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MEI Conference 2014

Introduction to S1:

Variation and Discrete

Random Variables

Clare Parsons clare.parsons@mei.org.uk

S1 Variation & Discrete Random Variables

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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

“divisor n” also called root mean square deviation OR population standard deviation

“divisor n – 1” also called sample standard deviation

2

2 2 2 2

xx i i i

xS x x x x nx

n

standard deviation

s = 1

xxS

n

variance,

s2 =

1

xxS

n

root mean square

deviation, rmsd = xxS

n

mean square deviation,

msd 2 = xxS

n

2

x x

n

2

1

x x

n

S1 Variation & Discrete Random Variables

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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

PRESS AC go to STAT (SHIFT 1) from menu below choose 4: Var

1:Type 2: Data 3: Sum 4: Var 5: MinMax

and simply choose what you want to know! ( x and 2x are on the 3:sum menu)

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

PRESS AC return to STAT, from menu below choose

1:Type 2: Data 3: Sum 4: Var 5: Reg 6: MinMax

4: Var for means, standard deviations 5: Reg for PMCC and coefficients of regression line y = a + bx

x FREQ

1

2

3

x y

1

2

3

NOTE IF A TABLE DOESN’T COME UP

YOU NEED TO SET IT UP AS A

DEFAULT USING:

S1 Variation & Discrete Random Variables

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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

(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.

List 1 List 2 List 3 List 4

1 0 37

2 1 52

3 2 48

4 3 34

5 4 17

6 5 12

Notes

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Frequency and Probability Distributions

FREQUENCY DISTRIBUTION

x is the difference in the value of the total score when 2 dice are thrown

Mean score, ∑

∑ Variance,

∑

……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………....

PROBABILITY DISTRIBUTION

X is ‘the difference when 2 dice are thrown’

Mean score, Variance,

E(X) = ∑ ( ) ( ) [( ) ] ( ) – [ ( )]

= = ∑ ( )

=

x 0 1 2 3 4 5 frequency, f 37 52 48 34 17 12

x 0 1 2 3 4 5 P(X=x)

OB

SE

RV

ED

DA

TA

T

HE

TH

EO

RY

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∑( ) ∑

∑

∑( )

∑ ∑ f

∑

0 1 2 3 4 5

37 52 48 34 17 12

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mean standard deviation √

variance mean square

deviation

number of data items

200

1.43113 2.0379 1.89 2.04814

378 407.58 1.42755 1122

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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.

S1 Variation & Discrete Random Variables

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1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

X

Probability, p(x)

1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

X

Probability, p(x)

1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

X

Probability, p(x)

1 2 3 4 5 6 7 8 9

0.1

0.2

0.3

0.4

0.5

X

Probability, p(x)

−2 −1 1 2

0.1

0.2

0.3

0.4

0.5

X

Probability, p(x)

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( )

18

18

0,1,2

(X ) 3,4

0

x x

P x x

otherwise

1P(X )

2nn

for n = 0, 1, 2, 3, 4, 5 ...

x 0 1 2 3 4 P(X=x) 0.1 0.1 0.5 0.1

x 1 2 3 4

P(X=x) 0.25

1P(X )

2nn

for n = 1, 2, 3, 4, 5 ….

x 0 1 2 3 4

P(X=x)

x -2 -1 0 1 2

P(X=x)

0 0

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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 (4, 0.5)B

This is an example of a DISCRETE UNIFORM probability distribution

over the interval [ ]

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X is the score

(labelled 1,2,3,4) when an unbiased 4 sided spinner is

spun

X is how many £s you are ‘up’, if you play 2 games of chance.

The probability of you winning

a game is

, you play £1 to

play each time and get £2 back

if you win.

X is the number of tosses of a coin until you get a head.

X is the number of heads when 4 unbiased coins are

tossed

X is the number of people sitting at a table for four in a

restaurant at lunchtime

X is the number of texts someone receives per hour.

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( )

( )

( )

( )

( )

( )

P(X=0) = 0.1353

( )

( )

( )

( ) ( ) 0.27

( )

S1 Variation & Discrete Random Variables

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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?