Title: Statistics%20for%20the%20Social%20Sciences
1Statistics for the Social Sciences
Introductions
2Outline (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
3Outline (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
4Describing distributions
- Distributions are typically described with three
properties - Shape unimodal, symmetric, skewed, etc.
- Center mean, median, mode
- Spread (variability) standard deviation, variance
5Describing distributions
- Distributions are typically described with three
properties - Shape unimodal, symmetric, skewed, etc.
- Center mean, median, mode
- Spread (variability) standard deviation, variance
6Which 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.
7Which 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
8The 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
9The 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?
10The 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?
Need to take into account the number of scores in
each mean
11The 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
12The 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
13The 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
14Describing distributions
- Distributions are typically described with three
properties - Shape unimodal, symmetric, skewed, etc.
- Center mean, median, mode
- Spread (variability) standard deviation, variance
15Variability 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
16Standard 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.
17Computing 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
18Computing 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
19Computing 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
20Computing 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
21Computing 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
22Computing 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
23Computing standard deviation (population)
- Step 4 Compute the standard deviation. Take the
square root of the population variance.
24Computing 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
25Computing 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
26Computing 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
27Computing standard deviation (sample)
- Step 2 Determine the sum of the squared
deviations (SS).
28Computing 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
29Computing 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
30Computing standard deviation (sample)
- Step 4 Determine the standard deviation
31Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
32Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
33Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
34Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
35Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
36Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
37Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
38Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
39Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
40Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
41Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
42Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
43Properties of means and standard deviations
Mean
Standard deviation
- Change/add/delete a given score
changes
changes
- Add/subtract a constant to each score
changes
44Properties 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
45Properties 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
46Properties 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
47Locating 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
48Locating a score
X1 - 100 62
X1 162
X2 57
X2 - 100 -43
49Locating a score
X1 - 100 62
X1 162
X2 57
X2 - 100 -43
50Transforming 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.
51Transforming 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
52Transforming 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.
53Properties of the z-score distribution
0
54Properties of the z-score distribution
150
50
0
Xmean 100
1
55Properties of the z-score distribution
150
50
1
0
Xmean 100
1
X1std 150
-1
56Properties 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.
57From 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