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Statistics%20for%20the%20Social%20Sciences

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Variables: IV, DV, scales of measurement. Discuss each variable and it's scale of measurement ... Shape: unimodal, symmetric, skewed, etc. Center: mean, median, mode ... – PowerPoint PPT presentation

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Title: Statistics%20for%20the%20Social%20Sciences


1
Statistics for the Social Sciences
  • Psychology 340
  • Fall 2006

Introductions
2
Outline (for week)
  • Variables IV, DV, scales of measurement
  • Discuss each variable and its scale of
    measurement
  • Characteristics of Distributions
  • Using graphs
  • Using numbers (center and variability)
  • Descriptive statistics decision tree
  • Locating scores z-scores and other
    transformations

3
Outline (for week)
  • Variables IV, DV, scales of measurement
  • Discuss each variable and its scale of
    measurement
  • Characteristics of Distributions
  • Using graphs
  • Using numbers (center and variability)
  • Descriptive statistics decision tree
  • Locating scores z-scores and other
    transformations

4
Describing distributions
  • Distributions are typically described with three
    properties
  • Shape unimodal, symmetric, skewed, etc.
  • Center mean, median, mode
  • Spread (variability) standard deviation, variance

5
Describing distributions
  • Distributions are typically described with three
    properties
  • Shape unimodal, symmetric, skewed, etc.
  • Center mean, median, mode
  • Spread (variability) standard deviation, variance

6
Which center when?
  • Depends on a number of factors, like scale of
    measurement and shape.
  • The mean is the most preferred measure and it is
    closely related to measures of variability
  • However, there are times when the mean isnt the
    appropriate measure.

7
Which center when?
  • Use the median if
  • The distribution is skewed
  • The distribution is open-ended
  • (e.g. your top answer on your questionnaire is 5
    or more)
  • Data are on an ordinal scale (rankings)
  • Use the mode if the data are on a nominal scale

8
The Mean
  • The most commonly used measure of center
  • The arithmetic average
  • Computing the mean
  • The formula for the population mean is (a
    parameter)
  • The formula for the sample mean is (a
    statistic)
  • Note your book uses M to denote the mean in
    formulas

9
The Mean
  • Number of shoes
  • 5, 7, 5, 5, 5
  • 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25,
    6, 35, 20, 20, 20, 25, 15
  • Suppose we want the mean of the entire group?

  • Can we simply add the
    two means together and divide by 2?
  • NO. Why not?

10
The Weighted Mean
  • Number of shoes
  • 5, 7, 5, 5, 5, 30, 11, 12, 20, 14, 12, 15, 8, 6,
    8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15
  • Suppose we want the mean of the entire group?
    Can we simply add the two means together and
    divide by 2?
  • NO. Why not?

Need to take into account the number of scores in
each mean
11
The Weighted Mean
  • Number of shoes
  • 5, 7, 5, 5, 5, 30, 11, 12, 20, 14, 12, 15, 8, 6,
    8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15

Lets check
12
The median
  • The median is the score that divides a
    distribution exactly in half. Exactly 50 of the
    individuals in a distribution have scores at or
    below the median.
  • Case1 Odd number of scores in the distribution

Step1 put the scores in order
Step2 find the middle score
  • Case2 Even number of scores in the distribution

Step1 put the scores in order
Step2 find the middle two scores
Step3 find the arithmetic average of the two
middle scores
13
The mode
  • The mode is the score or category that has the
    greatest frequency.
  • So look at your frequency table or graph and pick
    the variable that has the highest frequency.

so the mode is 5
so the modes are 2 and 8
Note if one were bigger than the other it would
be called the major mode and the other would be
the minor mode
14
Describing distributions
  • Distributions are typically described with three
    properties
  • Shape unimodal, symmetric, skewed, etc.
  • Center mean, median, mode
  • Spread (variability) standard deviation, variance

15
Variability of a distribution
  • Variability provides a quantitative measure of
    the degree to which scores in a distribution are
    spread out or clustered together.
  • In other words variabilility refers to the degree
    of differentness of the scores in the
    distribution.
  • High variability means that the scores differ
    by a lot
  • Low variability means that the scores are all
    similar

16
Standard deviation
  • The standard deviation is the most commonly used
    measure of variability.
  • The standard deviation measures how far off all
    of the scores in the distribution are from the
    mean of the distribution.
  • Essentially, the average of the deviations.

17
Computing standard deviation (population)
  • Step 1 To get a measure of the deviation we need
    to subtract the population mean from every
    individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
18
Computing standard deviation (population)
  • Step 1 To get a measure of the deviation we need
    to subtract the population mean from every
    individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
4 - 5 -1
19
Computing standard deviation (population)
  • Step 1 To get a measure of the deviation we need
    to subtract the population mean from every
    individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
6 - 5 1
4 - 5 -1
20
Computing standard deviation (population)
  • Step 1 Compute the deviation scores Subtract
    the population mean from every score in the
    distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
6 - 5 1
Notice that if you add up all of the deviations
they must equal 0.
4 - 5 -1
8 - 5 3
21
Computing standard deviation (population)
  • Step 2 Get rid of the negative signs. Square the
    deviations and add them together to compute the
    sum of the squared deviations (SS).

SS ? (X - ?)2
(-3)2
(-1)2
(1)2
(3)2
9 1 1 9 20
22
Computing standard deviation (population)
  • Step 3 Compute the Variance (the average of the
    squared deviations)
  • Divide by the number of individuals in the
    population.

variance ?2 SS/N
23
Computing standard deviation (population)
  • Step 4 Compute the standard deviation. Take the
    square root of the population variance.

24
Computing standard deviation (population)
  • To review
  • Step 1 compute deviation scores
  • Step 2 compute the SS
  • SS ? (X - ?)2
  • Step 3 determine the variance
  • take the average of the squared deviations
  • divide the SS by the N
  • Step 4 determine the standard deviation
  • take the square root of the variance

25
Computing standard deviation (sample)
  • The basic procedure is the same.
  • Step 1 compute deviation scores
  • Step 2 compute the SS
  • Step 3 determine the variance
  • This step is different
  • Step 4 determine the standard deviation

26
Computing standard deviation (sample)
  • Step 1 Compute the deviation scores
  • subtract the sample mean from every individual in
    our distribution.

2 - 5 -3
6 - 5 1
4 - 5 -1
8 - 5 3
27
Computing standard deviation (sample)
  • Step 2 Determine the sum of the squared
    deviations (SS).

28
Computing standard deviation (sample)
  • Step 3 Determine the variance

Recall
Population variance ?2 SS/N
The variability of the samples is typically
smaller than the populations variability
29
Computing standard deviation (sample)
  • Step 3 Determine the variance

Recall
Population variance ?2 SS/N
The variability of the samples is typically
smaller than the populations variability
To correct for this we divide by (n-1) instead of
just n
30
Computing standard deviation (sample)
  • Step 4 Determine the standard deviation

31
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
32
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

33
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

34
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

35
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

36
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
37
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
38
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
39
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
40
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
41
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
42
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
43
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
  • Add/subtract a constant to each score

changes
44
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
No change
changes
  • Add/subtract a constant to each score

45
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
No change
changes
  • Add/subtract a constant to each score
  • Multiply/divide a constant to each score

21 - 22 -1
(-1)2
23 - 22 1
(1)2
46
Properties of means and standard deviations
Mean
Standard deviation
  • Change/add/delete a given score

changes
changes
No change
changes
  • Add/subtract a constant to each score
  • Multiply/divide a constant to each score

changes
changes
42 - 44 -2
(-2)2
46 - 44 2
(2)2
Sold1.41
s
47
Locating a score
  • Where is our raw score within the distribution?
  • The natural choice of reference is the mean
    (since it is usually easy to find).
  • So well subtract the mean from the score (find
    the deviation score).
  • The direction will be given to us by the negative
    or positive sign on the deviation score
  • The distance is the value of the deviation score

48
Locating a score
X1 - 100 62
X1 162
X2 57
X2 - 100 -43
49
Locating a score
X1 - 100 62
X1 162
X2 57
X2 - 100 -43
50
Transforming a score
  • The distance is the value of the deviation score
  • However, this distance is measured with the units
    of measurement of the score.
  • Convert the score to a standard (neutral) score.
    In this case a z-score.

51
Transforming scores
  • A z-score specifies the precise location of each
    X value within a distribution.
  • Direction The sign of the z-score ( or -)
    signifies whether the score is above the mean or
    below the mean.
  • Distance The numerical value of the z-score
    specifies the distance from the mean by counting
    the number of standard deviations between X and ?.

X1 162
X2 57
52
Transforming a distribution
  • We can transform all of the scores in a
    distribution
  • We can transform any all observations to
    z-scores if we know either the distribution mean
    and standard deviation.
  • We call this transformed distribution a
    standardized distribution.
  • Standardized distributions are used to make
    dissimilar distributions comparable.
  • e.g., your height and weight
  • One of the most common standardized distributions
    is the Z-distribution.

53
Properties of the z-score distribution
0
54
Properties of the z-score distribution
150
50
0
Xmean 100
1
55
Properties of the z-score distribution
150
50
1
0
Xmean 100
1
X1std 150
-1
56
Properties of the z-score distribution
  • Shape - the shape of the z-score distribution
    will be exactly the same as the original
    distribution of raw scores. Every score stays in
    the exact same position relative to every other
    score in the distribution.
  • Mean - when raw scores are transformed into
    z-scores, the mean will always 0.
  • The standard deviation - when any distribution
    of raw scores is transformed into z-scores the
    standard deviation will always 1.

57
From z to raw score
  • We can also transform a z-score back into a raw
    score if we know the mean and standard deviation
    information of the original distribution.

X (-0.60)( 50) 100
X 70
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