Download Mean - Learnblock

Science & Statistics in
Psychology:Lecture 5
Variability and inferential
statistics
Dr Caleb Owens
Consultation: Wednesdays 9-10am
[email protected]
Lecture Plan
1. Theory & Evidence : Science and pseudoscience; the
importance of a rationale
2. The power of a name : Measurement and constructs
3. A thousand zeros : Types of research design; internal
and external validity
4. Predictions : Hypotheses in science; the null
hypothesis; the importance of a disprovable hypothesis
5. The coin toss : Understanding variability in sampling
and measurement; probability and the appeal to
ignorance
6. Too much of a good thing : Statistical power; Practical
significance
Build up knowledge about
current theories and evidence
Design experiment
Conceive of experimental hypothesis
Test null hypothesis
Decide if findings are
statistically significant
Decide if findings are
practically significant
Draw conclusions
Lecture 5 - Outline
• Descriptive statistics
– Central tendency
• Mean
• Mode
• Median
– Variability
• Range
• Variance
• Standard deviation
• Distributions
– Describing distributions (skew)
– The normal distribution
– Deviation from the mean and probability
• Inferential statistics
– The sampling distribution
– The P-value
– Making a statistical decision
Descriptive statistics
• Summarize a collection of data
– How many scores were there? (n)
– What is the difference between the highest
and lowest score (range)
– What is the most common score (mode)
– What is the ‘middle’ score (median)
– What number is least different to all the
scores? (mean/average)
Descriptive statistics: Measures of
Central Tendency
Number of children
in families
2
2
3
2
1
2
0
4
• How many scores?
(n=8)
• The range? (4-0=4)
• Mode? (2)
• Mean?
(2+2+3+2+1+2+0+4 / 8)
(16/8)
=2
• Median? (2)
Descriptive statistics just summarize a group of scores and nothing more
Numerical Indices of
Central Tendency
• Mode – Score with the highest frequency
(most common score)
X
4
f
1
X
4
f
2
5
6
7
2
2
3
5
6
7
1
1
1
8
9
10
0
1
1
8
2
Mode = 4, 8
Mode = 7
This set of scores
is ‘bimodal’
Advantages & Disadvantages of
the Mode
• Advantages:
– By definition it is a score that actually
occurred
– Represents the largest number of people
– Can be used with any scale
• Disadvantages
– Depends on how we group the data
• E.g. if all cancers are grouped together it becomes the
most common way to die; if considered separately road
fatalities begin to feature
Numerical Indices of
Central Tendency
• Median – The Middle Score
X
4
5
6
7
8
9
10
f
1
2
2
3
0
1
1
Median = 6
X
4
5
6
7
8
Median = 6.5
Median location = N+1 / 2
(where N is the number of numbers)
f
2
1
1
1
2
Advantages & Disadvantages of
the Median
• Advantages:
– Not disproportionately affected by extreme
scores
– Only requires an ordinal scale
• Disadvantages
– Not as easy to use in calculations as the
mean
– Not as efficient as the mean – the mean is
better for estimating population means by
using a sample
Numerical Indices of
Central Tendency
• Mean – the balance point of the distribution
X


n
•  = Sigma = Sum
• The mean () is equal to the sum of all
scores ( X) divided by the number of
scores (n)
A Worked Example
• X: 10, 10, 11, 12, 13, 14, 14
• n=7
• X = 10 + 10 + 11 + 12 + 13 + 14 + 14 =
84
X


n
84

 12
7
Advantages & Disadvantages of
the Mean
• Advantages:
– Easy to manipulate mathematically
– Good for estimating a population mean from
samples
• Disadvantages
– Easily affected by extreme values
– Requires at least an interval scale
Warning about Means
• Time taken to complete a test (minutes):
1
2
3
3
3
4
5
4
54
• Mean = Median = Mode = 3
• But if someone falls asleep!
1
2
3
3
3
• Mean = 10 (Median & Mode = 3)
Applying Central Tendency…
• With different scales of measurement
Scale of
Measurement
Nominal
Index of
Central Tendency
Mode
Ordinal
Mode, Median
Interval
Mode, Median, Mean
Ratio
Mode, Median, Mean
• Never over-interpret means
– Means do not tell you the nature of the
distribution
Females
Males
Frequency
Y
Performance
X
Understanding variability
• Scores on a hard
class quiz
• 16 students
• Mean = 94/16 = 5.875
9
7
8
10
7
5
6
6
2
7
6
5
1
4
5
6
How many students received each
score?
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
/
/
/
///
////
///
/
/
/
(Frequency)
Number of
people who got
each score
Frequency Distribution
Y
Scores possible
X
Shapes of distributions
Symmetrical Distributions
Y
X
Shapes of distributions
Skewed
Distributions
(Frequency distributions)
Positive Skew
Negative Skew
In a symmetric, unimodal
distribution…
Y
50% 50%
Mode
Median
Mean
• Mode = Median = Mean
X
In a skewed, unimodal
distribution…
Y
Mode
Median
X
Mean
• Median & Mean: dragged in direction of
skew
PSYC1001
Quiz Results
2010
Understanding variability
Y
•
•
•
X
Red line: scores spread out more, more variability
Black line: scores tightly packed around mean, less variability
These two different distributions have the same mean and mode, so how
can we describe them to show the difference?
Standard deviation
Y
Small standard deviation
Large standard deviation
X
• Standard deviation is the measure of the
spread of scores
Class quiz : no one studies
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
/
/
//
/
///
////
///
/
//
/
More variability: higher
standard deviation
Class quiz : many students copy
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
/
/////
/////////
///
/
Less variability: smaller
standard deviation
PSYC1001
Quiz Results
2010
Measures of variability
• Range
– Highest score minus the lowest
• Standard Deviation
– The average deviation of scores around the
mean
• Standard error
– The average deviation of means around a
population mean
Understanding variability
• Measurement is inexact (instruments,
errors)
• Noise is always present
• Behaviour is not consistent
• We struggle to find patterns in a chaotic
world
Challenges facing inferential
statistics
• We take a sample from a population
• Is the sample representative (Was it selected in an
unbiased manner? Is it large enough?)
• See Lecture 4 : External Validity
• How do we know if differences or effects in
the sample reflect real differences in the
population, or have just been caused by
sampling error?
– Understanding the variability we expect from
sampling alone vs the variability we expect if
there were a real effect present is the
challenge
The role of variability in inferential
statistics
•
•
•
You wish to test the usefulness of two different psychotherapies for relieving
depression
You randomly allocate ten participants to each therapy
You obtain the following improvement scores (a higher score means they
became less depressed)
Therapy A
0
2
5
3
Therapy B
5
12
-12 -4
4
6
-2
0
-1
3
5
0
13
-4
2
3
Therapy A
Mean = 2
SD = 2.5
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10 12
Mean = 2
Therapy B
SD = 7.2
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10 12
What is inferential statistics?
• We construct a hypothesis about an effect in the
population
• We select a sample from that population
• We measure the sample and use the values
we obtain from the sample to draw
inferences about the population
• We would like to say: “On the basis of what we
have observed in our sample, we can make this
conclusion about the population….”
The sampling distribution
• So far we have considered a spread of
raw scores within an individual experiment
• Since the question for inferential statistics
is: what conclusion can we make about
the population from our sample, we
need to consider the distribution of sample
means from each experiment
A distribution of raw scores
X
XXXXX
XXXXXXX
XXXXXXXXX
XXXXXXXXXXX
XXXXXXXXXXXX
XXXXXXXXXXXXXXXXX
Mean = M
Sample
mean
Raw scores
A distribution of sample means: The sampling distribution
M
MMM
MMMMM
MMMMMM
MMMMMMMM
MMMMMMMMMM
MMMMMMMMMMMMM
Sample means
Mean = 
Population
mean
• Raw score distribution • Sampling distribution
– Shows variability
across individuals
– Is real
– Shows us the
likelihood of obtaining
a particular score
– Shows variability
across experiments
– Is hypothetical: You
would never run the
same experiment so
many times
– Shows us the
likelihood of obtaining
an experiment result if
the null is true
See previous lecture
The coin toss
• We hypothesize that a particular coin is
“unbiased” – that is, we expect that if we toss it
there is a equal (50/50) chance that it will turn up
heads or tails
• We can take a ‘sample’ of the coins behaviour,
by tossing it 100 times
• If the coin is fair how many times would we
expect it to turn up heads?
• If we do not obtain that exact number from our
sample, is that a problem?
Y
Mean = 50
Population
mean
0
20
30
40
50
60
70
80
100
The experiment (toss a coin 100 times and record the
number of heads) is repeated 10,000 times.
This sampling distribution would result.
X
We tolerate a degree of variability
close to the expected mean
But beyond a certain
point we decide
something is
happening
Y
Mean = 50
Population
mean
0
20
30
40
50
60
70
80
100
The experiment (toss a coin 100 times and record the
number of heads) is repeated 10,000 times.
This sampling distribution would result.
X
The experiment (toss a coin 100 times and record the number of heads)
Null hypothesis: nothing unusual is
happening: population mean = 50
Y
Reject
40
Since this is a frequency distribution of all
possible sample means, the ‘height’ of the line
indicates the likelihood of getting a particular
result. That’s why there’s a bump in the middle
– the hypothesized population mean is the
most likely result!
Retain
Reject
50
60
X
The experiment (toss a coin 100 times and record the number of heads)
Null hypothesis: nothing unusual is
happening: population mean = 50
Y
Reject
40
p < 0.05
Retain
Reject
50
60
p > 0.05
p < 0.05
X
Possible outcomes
• Sample value: 53 heads
– Statistical test: p = 0.91
– We would retain our null hypothesis that it is
a fair coin. We have not found any evidence
to suggest it is not.
• Sample value: 65 heads
– Statistical test: p = 0.022
– We would reject our null hypothesis that it is a
fair coin. We have found evidence to suggest
that it is biased toward heads
The p-value
• A small p-value (0.002, 0.01, 0.045)
means: it is highly improbable that we
obtained our result by chance alone
• A large p-value (0.06, 0.12, 0.67) means: it
is probable that we obtained our result
by chance alone
• The usual cut off for psychology is 0.05.
This is a convention, there is no
mathematical reason for it.
The p-value
• The cutoff of 0.05 means, that if there is less
than a 5% chance that our result is due to
chance, we will say something is happening: we
will reject the null hypothesis (see Lecture 4).
• If there is more than 5% chance that our result is
simply due to chance, we will consider that the
result is unreliable and possibly represents
nothing more than sampling error: we will retain
the null hypothesis
Calculating the p-value
• Understand what the p-value is and how it is
used
• You need a lot of information to work out a pvalue (i.e. all the data)
– So no you don’t need to know how to do it
• Even in 2nd year you’ll be using software to do it
• But how is it that we can estimate the likelihood
of outcomes on hypothetical sampling
distributions?
– We use what we know about the normal curve
f
The Normal Distribution

The Normal Distribution
is:
• Symmetric
• Unimodal
• ‘Bell-shaped’
X
 Normal Distributions can have different means
and variances (or standard deviations)
 It is a standard shape with precise properties
 Approximately 2/3 of all scores lie within 1
SD of the mean
Some normal curves
Source: http://en.wikipedia.org/wiki/Normal_distribution
Source: http://en.wikipedia.org/wiki/Normal_distribution
There’s also a good diagram like this in the appendix of Weiten
Appendix B page A10
Are we certain?
• Science deals with probabilities
– Is it possible that for a lucky 100 tosses, we
just got more heads in our sample, and it is
still a fair coin?
• YES
– Is that probable?
• NO
• Science does not deal with certainty.
Science never proclaims certainty.
Nature of
evidence
_____________
Conclusion
Slight evidence
Modest evidence
Overwhelming
evidence
“Something is
happening!”
Too liberal – too
excited by
chance. Driven
by confirmation
bias to confirm a
belief.
Reasonable
scientific
conclusion.
Finding warrants
further
investigation.
Solid scientific
conclusion.
Phenomenon is
robust.
“Nothing is
happening!”
Reasonable
conclusion,
result is
attributed to
chance
Possibly too
conservative to
ignore
something not
caused by
chance.
Way too
conservative.
Denial of the
overwhelming
evidence may be
driven by a
strong belief.
The point at which we change
our mind is the challenge
inferential statistics faces