a few high values.
Normal distributions:
symmetrical, bell-shaped curve
mean, mode, and median will be similar
68.28% of cases ± 1 standard deviation of mean
95.46% of cases ± 2 standard deviations of mean
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25. Figure 8.2 Normal Distribution Curve
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Z Scores
A Z score represents the distance from the mean, in standard deviation units, of any value in a distribution.
The Z score formula is as follows:
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27. Areas Under the Normal Curve
Can determine what proportion of cases fall between two values or above/below a value
Steps:
Draw normal curve, marking mean and SD, and including lines to represent problem
Calculate Z score(s) for the problem
Look up value on Table 8.17, page 230
Solve problem. Recall that .5 of cases fall above the mean, and .5 below the mean
Convert proportion to percentage, if needed
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Other Distributions
Not all variables are normally distributed
Bimodal: two overlapping normally distributed plots
weight (females will have lower average rates)
Leptokurtic: little variability
distribution appears tall and peaked
Platykurtic: great deal of variability
distribution appears flat and wide
Having a normal distribution is important for doing tests of statistical significance (Table 8.18)
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29. Figure 8.4 Other Distributions
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30. Describing Relationships Among Variables
Involves three important steps:
Decide which variable is to be treated as dependent variable and independent variable
Decide on the appropriate procedure for examining the relationship
Perform the analysis
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Methods
Selection of statistical method depends upon the level of measurement of the dependent and independent variables
Contingency tables: Crosstabs
Comparing means: means analysis
Correlational analysis: correlation
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32. Contingency Tables: Crosstabs
A contingency table cross-classifies cases on two or more variables to show the relation between an independent and dependent variable
Uses a nominal dependent variable and an ordinal or nominal independent variable
A standard table looks like the one on the following slide.
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33. Table 8.19 Plans to Attend University by Size of Home Community
UNIVERSITY PLANS?
RURAL
TOWN UP
TO 5,000
TOWN OVER 5,000
TOTAL N %
Plans 69
52.3 44
48.9
102
73.9
215
59.7
No plans 63
47.7 46
51.1 36
26.1
145
40.3
132
100.0 90
138
360
If a test of significance is appropriate for the table, the value for the raw Chi-Square value (which will be introduced in Chapter 9), the degrees of freedom, whether the test is one- or two-tailed, and the probability level should be indicated.
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34. Rules for Constructing a Contingency Table
In table titles, name the dependent variable first
Place dependent variable on vertical plane
Place independent variable on horizontal plane
Use variable labels that are clear
Run percentages toward the independent variable
Report percentages to one decimal point
Report statistical test results below table
Interpret the table by comparing categories of the independent variable
Minimize categories in control tables
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35. Comparing Means: Means
Used when dependent variable is ratio
Comparison to categories of independent variable (nominal or ordinal)
Both t-test and ANOVA may be used (Chapter 9)
Presentation may be as shown on the following slide.
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36. Table 8.22 Mean Income by Gender
STANDARD DEVIATION
NUMBER OF CASES
Male
$37,052
12,061
142
Female
$34,706
10,474 37
COMBINED MEAN
$36,567
11,642
179
If a test of significance is appropriate for the table, the values for the t-test or F-test value, the degrees of freedom, whether the test is one- or two-tailed, and the probability level should be indicated.
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37. Correlational Analysis: Correlation
Correlational analysis is a procedure for measuring how closely two ratio level variables co-vary together
Basis for more advanced procedures: partial correlations, multiple correlations, regression, factor analysis, path analysis and canonical analysis
Advantage: can analyze many variables (multivariate analysis) simultaneously
Relies on having ratio level measures
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Two Basic Concerns
What is the equation that describes the relation between two variables?
What is the strength of the relation between the two?
Two visual estimations procedures
The linear equation: Y = a + bX
Correlation coefficient: r
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The Linear Equation
The linear equation, Y = a + bX, describes the relation between the two variables
Components:
Y - dependent variable (e.g., starting salary)
X - independent variable (e.g., years of post-secondary education)
a - the constant, which indicates where the regression line intersects the Y-axis
b - the slope of the regression line
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40. A. The Linear Equation: A Visual Estimation Procedure
Step 1: Plot the relation on a graph
Table 8.24:
Sample data set
X Y
2 3
3 4
5 4
7 6
8 8
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41. A. The Linear Equation (cont’d)
Step 2: Insert a straight regression line
From the regression line one can estimate how much one has to change the independent variable in order to produce a unit change in the dependent variable
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42. A. The Linear Equation (cont’d)
Step 3: Observe where the regression line crosses the Y axis; this represents the constant in the equation (a = 1.33 on Figure 8.6)
Step 4: Draw a line parallel to the X axis and one parallel to the Y axis to form a right-angled triangle
Measure the lines; divide the horizontal distance into the vertical distance to compute the b value (72/91 = 0.79)
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43. A. The Linear Equation (cont’d)
Step 5: If the slope of the regression is such that it is lower on the right-hand side, the b coefficient is negative, meaning the more X, the less Y.
If the slope is negative, use a minus sign in your equation
Y= a – bX
Step 6: Write the equation:
Y = (X)
The above formula is our visually estimated equation between the two variables
Equation used to predict the value of a Y variable given a value of the X variable
Done in regression analysis
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44. B. Correlation Coefficient: A Visual Estimation Procedure
Goal: to develop a sense of what correlations of different magnitudes look like
Correlation coefficient (r) is a measure of the strength of the association between two variables
Vary from +1 to –1
Perfect correlations are rare
Usually presented by a decimal point, as in .98, .56, –.32
Negative correlations ~ negative slope
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45. Figure 8.9 Eight Linear Correlations
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46. Figure 8.9 Eight Linear Correlations (cont’d)
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47. B. Correlation Coefficient (cont’d)
Figure 8.9 graphs 8 relationships
Graphing allows you to visually estimate the strength of the association
The closer the plotted points are to the regression line (e.g., Plots 1 and 2), the higher the correlation (.99 and .85)
Greater spread (e.g., Plots 3 and 4) ~ lower correlation (.53 and .36)
Would be difficult to draw regression line if r < .36
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48. B. Correlation Coefficient (cont’d)
Plots 5 and 6: curvilinear: not linear, hence r = 0
Procedure not appropriate for curvilinear relations
Plots 7 and 8: problem plots: deviant cases
This is one of the reasons it is important to plot relationships; extreme values indicate a non-linear relationship, therefore linear regression procedure are not appropriate for studying these relationships
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49. Calculating the Correlation Coefficient
The estimation of the correlation coefficient takes two kinds of variability into account:
Variations around the regression line
Variations around the mean of Y
r2 = 1 – variations around regression
variations around mean of Y
Can calculate (see p. 245); computer programs used by most researchers today
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50. Other Correlation Procedures
Spearman Correlation
Appropriate measure of association for ordinal level variables
Partial Correlation
Measures the strength of association between two variables while simultaneously controlling for the effects of one or more additional variables
Also varies from +1 to –1
Commonly used by social researchers
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