Intro to Research in Information
Studies
Inferential Statistics
Standard Error of the Mean
Significance
Inferential tests you can use
1
Do you speak the language?
—
XA
t=
2 (
SX ) 2
XA S
[(
A
n1
(n1-1)
- XB
+ SXB
)(
—
2 (
SX ) 2
-
B
n2
1
)] x (n
1
+
1
n2)
+ (n2-1)
2
Difference between
means
Don’t Panic !
—
XA
t=
2 (
SX ) 2
XA S
[(
A
n1
(n1-1)
- XB
+ SXB
)(
—
2 (
SX ) 2
-
B
n2
1
)] x (n
1
+
1
n2)
+ (n2-1)
Compare with SD formula
3
Basic types of statistical
treatment
o Descriptive statistics which summarize
the characteristics of a sample of data
o Inferential statistics which attempt to say
something about a population on the basis
of a sample of data - infer to all on the
basis of some
Statistical tests are inferential
4
Two kinds of descriptive statistic:
o Measures of central tendencyOr where about on the
measurement scale
most of the data fall
– mean
– median
– mode
Or how spread out they
are
o Measures of dispersion (variation)
– range
– inter-quartile range
– variance/standard deviation
The different measures have different sensitivity and
should be used at the appropriate times…
5
Symbol check

n
 xi
i 1
o
Sigma: Means the
‘sum of’
o
Sigma (1 to n) x of i:
means add all
values of i from 1 to
n in a data set
o
Xi = the ith data point
6
Mean
Sum of all observations divided by the number of
observations
n
In notation:
S
i =1
xi
Refer to handout on notation
n
See example on next slide
Mean uses every item of data but is sensitive to
extreme ‘outliers’
7
To overcome problems with range etc.
we need a better measure of spread
Variance and standard deviation
o
A deviation is a measure of how far
from the mean is a score in our
data
o
Sample: 6,4,7,5
o
Each score can be expressed in terms of distance
from 5.5
6,4,7,5, => 0.5, -1.5, 1.5, -0.5 (these are distances
from mean)
Since these are measures of distance, some are
8
positive (greater than mean) and some are
o
o
mean =5.5
Symbol check

x
o
Called ‘x bar’; refers
to the ‘mean’

(x  x)
o
Called ‘x minus xbar’; implies
subtracting the
mean from a data
point x. also known
as a deviation from
the mean
9
Two ways to get SD
2
(x

x)

sd 
n
•Sum the sq. deviations from the mean
•Divide by No. of observations
•Take the square root of the result
x
2

sd 
x
n
•Sum the squared raw scores
•Divide by N
•Subtract the squared mean
•Take the square root of the result
2
10
x
2
2
2
2
2
3
3
4
4
5
S x = 29
x2
4
4
4
4
4
9
9
16
16
25
S x 2 = 95
s=
=
2
x
S - x2
n
95
10
=
9.5
=
1.09
=
1.044
-
2
2.9
- 8.41
If we recalculate the
variance with the 60
instead of the 5 in the
data…
If we include a large outlier:
2
x
x
2
4
2
4
2
4
2
4
2
4
3
9
3
9
4
16
4
16
60
3600
S x = 84 S x 2 = 3670
Like the mean, the
standard deviation uses
every piece of data and
is therefore sensitive to
extreme values
Note increase in SD
s=
=
2
x
S - x2
n
3760
10
-
=
367 - 70.56
=
296.44
=
17.22
2
8.4
Mean
Two sets of data can have the same mean but different standard
deviations.
The bigger the SD, the more s-p-r-e-a-d out are the data.
On the use of N or N-1
2
(x  x)

sd 
n
2
(x  x)

sd 
n 1
o
When your
observations are the
complete set of
people that could be
measured
(parameter)
o
When you are
observing only a
sample of potential
users (statistic), the14
use of N-1 increases
Summary
Measures of Central Tendency
Most frequent
Mode •
observation. Use with
nominal data
‘Middle’ of data. Use with ordinal
Median •
data or when data contain outliers
‘Average’. Use with
interval and ratio data if
no outliers
Measures of Dispersion
Mean •
Range •
Interquartile Range •
Variance / Standard Deviation •
Dependent on two extreme
values
More useful than range.
Often used with median
Same conditions as mean.
With mean, provides excellent
summary of data
Andrew Dillon:
Move this to later in the course, after
distributions?
Deviation units: Z scores
Any data point can be expressed in terms of its
Distance from the mean in SD units:
xx
z
sd
A positive z score implies a value above the mean
A negative z score implies a value below the mean16
Interpreting Z scores
o
o
o
Mean = 70,SD = 6
Then a score of 82
is 2 sd [ (82-70)/6]
above the mean, or
82 = Z score of 2
Similarly, a score of
64 = a Z score of -1
o
o
By using Z scores, we
can standardize a set of
scores to a scale that is
more intuitive
Many IQ tests and
aptitude tests do this,
setting a mean of 100
and an SD of 10 etc.
17
Comparing data with Z scores
You score 49 in class A but 58 in class B
How can you compare your performance in both?
Class A:
Mean =45
SD=4
Class B:
Mean =55
SD = 6
49 is a Z=1.0
58 is a Z=0.5
18
With normal distributions
Mean,
SD and
Z tables
In combination provide powerful means of
estimating what your data indicates
19
Graphing data - the histogram
The frequency of
occurrence for
measure of
interest,
e.g., errors, time,
scores on a test
etc.
Graph gives instant
summary of data check spread,
similarity, outliers, etc.
100
90
80
70
60
Number
Of errors
50
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
The categories of data we are studying, e.g., task or
interface, or user group etc.
20
Very large data sets tend to have
distinct shape:
80
70
60
50
40
30
20
10
0
21
Normal distribution
o
Bell shaped, symmetrical, measures of
central tendency converge
o
o
o
mean, median, mode are equal in normal
distribution
Mean lies at the peak of the curve
Many events in nature follow this curve
o
IQ test scores, height, tosses of a fair coin,
user performance in tests,
22
The Normal Curve
f
NB: position of
measures of
central
tendency
50% of scores
fall below mean
Mean
Median
Mode
23
Positively skewed distribution
Note how the various measures of
central tendency separate now note the direction of the
change…mode moves left of other
two, mean stays highest,
indicating frequency of scores
less than the mean
f
Mode Median
Mean
24
Negatively skewed distribution
Here the tendency
to have higher
values more
common serves to
increase the value
of the mode
f
Mean Median
Mode
25
Other distributions
o
Bimodal
o
o
Data shows 2 peaks separated by trough
Multimodal
o
More than 2 peaks
o The shape of the underlying distribution determines
your choice of inferential test
26
Bimodal
Will occur in situations where
there might be distinct groups
being tested e.g., novices and
experts
Note how each mode is itself
part of a normal distribution
(more later)
f
Mode
Mean
Median
Mode
27
Standard deviations and the
normal curve
68% of observations
fall within ± 1 s.d.
f
95% of observations fall
within ± 2 s.d. (approx)
1 sd
1 sd
1 sd
1 sd
Mean
28
Z scores and tables
Knowing a Z score allows you to determine
where under the normal distribution it occurs
Z score between:
0 and 1 = 34% of observations
1 and -1 = 68% of observations etc.
Or 16% of scores are >1 Z score above mean
Check out Z tables in any basic stats book
29
Remember:
o
o
A Z score reflects position in a normal
distribution
The Normal Distribution has been
plotted out such that we know what
proportion of the distribution occurs
above or below any point
30
Importance of distribution
o
Given the mean, the standard deviation,
and some reasonable expectation of
normal distribution, we can establish the
confidence level of our findings
o
With a distribution, we can go beyond
descriptive statistics to inferential
statistics (tests of significance)
31
So - for your research:
o
o
o
Always summarize the data by graphing
it - look for general pattern of
distribution
Then, determine the mean, median,
mode and standard deviation
From these we know a LOT about what
we have observed
32
Inference is built on Probability
o
o
o
Inferential statistics rely on the laws of
probability to determine the
‘significance’ of the data we observe.
Statistical significance is NOT the same
as practical significance
In statistics, we generally consider
‘significant’ those differences that occur
less than 1:20 by chance alone
33
At this point I ask people to take out a
coin and toss it 10 times, noting the exact
sequence of outcomes e.g.,
Calculating probability
h,h,t,h,t,t,h,t,t,h.
Then I have people compare outcomes….
o
Probability refers to the likelihood of any
given event occurring out of all possible
events e.g.:
o
Tossing a coin - outcome is either head or
tail
o
o
o
Therefore probability of head is 1/2
Probability of two heads on two tosses is 1/4
since the other possible outcomes are two tails,
and two possible sequences of head and tail.
The probability of any event is expressed
as a value between 0 (no chance) and 134
Sampling distribution for 3 coin
tosses
3.5
3
2.5
2
1.5
1
0.5
0
0 heads
1 head
2 heads
3 heads
1
3
3
1
35
Probability and normal curves
o
Q? When is the probability of getting 10
heads in 10 coin tosses the same as getting 6
heads and 4 tails?
o
o
o
HHHHHHHHHH
HHTHTHHTHT
Answer: when you specify the precise order
of the 6 H/4T sequence:
o
o
(1/2)10 =1/1024 (specific order)
But to get 6 heads, in any order it is: 210/1024 (or
about 1:5)
36
What use is probability to us?
o
o
It tells us how likely is any event to
occur by chance
This enables us to determine if the
behavior of our users in a test is just
chance or is being affected by our
interfaces
37
Determining probability
o
o
o
o
Your statistical test result is plotted
against the distribution of all scores on
such a test.
It can be looked up in stats tables or is
calculated for you in EXCEL or SPSS
etc
This tells you its probabilityIntroduce
of
simple stats
occurrence
tables here :
The distributions have been determined
38
by statisticians.
What is a significance level?
o
o
o
In research, we estimate the probability
level of finding what we found by
chance alone.
Convention dictates that this level is
1:20 or a probability of .05, usually
expressed as : p<.05.
However, this level is negotiable
o
But the higher it is (e.g., p<.30 etc) the more likely
you are to claim a difference that is really just
39
occurring by chance (known as a Type 1 error)
What levels might we chose?
o
In research there are two types of errors
we can make when considering
probability:
o
o
o
Claiming a significant difference when
there is none (type 1 error)
Failing to claim a difference where there is
one (type 2 error)
The p<.05 convention is the ‘balanced’
case but tends to minimize type 1 errors40
Using other levels
o
o
Type 1 and 2 errors are interwoven, if
we lessen the probability of one
occurring, we increase the chance of
the other.
If we think that we really want to find
any differences that exist, we might
accept a probability level of .10 or
higher
41
Thinking about p levels
o
The p<.x level means we believe our results
could occur by chance alone (not because of
our manipulation) at least x/100 times
o
o
o
o
P<.10 => our results should occur by chance 1 in
10 times
P<.20=> our results should occur by chance 2 in
10 times
Depending on your context, you can take your
chances :)
In research, the consensus is 1:20 is high
42
Putting probability to work
o
o
o
Understanding the probability of gaining
the data you have can guide your
decisions
Determine how precise you need to be
IN ADVANCE, not after you see the
result
It is like making a bet….you cannot play
the odds after the event!
43
I find that this is the hardest part of stats for
novices to grasp, since it is the bridge
between descriptive and inferential
stats…..needs to be explained slowly!!
Sampling error and the mean
o
Usually, our data forms only a small part of all
the possible data we could collect
o
o
o
All possible users do not participate in a usability
test
Every possible respondent did not answer our
questions
The mean we observe therefore is unlikely to
be the exact mean for the whole population
o
The scores of our users in a test are not going to
be an exact index of how all users would perform 44
How can we relate our sample to
everyone else?
o
Central limit theorem
o
o
o
If we repeatedly sample and calculate means from
a population, our list of means will itself be
normally distributed
Holds true even for samples taken from a skewed
population distribution
This implies that our observed mean follows
the same rules as all data under the normal
curve
45
The distribution of the means forms a smaller normal
distribution about the true mean:
2
4
6
8
10
12
14
16
18
46
True for skewed distributions
too
Here the tendency
to have higher
values more
common serves to
increase the value
of the mode
Plot of means from
samples
f
Mean
47
How means behave..
o
o
o
A mean of any sample belongs to a
normal distribution of possible means of
samples
Any normal distribution behaves lawfully
If we calculate the SD of all these
means, we can determine what
proportion (%) of means fall within
specific distances of the ‘true’ or
48
population mean
But...
o
o
We only have a sample, not the
population…
We use an estimate of this SD of means
known as the Standard Error of the
Mean
SD
SE 
N
49
Implications
o
o
Given a sample of data, we can
estimate how confident we are in it
being a true reflection of the ‘world’ or…
If we test 10 users on an interface or
service, we can estimate how much
variability about our mean score we will
find within the intended full population of
users
50
Example
o
We test 20 users on a new interface:
o
o
o
Mean error score: 10, sd: 4
What can we infer about the broader user
population?
According to the central limit theorem,
our observed mean (10 errors) is itself
95% likely to be within 2 s.d. of the ‘true’
(but unknown to us) mean of the population
51
The Standard Error of the Means
s.d.(sample)
SE 
N
4
4


 0.89
20 4.47
52
If standard error of mean = 0.89
o
Then observed (sample) mean is within
a normal distribution about the ‘true’ or
population mean:
o
So we can be
68% confident that the true mean=10  0.89
o 95% confident our population mean = 10  1.78
o 99% confident it is within 10 2.67
This offers a strong method of interpreting of our data
o
o
53
Issues to note
o
If s.d. is large and/or sample size is small, the
estimated deviation of the population means
will appear large.
o
o
o
e.g., in last example, if n=9, SE mean=1.33
So confidence interval becomes 10  2.66 (i.e.,
we are now 95% confident that the true mean is
somewhere between 7.34 and 12.66.
Hence confidence improves as sample increases
and variability lessens
o
Or in other words: the more users you study, the more
sure you can be….!
54
Exercise:
o
If the mean = 10 and the s.d.=4, what is
the 68% confidence interval when
we
Answers:
9-11
have:
o
o
o
16 users?
9 users?
8.66-11.33
4-16
2-18
If the s.d. = 12, and mean is still 10,
what is the 95% confidence interval for
those N?
55
Exercise answers:
o
If the mean = 10 and the s.d.=4, what is the 68%
confidence interval when we have:
16 users?= 9-11 (hint: sd/n = 4/4=1)
9 users? = 8.66-11.33
o
If the s.d. = 12, and mean is still 10, what is the 95%
confidence interval for those N?
16 users: 4-16 (hint: 95% CI implies 2 SE either side of mean)
9 users: 2-18
56
Recap
o
o
o
Summarizing data effectively informs us
of central tendencies
We can estimate how our data deviates
from the population we are trying to
estimate
We can establish confidence intervals to
enable us to make reliable ‘bets’ on the
effects of our designs on users
57
This is the
beginning of
significance
testing
Comparing 2 means
o
The differences between means of
samples drawn from the same
population are also normally distributed
o
Thus, if we compare means from two
samples, we can estimate if they belong
to the same parent population
58
SE of difference between means
 [x 1x 2]   x 1 x 2
2
2
SEdiff.means SE(sample1)  SE(sample2)
2
2
This lets us set up confidence limits for the differences
between the two means
59
Regardless of population mean:
o
The difference between 2 true
measures of the mean of a population is
0
o
The differences between pairs of
sample means from this population is
normally distributed about 0
60
Consider two interfaces:
We capture 10 users’ times per task on
each.
The results are:
Interface A = mean 8, sd =3
Interface B = mean 10, sd=3.5
Q? - is Interface A really different?
How do we tackle this question?
61
Calculate the SE difference
between the means
SEa = 3/10 =
0.95
SEb= 3.5/ 10=1.11
SE a-b = (0.952+1.112) = (0.90+1.23)=1.46
Observed Difference between means= 2.0
95% Confidence interval of difference between means
is 2 x(1.46) or 2.92 (i.e. we expect to find difference
between 0-2.92 by chance alone).
suggests there is no significant difference at the p<.05
level.
62
But what else?
We can calculate the exact probability of finding
this difference by chance:
Divide observed difference between the means by
the SE(diff between means): 2.0/1.46 = 1.37
Gives us the number of standard deviation units
between two means (Z scores)
Check Z table: 82% of observations are within 1.37
sd, 18% are greater; thus the precise sig level of our
findings is p<.18.
Thus - Interface A is different, with rough odds of 5:1
63
Hold it!
o
Didn’t we first conclude there was no
significant difference?
o
o
Yes, no significant difference at p<.05
But the probability of getting the differences we
observed by chance was approximately 0.18
o
o
o
Not good enough for science (must avoid type 1 error),
but very useful for making a judgment on design
But you MUST specify levels you will accept BEFORE
not after….
Note - for small samples (n<20) t- distribution is
better than z distribution, when looking up probability
64
Why t?
o
o
o
Similar to the normal distribution
t distribution is flatter than Z for small degrees
of freedom (n-1), but virtually identical to Z
when N>30
Exact shape of t-distribution depends on
sample size
65
Simple t-test:
o
You want all users of a new interface to score
at least 70% on an effectiveness test. You
test 6 users on a new interface and gain the
following scores:
62
92
75
68
83
95
Mean = 79.17
Sd=13.17
66
T-test:
79.17  70 9.17
t  13.17 
 1.71
5.38
6
From t-tables, we can see that this value of t exceeds t
value (with 5 d.f.) for p.10 level
So we are confident at 90% level that our new interface
leads to improvement
67
T-test:
Sample mean
79.17  70 9.17
t  13.17 
 1.71
5.38
6
SE mean
Thus - we can still talk in confidence intervals, e.g.,
We are 68% confident the mean of population =79.17  5.38
68
Predicting the direction of the
difference
o
Since you stated that you wanted to see
if new Interface was BETTER (>70),
not just DIFFERENT (< or > 70%), this
is asking for a one-sided test….
o
For a two-sided test, I just want to see if
there is ANY difference (better or worse)
between A and B.
69
One tail (directional) test
o
o
o
o
Tester narrows the odds by half by testing for
a specific difference
One sided predictions specify which part of
the normal curve the difference observed
must reside in (left or right)
Testing for ANY difference is known as ‘twotail’ testing,
Testing for a directional difference (A>B) is
known as ‘one-tail’ testing
70
So to recap
o
o
o
If you are interested only in certain
differences, you are being ‘directional’
or ‘one-sided’
Under the normal curve, random or
chance differences occur equally on
both sides
You MUST state directional
expectations (hypothesis) in advance
71
Why would you predict the
direction?
o
Theoretical grounds
o
o
Experience or previous findings suggested
the difference
Practical grounds
o
You redesigned the interface to make it
better, so you EXPECT users will perform
better….
72