Inferential Statistics Standard Error of the Mean Significance Inferential tests you can use 1 Pengenalan kepada Statistik Inferensi

Inferential StatisticsStandard Error of the MeanSignificanceInferential tests you can use

1

Pengenalan kepada Statistik Inferensi

Perhatikan simbol/formula

2

t =

XA

—XB

—

XA 2 XA 2( )

n1

- XB 2 XB 2( )

n2

-( ) ( )+[ ]1

n1

1

n2

+( )x

-

(n1-1) + (n2-1)

3

Don’t Panic !

Don’t Panic !

t =XA

—XB

—

XA 2 XA 2( )

n1

- XB 2 XB 2( )

n2

-( ) ( )+[ ]1

n1

1

n2

+( )x

-

Compare with SD formula

(n1-1) + (n2-1)

Difference betweenmeans

Basic types of statistical treatment

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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 inferentialStatistical tests are inferential

Two kinds of descriptive statistic:

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o Measures of central tendency– mean

– median – mode

o Measures of dispersion (variation)– range – inter-quartile range– variance/standard deviation

Or where about on the measurement scale most of the data fall

Or where about on the measurement scale most of the data fall

Or how spread out they are

Or how spread out they are

The different measures have different sensitivity and should be used at the appropriate times…

The different measures have different sensitivity and should be used at the appropriate times…

Mean

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Sum of all observations divided by the number of observations

In notation:

Mean uses every item of data but is sensitive to extreme ‘outliers’

i = 1

n

xi

n

Refer to handout on notationRefer to handout on notation

See example on next slideSee example on next slide

Variance and standard deviation

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A deviation is a measure of how far from the mean is a score in our dataSample: 6,4,7,5 mean =5.5Each score can be expressed in terms of distance

from 5.56,4,7,5, => 0.5, -1.5, 1.5, -0.5 (these are distances

from mean)Since these are measures of distance, some are

positive (greater than mean) and some are negative (less than the mean)

TIP: Sum of these distances ALWAYS = 0

To overcome problems with range etc. we need a better measure of spread

To overcome problems with range etc. we need a better measure of spread

Symbol check

x

(x x)

Called ‘x bar’; refers to the ‘mean’

Called ‘x minus x-bar’; implies subtracting the mean from a data point x. also known as a deviation from the mean

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Two ways to get SD

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sd (x x)2

n

•Sum the sq. deviations from the mean•Divide by No. of observations•Take the square root of the result

•Sum the squared raw scores•Divide by N•Subtract the squared mean•Take the square root of the result

sd x2

n x

2

x x

2 4 2 4 2 4 2 4 2 4 3 9 3 9 4 16 4 16 5 25

x = 29 x = 952

2

s = -2

n x 2x

= -95

102.9

2

= -9.5 8.41

= 1.09

= 1.044 If we recalculate the variance with the 60 instead of the 5 in the data…

If we recalculate the variance with the 60 instead of the 5 in the data…

x x

2 4 2 4 2 4 2 4 2 4 3 9 3 9 4 16 4 16 60 3600

x = 84 x = 36702

2

s = -2

n x 2x

= -3760

108.4

2

= -367 70.56

= 296.44

= 17.22

If we include a large outlier:

Note increase in SD

Like the mean, the standard deviation uses every piece of data and is therefore sensitive to extreme values

Like the mean, the standard deviation uses every piece of data and is therefore sensitive to extreme values

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

sd (x x)2

n

When your observations are the complete set of people that could be measured (parameter)

When you are observing only a sample of potential users (statistic), the use of N-1 increases size of sd slightly

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sd (x x)2n 1

Summary

Mode •

Median •

Mean •

Range •

Interquartile Range •

Variance / Standard Deviation •

Most frequent observation. Use with nominal data‘Middle’ of data. Use with ordinal data or when data contain outliers‘Average’. Use with interval and ratio data if no outliers

Dependent on two extreme valuesMore useful than range. Often used with medianSame conditions as mean. With mean, provides excellent summary of data

Measures of Central Tendency

Measures of Dispersion

Deviation units: Z scores

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z x x

sd

Any data point can be expressed in terms of itsDistance from the mean in SD units:

A positive z score implies a value above the meanA negative z score implies a value below the mean

Interpreting Z scores

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Mean = 70,SD = 6Then 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

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.

Comparing data with Z scores

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You score 49 in class A but 58 in class B How can you compare your performance in both?

Class A: Class B:Mean =45 Mean =55SD=4 SD = 6

49 is a Z=1.0 58 is a Z=0.5

With normal distributions

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Mean, SD and Z tables

In combination provide powerful means of estimating what your data indicates

Graphing data - the histogram

0

10

20

30

40

50

60

70

80

90

100

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NumberOf errors

The categories of data we are studying, e.g., task or interface, or user group etc.

The frequency of occurrence for measure of interest,e.g., errors, time, scores on a test etc.

1 2 3 4 5 6 7 8 9 10Graph gives instant summary of data - check spread, similarity, outliers, etc.

Graph gives instant summary of data - check spread, similarity, outliers, etc.

Very large data sets tend to have distinct shape:

0

10

20

30

40

50

60

70

80

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Normal distribution

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Bell shaped, symmetrical, measures of central tendency convergemean, median, mode are equal in normal

distributionMean lies at the peak of the curve

Many events in nature follow this curveIQ test scores, height, tosses of a fair coin,

user performance in tests,

The Normal Curve

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MeanMedianMode

50% of scoresfall below meanf

NB: position of measures of central tendency

NB: position of measures of central tendency

Positively skewed distribution

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Mode Median Mean

f

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

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

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Negatively skewed distribution

Mean Median Mode

f

Here the tendency to have higher values more common serves to increase the value of the mode


Other distributions

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BimodalData shows 2 peaks separated by trough

MultimodalMore than 2 peaks

The shape of the underlying distribution determines your choice of inferential test

Bimodal

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f

MeanMedian

Mode Mode

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)

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)

Standard deviations and the normal curve

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Mean

1 sd

f

1 sd

68% of observationsfall within ± 1 s.d.

95% of observations fallwithin ± 2 s.d. (approx)

1 sd1 sd

Z scores and tables

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Knowing a Z score allows you to determine where under the normal distribution it occurs

Z score between:

0 and 1 = 34% of observations1 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

Remember:

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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

Importance of distribution

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Given the mean, the standard deviation, and some reasonable expectation of normal distribution, we can establish the confidence level of our findings

With a distribution, we can go beyond descriptive statistics to inferential statistics (tests of significance)

So - for your research:

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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

Inference is built on Probability

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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

Calculating probability

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Probability refers to the likelihood of any given event occurring out of all possible events e.g.:Tossing a coin - outcome is either head or tail

Therefore probability of head is 1/2Probability 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 1 (certain)

At this point I ask people to take out a coin and toss it 10 times, noting the exact sequence of outcomes e.g.,

h,h,t,h,t,t,h,t,t,h.

Then I have people compare outcomes….

At this point I ask people to take out a coin and toss it 10 times, noting the exact sequence of outcomes e.g.,

h,h,t,h,t,t,h,t,t,h.

Then I have people compare outcomes….

Sampling distribution for 3 coin tosses

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0

1

2

3

4

0 heads 1

1 head 3

2 heads 3

3 heads 1

Probability and normal curves

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Q? When is the probability of getting 10 heads in 10 coin tosses the same as getting 6 heads and 4 tails?HHHHHHHHHHHHTHTHHTHT

Answer: when you specify the precise order of the 6 H/4T sequence:(1/2)10 =1/1024 (specific order)But to get 6 heads, in any order it is:

210/1024 (or about 1:5)

What use is probability to us?

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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

Determining probability

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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 probability of occurrenceThe distributions have been determined by

statisticians.

Introduce simple stats tables here :

Introduce simple stats tables here :

What is a significance level?

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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 negotiableBut the higher it is (e.g., p<.30 etc) the more

likely you are to claim a difference that is really just occurring by chance (known as a Type 1 error)

What levels might we chose?

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In research there are two types of errors we can make when considering probability: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 errors

Using other levels

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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

Thinking about p levels

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The p<.x level means we believe our results could occur by chance alone (not because of our manipulation) at least x/100 timesP<.10 => our results should occur by chance 1 in

10 timesP<.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 enough…..

Putting probability to work

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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!

Sampling error and the mean

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Usually, our data forms only a small part of all the possible data we could collectAll possible users do not participate in a usability

testEvery possible respondent did not answer our

questions

The mean we observe therefore is unlikely to be the exact mean for the whole populationThe scores of our users in a test are not going to

be an exact index of how all users would perform

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!!

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!!

How can we relate our sample to everyone else?

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Central limit theoremIf 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

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2 4 6 8 10 12 14 16 18

The distribution of the means forms a smaller normal distribution about the true mean:

0 5 10 15 20

05

00

10

00

15

00

z

n = 5

mean of sample means = 10

SD of sample means = 2.41

0 5 10 15 20

02

00

40

06

00

80

0

z

n = 2



0 5 10 15 20

05

00

10

00

15

00

20

00

z

n = 15



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True for skewed distributions too

Mean

f

Plot of means from samples



How means behave..

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A mean of any sample belongs to a normal distribution of possible means of samples

Any normal distribution behaves lawfullyIf we calculate the SD of all these means,

we can determine what proportion (%) of means fall within specific distances of the ‘true’ or population mean

But...

49

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

SE SD

N

Implications

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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

Example

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We test 20 users on a new iPad:Mean error score: 10, sd: 4What 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

The Standard Error of the Means

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SE s.d .(sample)

N

4

20

4

4.470.89

If standard error of mean = 0.89

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Then observed (sample) mean is within a normal distribution about the ‘true’ or population mean:So we can be

68% confident that the true mean=10 0.89 95% confident our population mean = 10 1.78 99% confident it is within 10 2.67

This offers a strong method of interpreting of our data

Issues to note

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If s.d. is large and/or sample size is small, the estimated deviation of the population means will appear large.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 lessensOr in other words: the more users you study, the

more sure you can be….!

Exercise:

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If the mean = 10 and the s.d.=4, what is the 68% confidence interval when we have: 16 users? 9 users?

If the s.d. = 12, and mean is still 10, what is the 95% confidence interval for those N?

Answers:

9-11

8.66-11.33

4-16

2-18

Answers:

9-11

8.66-11.33

4-16

2-18

Exercise answers:

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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

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

Recap

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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

Comparing 2 means

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The differences between means of samples drawn from the same population are also normally distributed

Thus, if we compare means from two samples, we can estimate if they belong to the same parent population

This is the beginning of significance testing

This is the beginning of significance testing

SE of difference between means

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[x 1 x

2] 2

x 12

x 2

SEdiff .means SE(sample1)2 SE(sample2)2

This lets us set up confidence limits for the differences between the two means

Regardless of population mean:

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The difference between 2 true measures of the mean of a population is 0

The differences between pairs of sample means from this population is normally distributed about 0

Consider two interfaces:

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We capture 10 users’ times per task on each.

The results are:

Interface A = mean 8, sd =3Interface B = mean 10, sd=3.5

Q? - is Interface A really different?

How do we tackle this question?

Calculate the SE difference between the means

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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.

But what else?

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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 = 1.37Gives 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

Hold it!

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Didn’t we first conclude there was no significant difference?Yes, no significant difference at p<.05But the probability of getting the differences we

observed by chance was approximately 0.18 Not good enough for science (must avoid type 1 error),

but very useful for making a judgment on designBut 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

Why t?

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Similar to the normal distributiont 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

Simple t-test:

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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:

629275688395

Mean = 79.17Sd=13.17

T-test:

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t 79.17 70

13.17

6

9.17

5.381.71

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

T-test:

68

t 79.17 70

13.17

6

9.17

5.381.71

SE mean

Sample mean

Thus - we can still talk in confidence intervals, e.g., We are 68% confident the mean of population =79.17 5.38

Predicting the direction of the difference

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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….

For a two-sided test, I just want to see if there is ANY difference (better or worse) between A and B.

One tail (directional) test

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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 ‘two-tail’ testing,

Testing for a directional difference (A>B) is known as ‘one-tail’ testing

So to recap

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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

Why would you predict the direction?

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Theoretical groundsExperience or previous findings suggested the

differencePractical grounds

You redesigned the interface to make it better, so you EXPECT users will perform better….

Alternative and Null Hypotheses

Inferential statistics test the likelihood that the alternative (research) hypothesis (H1) is true and the null hypothesis (H0) is notin testing differences, the H1 would predict

that differences would be found, while the H0 would predict no differences

by setting the significance level (generally at .05), the researcher has a criterion for making this decision


If the .05 level is achieved (p is equal to or less than .05), then a researcher rejects the H0 and accepts the H1

If the the .05 significance level is not achieved, then the H0 is retained


If the .05 level is achieved (p is equal to or less than .05), then a researcher rejects the H0 and accepts the H1

If the the .05 significance level is not achieved, then the H0 is retained

Degrees of FreedomDegrees of freedom (df) are the way

in which the scientific tradition accounts for variation due to errorit specifies how many values vary

within a statistical testscientists recognize that collecting data can

never be error-freeeach piece of data collected can vary, or

carry error that we cannot account forby including df in statistical computations,

scientists help account for this errorthere are clear rules for how to calculate df

for each statistical test

Inferential Statistics: 5 StepsTo determine if SAMPLE means come from

same population, use 5 steps with inferential statistics1. State Hypothesis

Ho: no difference between 2 means; any difference found is due to sampling errorany significant difference found is not a

TRUE difference, but CHANCE due to sampling error

results stated in terms of probability that Ho is falsefindings are stronger if can reject Ho therefore, need to specify Ho and H1

Steps in Inferential Statistics 2. Level of Significance

Probability that sample means are different enough to reject Ho (.05 or .01)level of probability or level of

confidence

Steps in Inferential Statistics 3. Computing Calculated Value

Use statistical test to derive some calculated value (e.g., t value or F value)

4. Obtain Critical Valuea criterion used based on df and alpha

level (.05 or .01) is compared to the calculated value to determine if findings are significant and therefore reject Ho

Steps in Inferential Statistics

5. Reject or Fail to Reject HoCALCULATED value is compared to the

CRITICAL value to determine if the difference is significant enough to reject Ho at the predetermined level of significanceIf CRITICAL value > CALCULATED

value --> fail to reject HoIf CRITICAL value < CALCULATED

value --> reject Ho

If reject Ho, only supports H1; it does not prove H1

Testing Hypothesis

If reject Ho and conclude groups are really different, it doesn’t mean they’re different for the reason you hypothesized may be other reason

Since Ho testing is based on sample means, not population means, there is a possibility of making an error or wrong decision in rejecting or failing to reject Ho

Type I errorType II error

Testing Hypothesis

Type I error -- rejecting Ho when it was true (it should have been accepted)equal to alphaif = .05, then there’s a 5% chance of

Type I errorType II error -- accepting Ho when it

should have been rejectedIf increase , you will decrease the

chance of Type II error

Identifying the Appropriate Statistical Test of Difference

One variable One-way chi-square

Two variables(1 IV with 2 levels; 1 DV) t-test

Two variables(1 IV with 2+ levels; 1 DV) ANOVA

Three or more variables ANOVA

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Inferential Statistics Standard Error of the Mean Significance Inferential tests you can use 1 Pengenalan kepada Statistik Inferensi