MEI Conference 2014
Introduction to S1:
Variation and Discrete
Random Variables
Clare Parsons
clare.parsons@mei.org.uk
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May 2014 © MEI
S1 Variation & Discrete Random Variables
A Different Measure of Spread
Could there be more than one set of data with 22 values and a mean of 23 that would give this box plot?
Standard deviation
Standard deviation measures spread by calculating an “average” distance of the data values from the mean
x  x 
“divisor n” also called root mean
square deviation
OR
population standard deviation
2
n
x  x 
“divisor n – 1” also called sample
standard deviation
2
n 1
S xx    xi  x    xi 2
2
standard deviation
s =
root mean square
deviation, rmsd
=
 x

n
2
  xi 2  nx 2
S xx
n 1
variance,
S xx
n
mean square deviation,
2
msd
=
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s2 =
S xx
n 1
S xx
n
May 2014 © MEI
S1 Variation & Discrete Random Variables
STATISTICS FUNCTION ON BASIC CASIO CALCULATORS
You can use a basic scientific calculator with a statistics function to find the mean, ̅ ,
standard deviation, σ or s.
TO START :
MODE
2
(STAT)
1 : 1 - VAR
(for lots of just ‘x’ s given in a list OR in a frequency table)
put data into the table by inputting number followed by = and use the cursor to move between
columns and rows
x
NOTE IF A TABLE DOESN’T COME UP
YOU NEED TO SET IT UP AS A
DEFAULT USING:
FREQ
1
2
3
PRESS AC go to STAT (SHIFT 1) from menu below choose 4: Var
1:Type
3: Sum
5: MinMax
2: Data
4: Var
and simply choose what you want to know!
(  x and
 x are on the 3:sum menu)
2
On older models of basic scientific calculators you can also find the product moment correlation
coefficient, r, and the regression line of y on x: y = a + bx
2 : A + BX
(for bivariate data , pairs of x and y data)
put data into the table by inputting numbers followed by = and using the cursor to move
between columns and rows as before
x
y
1
2
3
PRESS AC
1:Type
3: Sum
5: Reg
4: Var
5: Reg
return to STAT, from menu below choose
2: Data
4: Var
6: MinMax
for means, standard deviations
for PMCC and coefficients of regression line y = a + bx
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May 2014 © MEI
S1 Variation & Discrete Random Variables
STATISTICS FUNCTION ON CASIO
fx-CG20 GRAPHICAL CALCULATOR
You can use a graphical calculator with a statistics function
to find the mean, ̅ , standard deviation, σ or s, product
moment correlation coefficient, r, the regression line of y on
x: y = a + bx, draw graphs and so much more.
Choose the STATS menu (2) Press EXE
Put data into the table by inputting number followed by
EXE and use the cursor to move between columns
and rows
1
2
3
4
5
6
List 1
0
1
2
3
4
5
List 2
37
52
48
34
17
12
List 3
List 4
(can store up to 26 lists with 100 entries)
1 - VAR
(for lots of just ‘x’ s given in a list OR in a frequency table)
Just press F1 to get the statistics!
Press F6 to get a histogram (slightly skew –whiff?)
To return to the table press EXE
You may have to change to 1 VAR to make sure that you are in single variable statistics if the
calculator has been used to calculate regression say (2 Variable) .
Make List 2 the frequency.
Notes
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May 2014 © MEI
Frequency and Probability Distributions
x
0 1 2 3 4 5
frequency, f 37 52 48 34 17 12
Mean score, ̅
∑
∑
Variance,
∑
̅
OBSERVED DATA
FREQUENCY DISTRIBUTION
x is the difference in the value of the total score when 2 dice are thrown
……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………....
PROBABILITY DISTRIBUTION
x
P(X=x)
Mean score,
(
E(X) = ∑
0
1
2
3
4
5
Variance,
[(
( )
)
= ∑
=
=
p5 of 14
) ]
(
May 2014 © MEI
(
)
) – [ ( )]
THE THEORY
X is ‘the difference when 2 dice are thrown’
0 1 2 3 4
37 52 48 34 17
∑(
̅)
∑(
̅)
̅
∑
∑ f
∑
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May 2014 © MEI
5
12
∑
∑
S1 Variation & Discrete Random Variables
mean
variance
standard
deviation
mean square number of
deviation
data items
√
200
1.43113
2.0379
1.89
2.04814
378
407.58
1.42755
1122
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May 2014 © MEI
Lesson Idea – Discrete Random Variables
It is important to link the concept of discrete random variables to real-life examples such as the number of
heads when we toss 4 coins or the total of the scores on two dice but it is also important that students
understand these concepts which refer to discrete random variables in general:




The total probability is 1.
There can be an infinite number of possible values
The possible values are usually positive integers but need not be
Probabilities can be given using a general formula.
In this card matching activity these concepts will need to be considered and there are separate pages
containing

8 tables/rules (2 are definitely NOT DRVs, the others can be made into DRVs with an appropriate
choice of probabilities)

6 graphs of discrete probability distributions

12 calculated probabilities

6 descriptions of possible scenarios

4 descriptions of named probability distributions
It is recommended that you give thought to which sets of cards you are going to use. The complete set
would be very suitable for revision, but for an introductory lesson on DRVs the first two, three or four sets
may suffice.
There are different ways in which you could introduce this particular activity, providing differing degrees of
scaffolding, including:
 Start with considering cards from a few categories as examples and lead a whole class discussion
about what they show.
 Ask all groups to first consider the table/rules cards and identify which could be DRVs and only then
give out the cards in the other categories gradually in the order of the list above
 Give out all the cards at once and see what happens
Different strategies are suitable for different groups of students.
Recording Results of Matching Activities
If this matching activity is the main part a lesson, you and your students may be concerned about not
having a permanent record of what has been done. Clearly plenary activities and subsequent work set can
provide a record, but if you have decided to not to make the resource permanent and have printed on
paper, simply getting individual students to glue some of the matchings onto paper or in their exercise
books, and annotating, works well. Similarly displaying the results as a poster means a record of the activity
is shared with all. Asking all students to annotate at least one matching on a large poster is a way of
assessing individual learning.
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S1 Variation & Discrete Random Variables
Probability, p(x)
0.5
0.4
0.3
0.2
0.1
X
1
0.5
Probability, p(x)
0.5
0.4
2
3
4
5
6
Probability, p(x)
0.4
0.3
0.3
0.2
0.2
0.1
X
1
2
3
4
5
6
0.1
7
X
1
0.5
2
3
4
5
6
7
Probability, p(x)
Probability, p(x)
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
X
X
1
2
3
4
5
6
7
8
−2
−1
1
9
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2
S1 Variation & Discrete Random Variables
(
)
1
P(X  n)  n
2
for n = 0, 1, 2, 3, 4, 5 ...
1
x
P(X=x)
x
0
2
1
x  0,1, 2
 18 x

P(X  x)   18
0

3
2
4
P(X=x)
4
1
P(X  n)  n
2
for n = 1, 2, 3, 4, 5 ….
x
P(X=x)
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otherwise
x
0 1
2
3
P(X=x) 0.1 0.1 0.5 0.1
4
0.25
3
x  3, 4
-2
-1
0
0
1
0
May 2014 © MEI
2
S1 Variation & Discrete Random Variables
This is example of the
GEOMETRIC distribution
with parameter
,
Geo (0.5)
This discrete random
variable has a POISSON
distribution with
parameter
( )
This Discrete random
variable has a BINOMIAL
distribution with parameters
n = 4, p= 0.5
X  B(4, 0.5)
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This is an example of a
DISCRETE UNIFORM
probability distribution
over the interval [ ]
May 2014 © MEI
S1 Variation & Discrete Random Variables
X is how many £s you are ‘up’,
if you play 2 games of chance.
X is the score
(labelled 1,2,3,4) when an The probability of you winning
a game is , you play £1 to
unbiased 4 sided spinner is
play each time and get £2 back
spun
if you win.
X is the number of heads
X is the number of tosses of when 4 unbiased coins are
a coin until you get a head.
tossed
X is the number of people
X is the number of texts
sitting at a table for four in a
someone receives per hour.
restaurant at lunchtime
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May 2014 © MEI
S1 Variation & Discrete Random Variables
(
)
(
)
(
)
(
(
(
)
(
)
(
)
)
)
(
P(X=0) = 0.1353
(
)
(
0.27
)
p13 of 14
)
(
)
May 2014 © MEI
S1 Variation & Discrete Random Variables
Linear Scaling
Complete these dot diagrams:
What do you see happening? Can you generalise these results?
Using a frequency table on your calculator, find the mean, range and standard deviation of the original set
of data and each of the transformed sets. Do the numbers confirm your generalisation?
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