6 More Gems
every teacher of S1& S2
should know
http://ciencies.udg.edu/w3/EGarcia/images/
Bivariate52.gif
Gem number 1
Visualising the bivariate normal
distribution
Mathematica demonstration
TheBivariateNormalDistribution.nbp
There is also a nice applet at
http://lstat.kuleuven.be/java/
From Wikipedia
Gem number 2
Anscombe’s data sets
“Francis John (Frank) Anscombe (1918-05-13–
2001-10-17) was an English statistician. Born in
Hove, he was educated at Cambridge
University. After wartime service, he joined
Rothamsted Experimental Station for two years
before returning to Cambridge to lecture. He
moved to Princeton University in 1956, and
became the founding chairman of the statistics
department at Yale University in 1963”
1
• Datasets published in Anscombe, F.J.,
Graphs in Statistical Analysis, American
Statistician, 1973
• The moral of the story – always plot the
data.
Gem number 3
The central limit theorem
The central limit theorem
6
5. 5
33
3
5. 3
66
67
4. 4
33
3
4. 3
66
67
3. 3
33
3
3. 3
66
67
2. 2
33
3
2. 3
66
67
0.1
0.08
0.06
0.04
0.02
0
1. 1
33
3
1. 3
66
67
Probability
6 throws of a fair die
• For a distribution with mean µ and
variance σ², the distribution of sample
means approaches a normal distribution,
⎛
σ 2 ⎞ as the sample size, n,
N ⎜ μ,
⎟
n ⎠
⎝
increases.
• This is true no matter what the shape of
the original distribution.
Mean score
Lots of resources including video
• http://www.gapminder.org/
• Gapcast_005.mov\Gapcast_005.mov
Gem number 4
Gapminder
2
Tossing a coin
•
A tosses a fair coin until he gets the
sequence HTT.
• B tosses a fair coin until she gets the
sequence HTH.
Which of the following is true?
1. On average, A takes longer than B.
2. On average, B takes longer than A.
3. On average, they take the same time.
Gem number 5
Streaks
One way to think about it
Write out the ways you can get the streak in exactly
3, 4, 5, … tosses:
No. of
tosses
Streaks of heads
• Tossing n coins.
• U n = number of ways of getting a sequence
without 5 (or more) heads in a row
3
4
Ends
HTT
HTT
Ends
HTH
HTH
HHTT
THTT
HHTH
THTH
5
6
• U1 = 1, U 2 = 2, U 3 = 4, U 4 = 16, U 5 = 31
• For more than 5 coins, if there are no
more than 4 heads in a row, the sequence
must end with one of these:
T or TH or THH or THHH or THHHH
• U 6 = U 5 + U 4 + U 3 + U 2 + U1
• U =
n
• coins.xls
Gem number 6
Are your friends more popular
than you?
3
• Sociologists did a survey:
– They asked a random sample of people to list
their friends. They found the average number
of friends (A).
– They contacted all the friends and asked them
to list their friends. They found the average
number of friends per friend (B).
– B was bigger than A. Why?
4
Correlation coefficient
Equation of regression line
Mean
Data set 1
x
y
10
8.04
8
6.95
13
7.58
9
8.81
11
8.33
14
9.96
6
7.24
4
4.26
12 10.84
7
4.82
5
5.68
9
Correlation coefficient
Equation of regression line
Mean
Data set 3
x
y
10
7.46
8
6.77
13 12.74
9
7.11
11
7.81
14
8.84
6
6.08
4
5.39
12
8.15
7
6.42
5
5.73
Correlation coefficient
Equation of regression line
0.816
Mean
Data set 2
x
y
10
9.14
8
8.14
13
8.74
9
8.77
11
9.26
14
8.1
6
6.13
4
3.1
12
9.13
7
7.26
5
4.74
7.5
Data set 4
x
y
8
6.58
8
5.76
8
7.71
8
8.84
8
8.47
8
7.04
8
5.25
19
12.5
8
5.56
8
7.91
8
6.89
Correlation coefficient
Equation of regression line
Mean
Person A
A
Bf
Cf
Df
Ef
F
G
Hf
I
Jf
Kf
L
Mf
Nf
O
P
Qf
Rf
Sf
T
Uf
V
W
Xf
Yf
Zf
friends 16
B
f
f
f
f
f
f
f
f
f
C
f
D
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
20 16
G H
f
f f
f
f
f
f
f
E F
f
f f
f f
f
f f
f
f
f f
f
f
f
f
f f
f
f
f
f
f
J
f
f
f
K
f
f
f
f
f
f
f
f
f
L
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
O P
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
Q R
f f
f
f
f
f
f f
S
f
f
f
T
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
M N
f f
f
f
f
f
f
f
f f
f
7 8 20
I
f
f
f
f
3 12 16 15
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
U
f
f
V
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
9 13 13 17 15 12 17 20 16 13
f
f
f
f
f
f
Y
f
f
f
f
f
f
f
f
f
f
f
f
W X
f
f f
f f
Z
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
9 12 13 14 13 16
no.of
friends
16
20
16
7
8
20
3
12
16
15
9
13
13
17
15
12
17
20
16
13
9
12
13
14
13
16