Download Social Science Reasoning Using Statistics

Reasoning in Psychology
Using Statistics
Psychology 138
2015
• Quiz 3 due Friday, Feb. 20 at 11:59 pm
– Covers
• Tables and graphs
• Measures of center
• Measures of variability
• You don’t need SPSS, but may want to have
a calculator handy
Announcements
Reasoning in Psychology
Using Statistics
• In addition to pictures of the distribution,
numerical summaries are also typically presented.
– Numeric Descriptive Statistics
• Shape: skew and kurtosis
• Measures of Center: mode, median, mean
• Measures of Variability (Spread): Range, Inter-quartile range,
Standard Deviation (& variance)
Descriptive statistics
Reasoning in Psychology
Using Statistics
• In addition to pictures of the distribution,
numerical summaries are also typically presented.
– Numeric Descriptive Statistics
Today’s
focus
• Shape: skew and kurtosis
• Measures of Center: mode, median, mean
• Measures of Variability (Spread): Range, Inter-quartile range,
Standard Deviation (& variance)
Descriptive statistics
Reasoning in Psychology
Using Statistics
• Quantitative measure of how spread out or
clustered scores are
– the degree of “differentness” of the scores in the
distribution.
• High variability: scores
differ a lot
• Low variability: scores are all similar
Variability of a distribution
Reasoning in Psychology
Using Statistics
• Simplest measure of variability
– Range = Maximum - Minimum
– Disadvantage: based solely on two most extreme values
Range
Reasoning in Psychology
Using Statistics
• IQR: Alternative measure of variability
– Median (50%tile): 1/2 distribution on one side & 1/2 on
other side.
– 25%tile?
– 75%tile?
Median
25%tile
75%tile
25%
25%
Inter-Quartile Range
Reasoning in Psychology
Using Statistics
25%
25%
• IQR: Alternative measure of variability
– IQR = 3rd quartile - 1st quartile
– Middle 50% of the scores
Median
25%tile
75%tile
25%
25%
25%
25%
IQR
Inter-Quartile Range
Reasoning in Psychology
Using Statistics
• IQR: Alternative measure of variability
– IQR = 3rd quartile - 1st quartile
– Middle 50% of the scores
– Works well for skewed distributions
Inter-Quartile Range
Reasoning in Psychology
Using Statistics
• IQR: Alternative measure of variability
IQR
- IQR more representative than Range
- Extreme scores (i.e., outliers) have
less influence (robust)
- Disadvantage: all scores not
represented (only the middle 50%)
25% 25% 25% 25%
IQR
Inter-Quartile Range
Reasoning in Psychology
Using Statistics
• Most utilized & important measure of variability
– Standard deviation measures how far off all of
the scores are from a standard
– Standard deviation =
average of deviations
Typically mean is
used as standard.

Standard Deviation & Variance
Reasoning in Psychology
Using Statistics
• Step 1: For measure of deviation, subtract
population mean from every score.
Our population
2, 4, 6, 8
å X 2 + 4 + 6 + 8 20
m=
=
= = 5.0
N
4
4
X - μ = deviation scores
-3
1 2 3 4 5 6 7 8 9 10

2 - 5 = -3
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Step 1: For measure of deviation, subtract
population mean from every score.
Our population
2, 4, 6, 8
å X 2 + 4 + 6 + 8 20
m=
=
= = 5.0
N
4
4
X - μ = deviation scores
-1
1 2 3 4 5 6 7 8 9 10

2 - 5 = -3
4 - 5 = -1
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Step 1: For measure of deviation, subtract
population mean from every score.
Our population
2, 4, 6, 8
å X 2 + 4 + 6 + 8 20
m=
=
= = 5.0
N
4
4
1
1 2 3 4 5 6 7 8 9 10
X - μ = deviation scores
2 - 5 = -3
4 - 5 = -1

6 - 5 = +1 (balances off -1)
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Step 1: For measure of deviation, subtract
population mean from every score.
Our population
2, 4, 6, 8
å X 2 + 4 + 6 + 8 20
m=
=
= = 5.0
N
4
4
X - μ = deviation scores
2 - 5 = -3
4 - 5 = -1
3
1 2 3 4 5 6 7 8 9 10

6 - 5 = +1
8 - 5 = +3
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Characteristic of average of
deviation scores:
Total of the deviations
must equal 0
X - μ = deviation scores
2 - 5 = -3
6 - 5 = +1
4 - 5 = -1
8 - 5 = +3
-3 + -1 + 1 + 3 =
-4 + 4 = 0
Mean is the balancing
point, so 2 sides are
equal, that is, total
deviations on each side
are equal
1 2 3 4 5 6 7 8 9 10

Good check of computations
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Step 2: Preserve all deviations, positive &
negative. How?
– Square the deviations!
– Add to get Sum of Squared
deviations, “Sum of Squares”
(SS)
SS = Σ (X – μ)2
MEMORIZE!
Note Order Of Operations!
X - μ = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
SS = (-3)2 + (-1)2 + (+1)2 + (+3)2
= 9 + 1 + 1 + 9 = 20
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Step 3: Compute Variance.
– To get mean squared deviation, divide SS by
number of scores
Variance = σ2 = SS/N
MEMORIZE!
(Population)
Our population
So N = 4
2, 4, 6, 8
SS = å ( X - m ) = 20
2
= 20/4 = 5.0
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• Step 4: Compute the Standard Deviation
– Variance = average squared deviation score
– Get average deviation score by taking square root of
variance
MEMORIZE!
Standard Deviation = σ = s 2 =
(Population)
å( X - m)
2
N
s = 5.0 = 2.24
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• A good idea: check your answer, does it appear to
be in the right ballpark?
Our population
2, 4, 6, 8
+2.24
-2.24
1 2 3 4 5 6 7 8 9 10
X - μ = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3

s = 5.0 = 2.24
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• To review (computing standard deviation for a population):
–
–
–
–
Step 1: compute deviation scores
Step 2: compute SS
Step 3: determine variance ( σ2 squared standard deviation)
Step 4: determine standard deviation (square root of
variance)
Standard Deviation = σ =
2
X
m
(
)
å
N
Computing standard deviation (population)
Reasoning in Psychology
Using Statistics
• For a sample: The basic procedure is the same.
–
–
–
–
Step 1:
Step 2:
Step 3:
Step 4:
compute deviation scores
compute SS
This step is different
determine variance
determine standard deviation
Computing standard deviation (sample)
Reasoning in Psychology
Using Statistics
• Step 1: Compute deviation scores
– subtract sample mean from every score
Our sample
2, 4, 6, 8
Notational differences
å X 2 + 4 + 6 + 8 20
X=
=
= = 5.0
n
4
4
X - X = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
1 2 3 4 5 6 7 8 9 10
X
Numerically everything is
the same as before
Computing standard deviation (sample)
Reasoning in Psychology
Using Statistics
• Step 2: Compute Sum of Squared deviations (SS).
X - X = deviation scores
2 - 5 = -3
4 - 5 = -1
6 - 5 = +1
8 - 5 = +3
SS = Σ (X - X)2
= (-3)2 + (-1)2 + (+1)2 + (+3)2
= 9 + 1 + 1 + 9 = 20
Apart from notational differences, procedure
same as before
Computing standard deviation (sample)
Reasoning in Psychology
Using Statistics
• Step 3: Determine Variance
Recall:
Population variance = σ2 = SS/N
Samples’ variability smaller than
population’s
X4
X1  X3
X2
Computing standard deviation (sample)
Reasoning in Psychology
Using Statistics
• Step 3: Determine Variance
Recall:
Population variance = σ2 = SS/N
Samples’ variability smaller than
population’s
To estimate population variance (what we are
interested in) from sample, correct by dividing by (n1) instead of just n
Sample variance =
s2
SS
=
( n -1)
Computing standard deviation (sample)
Reasoning in Psychology
Using Statistics
• Step 4: Determine the standard deviation
Standard Deviation = s = s =
2
(Sample)
å( X - X )
2
n -1
MEMORIZE!
MOST IMPORTANT!!
NOTE: SPSS, Excel, other spreadsheets, and
calculators use this formula for standard deviation.
Computing standard deviation (sample)
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
– May change the mean and (if adding or subtracting) the
2
number of scores (n or N)
( X - m )2
X-X
å
N
å(
n -1
)
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– All of the scores change by the same constant.
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– All of the scores change by the same constant.
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– All of the scores change by the same constant.
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– All of the scores change by the same constant.
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– All of the scores change by the same constant.
– But so does the mean
Xnew
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– It is as if you just pick up the distribution and move it over, but the
spread (variability) stays the same
Xold Xnew
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the standard
deviation will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
– Looking at a numerical example. (subtract 1 from every score)
Original sample Original mean
2, 4, 6, 8
5
New sample
New mean
1, 3, 5, 7
4
2 - 5 = -3
4 - 5 = -1
1 - 4 = -3
3 - 4 = -1
6 - 5 = +1 Original SS
8 - 5 = +3
20
5 - 4 = +1 Original SS
7 - 4 = +3
20
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the mean
will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
• Multiply (or divide) each score by a constant, then
the standard deviation will change by being
multiplied by that constant.
20 21 22 23 24
21 - 22 = -1
23 - 22 = +1
s=
X
(-1)2
(+1)2
å( X - X )
n -1
2
= 2 = 1.41
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Change/add/delete a given score, then the mean
will change.
• Add/subtract a constant to each score, then the
standard deviation will NOT change.
• Multiply (or divide) each score by a constant, then
the standard deviation will change by being
multiplied by that constant.
40 42 44 46 48
42 - 44 = -2
46 - 44 = +2
s=
X
å( X - X )
n -1
(-2)2
(+2)2
2
= 8 = 2.82
Sold=1.41
Characteristics of a standard deviation
Reasoning in Psychology
Using Statistics
• Extreme scores: Range is most affected, IQR is
least affected
• Sample size: Range tends to increase as n
increases, IQR & s do not
• With open-ended distributions, one cannot even
compute the Range or σ, so the IQR is the only
option
• Range is unstable when you repeatedly sample
from the same population, but the IQR & σ are
stable and tend not to fluctuate.
When to use which
Reasoning in Psychology
Using Statistics
• Today’s lab: compute several measures of
variability both by hand and using SPSS
• Questions?
Wrap up
Reasoning in Psychology
Using Statistics