using statistical functions on a scientific calculator (1)

Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St 1

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR

Objective:

1. Improve calculator skills needed in a multiple choice statistical examination where the

exam allows the student to use a scientific calculator.

2. Helps students identify where and what their problem may be.

Target group: This worksheet is for students who do not have a sound mathematical

background and are doing a statistics module; students who did maths literacy at school;

students who have a supplementary for a statistics exam.

FUNCTION

(normal mode)

CASIO fx-991ES PLUS SHARP EL-532WH MY NOTES

FROM QL

WORKSHOP

Proper fractions 1

4

Improper fractions 10

3

1 4 =1 4

10 3 = 3 1 3

1 bac

4 = 1 4

10 bac

3 = 3 1 3

(the answer is 1

33

)

Mixed fractions 1

43

4 1 3 = 4 1 3

SHIFT SD = 13 3

4 bac

1 bac

3 = 4 1 3

2ndF bac

answer 13 3

Converting between

fractions and decimals

Follows from above

13 3 SD answer 4.333

4.333 SD answer 4 1 3

Follows from above

13 3 bac

answer 4.3333

4.3333 bac

answer 4 1 3


Having difficulty simplifying 4

12 without the use of a calculator? Did you know it is much quicker to do

it mentally than reach for a calculator?! Try doing a worksheet or QL workshop on

“FRACTIONS”

!n n factorial

! 1 2 1n n n

5! 5 4 3 2 1

The factorial notation is

used to represent the

product of the first

n natural numbers,

where 0n

5 SHIFT 1x = 120

The button with orange

SHIFT above it engages the

orange colour stats

functions above each

numerical pad button.

5 2ndF 4 =120

The orange 2ndF button

engages the orange colour

stats functions in the left

hand corner of each

numerical pad button.

Fractions using !n 5!

3!

5 SHIFT 1x 3 SHIFT 1x

= 20

5 2ndF 4 3 2ndF 4

= 20

Fractions with two

factorials in the

denominator 10!

7!3!

10 SHIFT 1x ( 7 SHIFT

1x 3 SHIFT 1x ) =

120 Brackets NB! Why?

10 2ndF 4 ( 7 2ndF

4 3 2ndF 4 ) = 120

Brackets NB! Why?

Permutations nPr

(no repeats; order is

important)

!

( )!

nnPr

n r

9!9 6

(9 6)!P

9 SHIFT 6=60480

Before pushing equal the

screen will have9 6P .

Else

9 SHIFT 1x ( 9 – 6 )

SHIFT 1x = 60480

9 2ndF 6 6=60480

After pushing equal the

screen will have9 6P .

Else

9 2ndF 4 ( 9 – 6 )

2ndF 4 = 60480

Combinations nCr

(no repeats; order is

NOT important)

9 SHIFT 6=84

Before pushing equal the

screen will have9 6C .

Else

9 2ndF 5 6=84

After pushing equal the

screen will have9 6C .

Else


!

!( )!

nnCr

r n r

9!9 6

6!(9 6)!C

9 SHIFT 1x ( 6 SHIFT

1x ( 9 – 6 ) SHIFT 1x

) = 84

Both sets of brackets NB!

Why?

9 2ndF 4 ( 6 2ndF

4 ( 9 – 6 ) 2ndF 4

) = 84

Both sets of Brackets NB!

Why?

Counting r objects in

n different ways rn

(repeats/repetition) 65

5 x 6 ) = 15625

5 xy 6 = 15625

See diagram 1 for determining the differences between PERMUTATIONS & COMBINATIONS.

(Appendix A attached at the end of this QL worksheet)

Exponents in a bracket

(9 6)0.75

(9 6)0.75 2

0.75 x 9 – 6 ) = 0.422

The calculator already

opens with a bracket, it just

needs to be closed. Why

and when?

You get the answer without

closing the bracket, BUT

what would happen if you

multiplied the expression by

2?

0.75 x 9 – 6 ) 2 = 0.844

Now closing the bracket is

important. Why?

0.75 xy ( 9 – 6 ) = 0.422

Multiply the expression by 2

before pressing the equal

sign,

0.75 xy ( 9 – 6 ) 2 =0.844

Brackets NB! Why?

Powers with base e xe

1

3e

SHIFT ln 1 3 ) = 1.396

Alternatively,

2ndF ln 1 bac

3 = 1.396

Alternatively,


3e

3

1

e

2

34e

SHIFT ln 1 3 ) = 1.396

SHIFT ln ( ) 3 ) = 0.05

1 SHIFT ln 3 ) = 0.05

Alternatively,

1 SHIFT ln 3 ) = 0.05

Note: 3

3

1e

e. Why?

But: 1

33e e .

( ) 4 SHIFT ln ( ) 2

3 ) = -2.054

Alternatively replace with

.

2ndF ln ( 1 3 ) = 1.396

2ndF ln 3 = 0.05

The calculator automatically

puts brackets around the first

term. Why is there no need to

put brackets in this situation?

1 2ndF ln ( 3 ) = 0.05

Alternatively,

1 bac

2ndF ln ( 3 ) =

0.05

Note: 3

3

1e

e. Why?

But: 1

33e e .

4 2ndF ln ( 2

3 ) = -2.054


bac

.

Discrete probabilty distributions make use of the universal constant e . Try a worksheet or QL workshop

on

“DISCRETE PROBABILITY DISTRIBUTION FUNCTIONS”

Square root x

When to use brackets

6

6 ) = 2.45


( 6 ) = 2.45

Brackets are not required


Two terms under the

square root sign

64 20

64 20

2

Square root as a

denominator

1

6

Square root as a

fraction in the

denominator, where

is in the numerator

puts brackets around the

first term.

6 = 2.45

When there is one term

brackets are not necessary.

Why?

6 4 – 2 0 ) = 6.63

Leaving out the closing

brackets gives the same

answer.

Why the emphasis on

brackets?????

6 4 – 2 0 ) 2 = 3.32

The bracket must be closed.

Why?


.

1 6 ) = 0.41

Similar to 6 above.

1 ( 6 ) 2 ) =

0.82


when there is one term.

6 = 2.45

Why?

( 6 4 – 2 0 ) = 6.63

Can you see why brackets are

important in this situation?

( 6 4 – 2 0 ) 2 = 3.32

The brackets tell the square

root sign what needs to be

beneath the square root.

1 ( 6 ) = 0.41

Similar to 6 above.

1 ( ( 6 ) 2 ) =

0.82

The outer red bolded set of

brackets (arrows above) is

Brackets for

Denominator

Brackets for

Denominator


of the fraction

1

6

2

Square root as a

fraction in the

denominator, where

is in the

denominator of the

fraction 1

3.5

10

Denominator having a

square root with two

terms 6

24 12

Brackets required above

and below

7.3 3.5

3.5

10

opens the square root sign

with a bracket. Now the

bracket must be closed.

Why? The denominator

also requires brackets.

Why?

1 ( 3 . 5 10 ) ) =

0.9035

Alternatively ,

1 3 . 5 1 0 ) =

0.9035

Using the function

means that there is no need

for the fraction to go into

brackets.

6 ( 2 4 – 12 ) = 1.73


.

The calculator always opens

the square root sign. In this

situation it is not essential

to close bracket but it

creates a habit.

( 7.3 – 3.5 ) ( 3.5

10 ) ) = 3.433

Get into the habit of putting

brackets at the top of a

division line and at the

bottom. Why?

necessary. (a must!) The inner

is not (see 6 above). Why?

1 ( 3 . 5 ( 10 ) ) =

0.9035

Alternatively,

1 3 . 5 bac

10 = 0.9035

Using the function bac

means that there is no need

for the fraction to go into

brackets. Yes you can use

bac

for both but lower

one requires brackets. Why?

6 ( 2 4 – 1 2 ) = 1.73


bac

.

It is essential to open with a

bracket under the square

root sign. In this situation it is

not essential to close bracket

but it creates a habit.

( 7.3 – 3.5 ) ( 3.5

10 ) = 3.433

Get into the habit of putting

brackets at the top of a

division line and at the

bottom. Why?

Brackets for

Denominator

Brackets for

Denominator


7.3 3.5

3.5

10

The calculator opened the

bracket beneath the square

root sign, hence the double

closed bracket before the

equal sign. The bottom

brackets can be removed

ONLY if the fraction function

is used to replace the

second division sign. Try

this!

7.3 3.5

3.5

10

brackets are not required

beneath the square root sign

because there is only one

term. The bottom brackets

can be removed ONLY if the

fraction function

bac

to replace the

second division sign. Try this!

Having difficulty answering all or some of the ‘why?’ above ….. try a worksheet or QL workshop on

“ORDER OF OPERATION (BODMAS)”

percentage sign %

converting from 67% to

a probability

6 7 SHIFT ( = 0.67

6 7 2ndF 1 = ERROR 1

Begin by doing 6 7 1 = 67

now 2ndF 1 = 0.67

Why does this work? The

calculator requires an ANS

then the 2ndF 1 = 0.67

Having difficulty doing the above without a calculator, then try a worksheet or QL workshop on

“PERCENTAGES”

The next sections deal with using the MODE function on a scientific calculator.

ONE variable of data

x

ENTER Data for x :

MODE

Choose option 3:STAT by

pushing the keypad for 3

Choose option 1: 1-VAR by

MODE

Choose option 1

STAT by



16; 15; 24; 7; 9; 21; 34 pushing the keypad for 1

Enter the data as follows:

16 = 15= 24= 7= 9= 21=

34= using the toggle button

move the cursor up to the

last value, i.e. 34 on the

screen then push AC

Choose option 0

SD by


Notice the following on the

screen 0Stat


16 M+ 15 M+ 24 M+ 7 M+ 9

M+ 21 M+ 34 M+

Notice the screen says

DATA SET 7

There are 7 values in the

dataset.

SUM of x

2x

3: 2:

1 3 2

126

STAT SUM x

SHIFT

23: 1:

1 3 1

2784

STAT SUM x

SHIFT

Use either the teal ALPHA

button or the RCL button to

recall the information

126

x

ALPHA

2

2784

x

ALPHA

The ALPHA or the RCL button

engages the teal colour stats

functions in the right hand

corner of each numerical pad

button.

Combining sum and

variable functions

functions

x

n

3: 2:

4: 1:

1 3 2

1 4 1

18

STAT SUM x

STAT VAR n

SHIFT

SHIFT

0

18

x n

ALPHA ALPHA

REPLAY


2

2

1

xx

n

n

Before beginning such a

problem, insert the brackets

2

2

1

xx

n

n

Brackets inserted:

underneath the square root

sign; at the top (numerator)

and the bottom

(denominator) of a fraction.

2

calculatoropens this one

3: 1:

3: 2:

2

4: 1:

4: 1:

1 3 1

1 3 2

1

4 1

1 4 1 1

= 9.274

STAT SUM x

STAT SUM x

STAT

VAR n

STAT VAR n

SHIFT

SHIFT

x SHIFT

SHIFT

See the importance of

brackets? Brackets create

steps.

Before beginning such a

problem, insert the brackets

2

2

1

xx

n

n

Brackets inserted:

underneath the square root

sign; at the top (numerator)

and the bottom

(denominator) of a fraction

2

2

0

0 1

= 9.274

x

x

n

n

ALPHA

ALPHA

x ALPHA

ALPHA

See the importance of

brackets? Brackets create

steps.

SIZE sample n

Sample MEAN x

4: 1:

1 4 1

7

STAT VAR n

SHIFT

4: 2:

1 4 2

18

STAT VAR x

SHIFT

0 7

n

ALPHA

4 18

x

ALPHA


Sample STD DEVIATION

sx

Sample VARIANCE

2s

4: 4:

1 4 4

9.274

STAT VAR sx

SHIFT

Option 3: x is for the

POPULATION standard

deviation. What is the

difference between a

sample and population

standard deviation?

4: 4:

2

1 4 4

86

STAT VAR sx

SHIFT

x

Notice that when using the

calculator, the sample

standard deviaiton is

obtained first, squaring the

value gives the sample

variance.

2standard deviation variance

5 9.274

sx

ALPHA

6

x

ALPHA is for the

POPULATION standard

deviation. What is the

difference between a sample

and population standard

deviation?

2 5 86

sx

ALPHA x

Notice that when using the

calculator, the sample

standard deviaiton is

obtained first, squaring the

value gives the sample

variance.


It is one thing to learn to use a calculator, but a sound understanding of Descriptive Statistics is needed.

Try do worksheets or QL workshops on the following: “DESCRIPTIVE STATISTICS”

ONE variable of data

using FREQUENCY

option

Example: pdf of a

discrete random

variable

x 0 1 2 3

( )P x 0.25 0.4 0.2 0.15

SHIFT MODE toggle down

using

4:

1:

?4 1

STATON

Frequency

MODE

1:13:

3 1VARSTAT


0 = 1= 2= 3=

MODE

Choose option 1

STAT by


Choose option 0

SD by



screen 0Stat


REPLAY


The expected value of

x

The standard deviation

of x

The variance of x

Then toggle to the top

where x is 0

then move to

the freq column using the

right arrow on the replay

button.

0.25 = 0.4 = 0.2 =0.15 =using

the toggle button move

the cursor up to the last line

where x=3 and freq=0.15 on

the screen then push AC

x in this situation is the

variables and freq is the

probability of x , i.e. P x .

4: 2:

1 4 2

1.25

STAT VAR x

SHIFT

4: 3:

1 4 3

0.9937

STAT VAR x

SHIFT

Notice that

4: 4:

1 4 4 STAT VAR sx

SHIFT ERROR

In this situation the

POPULATION standard

deviation is used. Why?

4: 3:

2

1 4 3

0.9875

STAT VAR x

SHIFT

x


Remove frequency from the

screen:

0 STO 0.25 M+

1 STO 0.4 M+

2 STO 0.2 M+

3 STO 0.15 M+


DATA SET 4

There are 4 values in the

dataset.

x in this situation is the

variables and y is the

probability of x , i.e. P x

4 1.25

x

ALPHA

6 0.9937

x

ALPHA

Notice that

5 2

sx

ALPHA ERROR . In this

situation the POPULATION

standard deviation is used.

Why?

2 6 0.9875

x

ALPHA x






button.

REPLAY


SHIFT MODE toggle down

using

4:

2:

?4 2

STATOFF

Frequency

It is not sufficient to learn only the calculator version. Do a worksheet or QL workshop on

“DISCRETE PROBABILITY DISTRIBUTION FUNCTIONS”

TWO variables of data

;x y

For example:

2.6 5.6

2.6 5.1

3.2 5.4

3.0 5.0

2.4 4.0

3.7 5.0

3.7 5.2

x y

MODE 2:3:

3 2A BXSTAT


2.6 = 2.6= 3.2= 3.0= 2.4=

3.7= 3.7=

Then toggle to the top

to the very first

x equal to 2.6

then move to

the freq column.

5.6 = 5.1 = 5.4 = 5.0= 4.0 =

5.0= 5.2=

using the toggle button

move the cursor up to the

last line where x=3.7 and

y=5.2 on the screen then

push AC

Make sure that the variable

x is entered underneath

the x column and the

y variable underneath the

y column.

MODE

Choose option 1

STAT by


Choose option 1

LINE by



screen 1Stat


Always the x variable first,

then the y variable. NB!!

2.6 STO 5.6 M+

2.6 STO 5.1 M+

3.2 STO 5.4 M+

3.0 STO 5.0 M+

2.4 STO 4.0 M+

3.7 STO 5.0 M+

3.7 STO 5.2 M+

REPLAY

REPLAY


Notice that a bx is the

mathematical equation for a

straight line.


DATA SET 7

There are 7 paired values in

the dataset.

SIZE sample n

Sample MEAN

x

y

Sample STD DEVIATION

sx

sy

4: 1:

1 4 1

7

STAT VAR n

SHIFT

4: 2:

1 4 2

3.029

STAT VAR x

SHIFT

4: 5:

1 4 5

5.043

STAT VAR y

SHIFT

4: 4:

1 4 4

0.531

STAT VAR sx

SHIFT

4: 7:

1 4 7

0.509

STAT VAR sy

SHIFT

0 7

n

ALPHA

4 3.029

x

ALPHA

7 5.043

y

ALPHA

5 0.531

sx

ALPHA

8 0.509

sy

ALPHA





button.

SUM of x

3: 2:

1 3 2

21.2

STAT SUM x

SHIFT

Use either the teal ALPHA

button or the RCL button to

recall the information

21.2

x

ALPHA


y

2x

2y

xy

3: 4:

1 3 4

35.3

STAT SUM y

SHIFT

23: 1:

1 3 1

65.9

STAT SUM x

SHIFT

23: 3:

1 3 3

179.57

STAT SUM y

SHIFT

3: 5:

1 3 5

107.44

STAT SUM xy

SHIFT

2 35.3

y

ALPHA

2

65.9

x

ALPHA

2

3 179.57

y

ALPHA

1 107.44

xy

ALPHA





button.

Regression coefficients

0 1

slopeintercept

y b b x

Subscript increases 0 to

1

On calculator

intercept slope

y a b x

Alphabet increases from

a to b.

Hence,

Intercept 0b a

Intercept: 0b a

1:5:Re

1 5 1

4.093

STAT Ag

SHIFT

Slope: 1b b

2:5:Re

1 5 1

0.314

STAT Bg

SHIFT

Correlation coefficient: r

5:Re 3:

1 5 3

0.327

STAT g r

SHIFT

Intercept: 0b a

( 4.093

a

ALPHA

Slope: 1b b

) 0.314

b

ALPHA

Correlation coefficient: r

0.327

r

ALPHA


Slope 1b b

Coefficient of

Determination: 2r

5:Re 3:

2

1 5 3

0.107

STAT g r

SHIFT

x

Estimated value of y i.e. y

when 3.1x :

1:5:Re

2:5:Re

1 5 1

1

5 2 3.1

=5.065

STAT Ag

STAT

Bg

SHIFT

SHIFT

Alternatively,

ˆ5:Re 5:

3.1 1

5 5 5.065

STAT

g y

SHIFT

Coefficient of

Determination: 2r

2 x 0.107

r

ALPHA

Estimated value of y i.e. y

when 3.1x :

(

)

3.1 5.065

a

b

ALPHA

ALPHA

Alternatively,

'

3.1 2 ) 5.065

y

ndF

The above section requires a sound understanding of correlation and simple regression. Try a worksheet

or a QL workshop on “CORRELATION AND SIMPLE REGRESSION”.

Contact [email protected] or a Quantitative Literacy Facilitator in your region

mailto:[email protected]


Appendix A:

Note that nCr nPr . Why?

!

!( )!

! 1

!( )! !

!

ncombination

r n r

ncombination permutation

r n r r

permutation r combination

So there will be !r permutations for every possible combination.

Diagram 1: Counting: Permutations and Combinations

Repetition

(repeats)

Yes

No

rn

ORDER important

Yes

Permutation ( nPr )

No

Combination ( nCr )

using statistical functions on a scientific calculator (1)

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