Sociology 690

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Sociology 690 Data Analysis

• Simple Quantitative
• Data Analysis

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Four Issues in Describing Quantity

• 1. Grouping/Graphing Quantitative Data
• 2. Describing Central Tendency
• 3. Describing Variation
• 4. Describing Co-variation

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1. Grouping Quantitative Data
If there are a large number of quantitative
scores, one would not simply create a raw score
frequency distribution, as that would contain too
many unique scores and, therefore, not fulfill
the data reduction goal.

• Intervals and Real Limits
• Widths and midpoints
• Graphing grouped data

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Grouping Data - Intervals

• To group quantitative data, three rules are
  followed
• 1. Make the intervals no greater than the most
  amount of information you are willing to lose.
• 2. Make the intervals in multiples of five.
• 3. Make the distribution intervals few enough to
  be internalized at a glance.

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Grouping Data Intervals Example

• If these are the scores on a midterm
• 9,13,18,19,22,25,31,34,35,36,36,38,41,43,44,45
• The corresponding grouped frequency distribution
  would look like
• i fi
• 01-10 1
• 11-20 3
• 21-30 2
• 31-40 6
• 41-50 4
• Total 16

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Grouping Data - Real Limits

• This implies the need for real limits as there
  are gaps in these intervals. The real limits
  of an interval are characterized by numbers that
  are plus and minus one-half unit on each side of
  stated limits
• For example
• the interval 11-20 becomes 10.5 20.5
• the interval 3.5 4.5 becomes 3.45 4.55

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Grouped Data Width and Midpoint

• The width of an interval is simply the difference
  between the upper and lower real limits.
• e.g. 11-20 ? 20.5 10.5 10
• The midpoint is determined by calculating the
  interval width, dividing it by 2, and adding that
  number to the lower real limit.
• e.g. 10/2 10.5 15.5

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Graphing Grouped Data

• A Quantitative version of a bar graph is called
  an Histogram
• When the frequencies are connected via a
  line, it is call a frequency polygon

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2. Describing Central Tendency
But we can do more than simply create a frequency
distribution. We can also describe how these
observations bunch up and how they
distribute. Describing how they bunch up
involves measures of

• Modes
• Medians
• Means
• Skew

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Central Tendency - Modes

• The mode for raw data is simply the most frequent
  score e.g. 2,3,5,6,6,8. The mode is 6.
• The mode for grouped data is the midpoint of the
  interval containing the highest frequency
  (35.5 here)

i fi 01-10
1 11-20 3 21-30 2 31-40
6 41-50 4 Total 16
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Central Tendency - Medians

• The median for raw data is simply the score
  at the middle position. This involves taking the
  (N1)/2 position and stating the associated value
  attached to it
• e.g. 2,3,5,6,8 (51)/2 ? the third position
  score
• The third position score is 5.
• e.g. 2,3,5,8 (41)/2 ? the 2.5 position
  score
• The 2.5 position score is (35)/2 4

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Medians for Grouped Data

• The median for grouped data is
• For our previous distribution of scores,
• the answer would be
• 30.5 ((16/2-6)/6)10
• 30.5 3.33 33.83

i fi 01-10
1 11-20 3 21-30 2 31-40
6 41-50 4 Total 16
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Central Tendency - Mean

• For raw data, the mean is simply the sum of the
  values divided by N
• Suppose Xi 2,3,5,6
• The mean would be 16/4 4

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Means for Grouped Data

• For grouped data, the mean would be the sum of
  the frequencies times midpoints for each
  interval, that sum divided by N
• For our previous distribution,
• the answer would be
• i fi
• 01-10 1
• 11-20 3
  1(5.5)3(15.5)2(25.5)6(35.5)
• 21-30 2 4(45.5)
  498 / 16 31.125
• 31-40 6
• 41-50 4
• Total 16

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3. Describing Variation

• Range
• Mean Deviation
• Variance
• Standard Scores (Z score)

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Describing Variation - Range

• The Range for raw scores is the highest minus the
  lowest score, plus one (i.e. inclusive)
• The Range for grouped scores is the upper real
  limit of the highest interval minus the lower
  real limit of the lowest interval. In the case
  of our
• previous distribution this would be
• 50.5 - .5 50

i fi 01-10
1 11-20 3 21-30 2 31-40
6 41-50 4 Total 16
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Describing Variation Mean Deviation

• The mean deviation is the sum of all deviations,
  in absolute numbers, divided by N.
• Consider the set of observations, 6,7,9,10 The
  mean is 8 and the MD is (6-87-89-810-8)
  /4 6/4 1.5

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Mean Deviation for Grouped Data

• Again grouped data implies we substitute
  frequencies and midpoints for values

The mean would be 50,000 (satisfy yourself that
that is true) and the MD would be (638-50)
(843-50) (1248-50) (1253-50)
(858-50) (463-50) 725624366452
304/50 6.080 x 1000 6,080
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Variation The Variance

• The variance for raw data is the sum of the
  squared deviations divided by N
• Consider the set Xi 6,7,9,10 The mean is 8
  and the variance is ((6-8)2(7-8)2(9-8)2(10-8)2)
  /4 2.5

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Variance for Grouped Data

• Frequencies and midpoints are still
  substituted for the values of Xi.

Again the mean is 50 and the Variance is
6(38-50)2 8(43-50)2 12(48-50)2 12 (53-50)2
8(58-50)2 4(63-60)2 1014 392 48 108
512 676 2690 / 50 53.8 x 1000 53,800.
The Standard Deviation is the sq root of this.
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4. Covariance and Correlation

• The Definition and Concept
• The Formula
• Proportional Reduction in Error and r2

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Correlation Definition and Concept

• Visually we can observe the co-variation of
  two variables as a scatter diagram where the
  abscissa and ordinate are the quantitative
  continua and the points are simultaneously
  mapping of the pairs of scores.

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Correlation - Formula

• Think of the correlation as a proportional
  measure of the relationship between two
  variables. It consists of the co-variation
  divided by the average variation

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Correlation and P.R.E.
Consider this scatter diagram. The proportion of
variation around the Y mean (variation before
knowing X), less the proportion of variation
around the regression line (variation after
knowing x) is r2
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Partial Correlation
IV. Quantitative Statistical Example of
Elaboration
Step 1 Construct the zero order

Pearsons correlations (r).
Assume rxy .55 where x divorce rates and y
suicide rates.
Further, assume that unemployment rates (z) is
our control variable and that rxz .60 and ryz
.40
Step 2 Calculate the partial correlation
(rxy.z)


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Therefore, Z accounts for (.30-.18) or 12 of Y
and (.12/.30) or 40 of the relationship between
XY
Before z (rxy)2 .30
Step 3 Draw conclusions
After z (rxy.z)2 .18