Students’

Misconceptions in

Statistics

HK Mathematics Education Conference 2013

20-June-2013

NG Douglas, CHU Carlin, TSANG Kin Fun

• Common misconceptions

i. Probability/conditional probability

ii. Expectation and Variance

iii. Relationship between two variables

iv. Confidence Interval

v. Standardization and Normal distribution

vi. Relationship among distributions

Background

• Teacher:

– Share the statistical concepts in a symmetrical way

– Follow a logical sequence to introduce statistical

concepts

• Student:

– May not pay attention to all the pieces

– Learn fragmented parts

– Partial understanding of the subject matter

• Misconception:

– a wrong belief or opinion as a result of not fully understanding

something

• Cause:

– some statistical concepts are interrelated

– known portions to fill in the missing/unknown part

– similar terminologies

• Remedy:

– Spot out the related components for reinforcement

i) Probability/conditional probability

• Find P(A and B), given P(A), P(B) and P(A or B)

Probability of

compound event

P(A and B)=P(A) P(B)

Independence

condition

i) Probability/conditional probability

• Fill the missing piece

– Both independent event and dependent event

Independence

condition

Independent event:

P(A and B)= P(A)P( B)

Dependent event:

P(A and B)= P(A|B) P(B)

or P(B|A) P(A)

ii) Expectation and Variance

• Evaluate Var(2X)

As E(2X)=2E(X)

Var(2X)= 2Var(X)

Confusion:

Expectation and

Variance

ii) Expectation and Variance

• Fill the missing piece

– Relationships between Expectation and Variance

– Related mathematical proofs https://en.wikipedia.org/wiki/Variance

Confusion:

Expectation and

Variance

Relationship:

Var(X)=E[X-E(X)]2

=E[X-µ] 2

Proof:

Var(2X)=E[2X-E(2X)]2

= 4E[X-µ] 2

= 4Var(X)

Var(2X)= E[2X-E(2X)] 2

= E[2X-2µ] 2

= E[2X-2µ] 2

= E[4X 2 +4µ 2-8Xµ ]

= 4E[X 2 +µ 2-2Xµ ]

= 4E[X-µ] 2

= 4Var(X)

ii) Expectation and Variance

• Evaluate Var(X+Y) and Var(X-Y) (Out of DSE syllabus)

– Some simple equations may …

E(X+Y)=E(X)+E(Y)

Var(X+Y)=Var(X)+Var(Y)

Var(X-Y) =Var(X)+Var(Y)

When X, Y are independent

Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)

Var(X-Y) =Var(X)+Var(Y)-2Cov(X,Y)

When X, Y are not independent

Expectation and

Variance have

different

equations

iii) Relationship between two variables

• What is the conclusion of zero correlation?

Independence => zero correlation

Therefore,

zero correlation =>Independence

Zero correlation

iii) Relationship between two variables

• Fill the missing piece

– Mathematical example http://mathforum.org/library/drmath/view/64808.html

http://en.wikipedia.org/wiki/Correlation_and_dependence

http://www.purplemath.com/modules/scattreg2.htm

Zero correlation

does not imply

independence

Pictorial examples

Other example : Y=X2

X and Y are clearly

dependent

Their correlation is zero

iv) Confidence Interval

"we are 95% confident that the true

value of the parameter is in our

confidence interval”

Confident => Probability ?

Which position is a more

reasonable guess of the

true parameter value ?

P1 P2

iv) Confidence Interval

• Fill the missing piecehttp://en.wikipedia.org/wiki/Confidence_interval

Concepts of

Confidence

interval

Meaning:

Drawing N sample and construct N

different CI, 95% of the observed

confidence intervals will hold the

true value of the parameter.

After a sample is taken, the population

parameter is either in the interval made or not,

there is no chance.

i.e. Prob (θ| sample observation) = 0 or 1

iv) Confidence Interval (Advanced)

If the 95% CI that’s constructed for one

sample partially overlaps the 95% CI

that’s constructed from a second

independent sample, the two samples

statistics are not significantly different

from each other at α = 0.05.

CI of first sample CI of second sample

Compare the

CI ?

iv) Confidence Interval (Advanced)

• Fill the missing piece

– Mathematical example http://www.statisticalmisconceptions.com/MiscAndInvite07b.html

http://www.cscu.cornell.edu/news/statnews/stnews73.pdf

http://www.measuringusability.com/blog/ci-10things.php

Concepts of

Confidence

interval

Applications:

Hypothesis testing

v) Standardization and Normal distribution

Is standardized data

normally distributed ?

Standardization is used

together with Normal

distribution most of the time

v) Standardization and Normal distribution

Confusion

Standardization ensure

Mean=0

SD=1

W=(X-µ)/ σ

Shifting: using µ

Scaling: using σ

vi) Relationship among distributions

Normal

distribution

Poisson

distributionGeometric

distribution

Bernoulli

distribution

Binomial

distribution

• Ref: http://math.wustl.edu/~jmding/math493/dist.pdf

• Ref: http://math.wustl.edu/~jmding/math493/dist.pdf

Out of DSE syllabus

• Ref:

http://curricular.providen

ce.edu/~rgoldstein/statis

tics/BinPoissNorm.pdf

• Poisson or Binomial distribution• If a mean or average probability of an event happening per unit time/per page/per mile cycled etc.,

is given, and you are asked to calculate a probability of n events happening in a given

time/number of pages/number of miles cycled, then the Poisson Distribution is used.

• If, on the other hand, an exact probability of an event happening is given, or implied, in the

question, and you are asked to caclulate the probability of this event happening k times out of n,

then the Binomial Distribution must be used.

• Ref: http://personal.maths.surrey.ac.uk/st/J.Deane/Teach/se202/poiss_bin.html

• http://curricular.providence.edu/~rgoldstein/statistics/BinPoissNorm.pdf

Reference

• http://www.statisticalmisconceptions.com/

• http://blog.minitab.com/blog/real-world-quality-improvement/3-common-and-dangerous-statistical-

misconceptions

• http://statswithcats.wordpress.com/2011/02/20/six-misconceptions-about-statistics-you-may-get-

from-stats-101/

• https://en.wikipedia.org/wiki/Variance

• http://mathforum.org/library/drmath/view/64808.html

• http://en.wikipedia.org/wiki/Correlation_and_dependence

• http://www.purplemath.com/modules/scattreg2.htm

• http://www.measuringusability.com/blog/ci-10things.php

• http://math.wustl.edu/~jmding/math493/dist.pdf

• http://personal.maths.surrey.ac.uk/st/J.Deane/Teach/se202/poiss_bin.html

• http://curricular.providence.edu/~rgoldstein/statistics/BinPoissNorm.pdf

iphone Statistics App(free)

Probability/conditional probability

If someone is diagnosed as having a

very rare and fatal disease, and if the

procedure used to come up with this

diagnosis is 99 percent accurate, then

the person who’s been diagnosed has a

right to feel that “the end is near.”

• Fill the missing piece

– http://www.statisticalmisconceptions.com/MiscAndInvite05f.html