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
CASIO fx-991ES PLUS
MY NOTES
FROM QL
WORKSHOP
SHARP EL-532WH
(normal mode)
Proper fractions
1
1
4
Improper fractions
10
3
10
4 =1
4
3=3
1
1
3
ab
10
4=1 4
c
ab
3=3
c
1
3
1
(the answer is 3 )
3
Mixed fractions 4
1
3
4
1
3=4
1
3
4
ab
c
2ndF
SHIFT SD = 13
Converting between
fractions and decimals
1
ab
ab
c
c
3=4
1
3
answer 13 3
3
Follows from above
Follows from above
13 3 SD answer 4.333
13 3 a b
c
4.333 SD answer 4 1 3
4.3333 a b
answer 4.3333
c
answer 4 1 3
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4
without the use of a calculator? Did you know it is much quicker to do
12
Having difficulty simplifying
it mentally than reach for a calculator?! Try doing a worksheet or QL workshop on
“FRACTIONS”
n!  n factorial
5 SHIFT
n! n n 1  2 1
5! 5 4 3 2 1
The factorial notation is
used to represent the
product of the first
n natural numbers,
where n
x
1
5 2ndF
= 120
The button with orange
SHIFT above it engages the
orange colour stats
functions above each
numerical pad button.
4 =120
The orange 2ndF button
engages the orange colour
stats functions in the left
hand corner of each
numerical pad button.
0
Fractions using n!
5!
3!
Fractions with two
factorials in the
denominator
10!
7!3!
5 SHIFT x
= 20
1
10 SHIFT x
x
1
120
3 SHIFT x
1
1
( 7 SHIFT
3 SHIFT x
1
) =
Brackets NB! Why?
5 2ndF
= 20
4
10 2ndF 4
4
3 2ndF
( 7 2ndF
3 2ndF 4
) = 120
Brackets NB! Why?
Permutations  nPr
9 SHIFT
(no repeats; order is
important)
Before pushing equal the
screen will have 9P6 .
After pushing equal the
screen will have 9P6 .
Else
Else
nPr
9P6
n!
(n r )!
9!
(9 6)!
6=60480
9 SHIFT x
SHIFT x
1
1
( 9–6 )
= 60480
9 2ndF 6 6=60480
9 2ndF 4
2ndF 4
( 9–6 )
= 60480
Combinations  nCr
9 SHIFT
(no repeats; order is
NOT important)
Before pushing equal the
screen will have 9C6 .
After pushing equal the
screen will have 9C6 .
Else
Else
6=84
4
9 2ndF 5 6=84
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nCr
n!
r !(n r )!
9 SHIFT x
x
9C 6
9!
6!(9 6)!
1
1
( 6 SHIFT
( 9 – 6 ) SHIFT x
1
9 2ndF
4
( 6 2ndF
4 ( 9 – 6 ) 2ndF 4
) = 84
) = 84
Both sets of brackets NB!
Why?
Both sets of Brackets NB!
Why?
5 x 6 ) = 15625
5 y x 6 = 15625
Counting r objects in
n different ways  n r
(repeats/repetition)
56
See diagram 1 for determining the differences between PERMUTATIONS & COMBINATIONS.
(Appendix A attached at the end of this QL worksheet)
Exponents in a bracket
0.75(9
0.75(9
0.75 x 9 – 6 ) = 0.422
6)
6)
2
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 )
0.75 y x ( 9 – 6 ) = 0.422
Multiply the expression by 2
before pressing the equal
sign,
0.75 y x ( 9 – 6 )
2 =0.844
Brackets NB! Why?
2 = 0.844
Now closing the bracket is
important. Why?
Powers with base e 
ex
SHIFT ln 1
3 ) = 1.396
2ndF ln
1
ab
c
3 = 1.396
1
e3
Alternatively,
Alternatively,
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SHIFT ln 1
3
e
3 ) = 1.396
SHIFT ln ( ) 3 ) = 0.05
2ndF ln ( 1
2ndF ln
3 ) = 1.396
3 = 0.05
The calculator automatically
puts brackets around the first
term. Why is there no need to
put brackets in this situation?
1
e3
1
SHIFT ln 3 ) = 0.05
Alternatively,
1
1
SHIFT ln 3 ) = 0.05
But: e
2
3
2ndF ln ( 3 ) = 0.05
Alternatively,
1
e3
Note:
4e
1
1
3
( ) 4
ab
c
2ndF ln ( 3 ) =
0.05
e 3 . Why?
1
e3
Note:
e 3.
SHIFT ln ( ) 2
But: e
1
3
e 3 . Why?
e 3.
3 ) = -2.054
4
Alternatively replace
with
.
2ndF ln (
2
3 ) = -2.054
Alternatively replace
.
ab
with
c
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
The calculator automatically
( 6 ) = 2.45
Brackets are not required
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puts brackets around the
first term.
when there is one term.
6 = 2.45
6 = 2.45
Why?
When there is one term
brackets are not necessary.
Why?
Two terms under the
square root sign
64 20
6 4 – 2 0 ) = 6.63
( 6 4 – 2 0 ) = 6.63
Leaving out the closing
brackets gives the same
answer.
Can you see why brackets are
important in this situation?
Why the emphasis on
brackets?????
64 20
2
64–20 )
( 64–20 )
2 = 3.32
The bracket must be closed.
Why?
Alternatively replace
with
2 = 3.32
The brackets tell the square
root sign what needs to be
beneath the square root.
.
Square root as a
denominator
1
6 ) = 0.41
Similar to
1
6
6 above.
1
Similar to
is in the numerator
1
(
0.82
6 )
6 above.
Brackets for
Denominator
Brackets for
Denominator
Square root as a
fraction in the
denominator, where
( 6 ) = 0.41
2 ) =
The calculator automatically
1
(
0.82
( 6 )
2 ) =
The outer red bolded set of
brackets (arrows above) is
Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St
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of the fraction
1
6
2
opens the square root sign
with a bracket. Now the
bracket must be closed.
Why? The denominator
also requires brackets.
Why?
1
( 3.5
0.9035
is in the
denominator of the
Alternatively ,
1
3.5
10
is not (see
1 3.5
0.9035
6 above). Why?
Brackets for
Denominator
Brackets for
Denominator
Square root as a
fraction in the
denominator, where
fraction
necessary. (a must!) The inner
10 ) ) =
1
( 3.5
0.9035
( 10 ) ) =
Alternatively,
1
10) =
3.5
ab
c
10 = 0.9035
Using the function
Using the function
means that there is no need
for the fraction to go into
brackets.
ab
c
means that there is no need
for the fraction to go into
brackets. Yes you can use
but lower
a b for both
c
one requires brackets. Why?
Denominator having a
square root with two
terms
6
24 12
6
( 2 4 – 12 ) = 1.73
Alternatively replace
with
6
( 2 4 – 1 2 ) = 1.73
Alternatively replace
ab .
with
c
.
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
Brackets required above 10 ) ) = 3.433
and below
Get into the habit of putting
7.3 3.5
brackets at the top of a
3.5
division line and at the
10
bottom. Why?
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 )
10 ) = 3.433
( 3.5
Get into the habit of putting
brackets at the top of a
division line and at the
bottom. Why?
Quantitative Literacy (QL) UNISA | Durban Learning Centre, 221 Dr Pixley Ka Seme St
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7.3 3.5
3.5
10
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!
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
ab
c
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
MODE
MODE
 x
Choose option 3:STAT by
pushing the keypad for 3
Choose option
Choose option 1: 1-VAR by
pushing the keypad for 1
ENTER Data for x :
STAT
by
1
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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
REPLAY
Choose option
SD
by
0
pushing the keypad for 0
Notice the following on the
screen Stat 0
Enter the data as follows:
move the cursor up to the
last value, i.e. 34 on the
screen then push AC
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.
Use either the teal ALPHA
button or the RCL button to
recall the information
SUM of 
x
SHIFT 1
3
 STAT  3:SUM
2
2:
x
126

x
ALPHA
126
x2
x
2
SHIFT 1
3
 STAT  3:SUM

1
x2
1:
2784
Combining sum and
variable functions
functions
x
n
ALPHA
2784
The ALPHA or the RCL button
engages the teal colour stats
functions in the right hand
corner of each numerical pad
button.
SHIFT 1
3
 STAT  3:SUM
SHIFT 1
4
2
2:
1

x
ALPHA
18
x
n
ALPHA 0
 STAT  4:VAR 1:n
18
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x
x
2
2
Before beginning such a
problem, insert the brackets
n
n 1
x
2
x
2
Before beginning such a
problem, insert the brackets
x
n
x
2
2
n
n 1
n 1
Brackets inserted:
underneath the square root
sign; at the top (numerator)
and the bottom
(denominator) of a fraction.
Brackets inserted:
underneath the square root
sign; at the top (numerator)
and the bottom
(denominator) of a fraction
ALPHA
SHIFT

calculator
opens this one
1
x

3
 STAT  3:SUM
2
ALPHA
1
1:
SHIFT 1
x
2
3

2
 STAT  3:SUM
2:
x
x
x2
ALPHA

0
n
x2
SHIFT 1
 STAT 
4
1
SHIFT
4:VAR 1:n
1
4
1
n
ALPHA 0
1
= 9.274
1
 STAT  4:VAR 1:n
= 9.274
See the importance of
brackets? Brackets create
steps.
SIZE sample  n
SHIFT 1
4
1
STAT  4:VAR 1:n
7
Sample MEAN  x
SHIFT 1
4
2
See the importance of
brackets? Brackets create
steps.
n
ALPHA 0
7
x
ALPHA 4
18
STAT  4:VAR 2:x
18
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9
Sample STD DEVIATION
 sx
SHIFT 1
4
 STAT  4:VAR
4
4:sx
9.274
Sample VARIANCE
9.274
x
ALPHA 6 is for the
Option 3: x is for the
POPULATION standard
deviation. What is the
difference between a
sample and population
standard deviation?
POPULATION standard
deviation. What is the
difference between a sample
and population standard
deviation?
SHIFT 1

ALPHA 5 x 2
4
 STAT  4:VAR
x2
 s2
sx

ALPHA 5
4
4:sx
sx
86
86
Notice that when using the
calculator, the sample
standard deviaiton is
obtained first, squaring the
value gives the sample
variance.
standard deviation
2
variance
Notice that when using the
calculator, the sample
standard deviaiton is
obtained first, squaring the
value gives the sample
variance.
standard deviation
2
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
SHIFT MODE toggle down
using
REPLAY
4
4:STAT
3
Frequency ?
1
1:ON
MODE
MODE
Choose option
STAT
by
1
pushing the keypad for 1
Choose option
SD
by
0
pushing the keypad for 0
P ( x) 0.25 0.4 0.2 0.15
3
1
3:STAT
1:1 VAR
Enter the data as follows:
Notice the following on the
screen Stat 0
Enter the data as follows:
0 = 1= 2= 3=
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Then toggle to the top
0 STO 0.25 M+
where x is 0
1 STO 0.4 M+
REPLAY
then move to
the freq column using the
right arrow on the replay
button.
2 STO 0.2 M+
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
DATA SET 4
3 STO 0.15 M+
Notice the screen says
There are 4 values in the
dataset.
x in this situation is the
x in this situation is the
variables and y is the
variables and freq is the
probability of x , i.e. P x
probability of x , i.e. P x .
The expected value of
SHIFT 1
4
2
x
ALPHA 4
1.25
x
ALPHA 6
0.9937
STAT  4:VAR 2:x
x
1.25
The standard deviation
of x
SHIFT 1
4
3
STAT  4:VAR 3: x
0.9937
Notice that
sx

ALPHA 5
Notice that
SHIFT 1
4
4
ERROR
STAT  4:VAR 4:sx
In this situation the
POPULATION standard
deviation is used. Why?
ERROR 2 . In this
situation the POPULATION
standard deviation is used.
Why?
x
ALPHA 6 x 2
The variance of x
SHIFT 1
4
3
standard deviation
2
0.9875
variance
 STAT  4:VAR 3: x
x
2
0.9875
standard deviation
2
variance
Remove frequency from the
screen:
The ALPHA or the RCL button
engages the teal colour stats
functions in the right hand
corner of each numerical pad
button.
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SHIFT MODE toggle down
using
REPLAY
Frequency ?
2
4
4:STAT
2:OFF
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
MODE 3

For example:
Enter the data as follows:
Choose option
x
2.6
2.6
3.2
3.0
2.4
3.7
3.7
2.6 = 2.6= 3.2= 3.0= 2.4=
3.7= 3.7=
pushing the keypad for 1
y
5.6
5.1
5.4
5.0
4.0
5.0
5.2
3:STAT
2
2: A BX
Then toggle to the top
to the very first
REPLAY
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
STAT
by
1
LINE
by
1
pushing the keypad for 1
Notice the following on the
screen Stat 1
Enter the data as follows:
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+
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Notice the screen says
Notice that a bx is the
mathematical equation for a
DATA SET 7
straight line.
There are 7 paired values in
the dataset.
SIZE sample  n
SHIFT 1
4
1
STAT  4:VAR 1:n
n
ALPHA 0
7
x
ALPHA 4
3.029
y
ALPHA 7
5.043

ALPHA 5
0.531

ALPHA 8
0.509
7
Sample MEAN 
x
SHIFT 1
4
2
STAT  4:VAR 2:x
3.029
y
SHIFT 1
4
5
STAT  4:VAR 5: y
5.043
sx
Sample STD DEVIATION
sx

SHIFT 1
4
 STAT  4:VAR
4
4:sx
0.531
sy
sy
SHIFT 1
4
7
 STAT  4:VAR 7:sy
The ALPHA or the RCL button
engages the teal colour stats
functions in the right hand
corner of each numerical pad
button.
0.509
Use either the teal ALPHA
button or the RCL button to
recall the information
SUM of 
x
SHIFT 1
3
 STAT  3:SUM
21.2
2
2:
x

ALPHA
x
21.2
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13
SHIFT 1
y
3
 STAT  3:SUM
4
4:
y

ALPHA 2
y
35.3
35.3
x2

x2
SHIFT 1
3
 STAT  3:SUM
1
x2
1:
ALPHA
65.9
65.9
y2
y2
SHIFT 1
3
 STAT  3:SUM

ALPHA 3
3
179.57
y2
3:
179.57
xy
xy
SHIFT 1
3
 STAT  3:SUM
5
5:
xy
ŷ
b0

b1 x
slope
intercept
Subscript increases 0 to
1
Intercept: b0
a
SHIFT 1
5
Intercept: b0
1
STAT  5:Re g 1: A
a

ALPHA ( 4.093
a
4.093
Slope: b1
On calculator
107.44
The ALPHA or the RCL button
engages the teal colour stats
functions in the right hand
corner of each numerical pad
button.
107.44
Regression coefficients

ALPHA 1
Slope: b1
b
b

ALPHA ) 0.314
b
ŷ
a
b x
intercept
slope
SHIFT 1
5
 STAT  5:Re g
1
2:B
0.314
Alphabet increases from
a to b.
Correlation coefficient: r
Hence,
SHIFT 1
Intercept b0
a
5
STAT  5:Re g
3
3:r
Correlation coefficient: r
r
ALPHA
0.327
0.327
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Slope b1
b
Coefficient of
Coefficient of
Determination: r
SHIFT 1
2
5
 STAT  5:Re g
x
2
Determination: r 2
3
3:r
r
x2
ALPHA
0.107
0.107
Estimated value of y i.e. ŷ
Estimated value of y i.e. ŷ
when x
3.1 :
SHIFT 1
5
SHIFT 1
 STAT 
5
2
2:B
3.1 :

ALPHA (
b
ALPHA )
3.1
5.065
a
1
 STAT  5:Re g 1: A
5:Re g
when x
3.1
=5.065
Alternatively,
Alternatively,
3.1 SHIFT 1
 STAT 
5
5:Re g
5
5.065
y'

3.1 2ndF )
5.065
5: yˆ
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 acalit@unisa.ac.za or a Quantitative Literacy Facilitator in your region
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15
Appendix A:
Repetition
(repeats)
Yes
nr
Note that nCr
combination
No
ORDER important
Yes
No
Permutation ( nPr )
Combination ( nCr )
nPr . Why?
n!
r !(n r )!
n!
1
permutation
r !(n r )! r !
permutation r !combination
combination
So there will be r !permutations for every possible combination.
Diagram 1: Counting: Permutations and Combinations
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