Short Communication
Scientific calculators: How best to use in statistical
problem solving
K. Balasubramanian
Retired Professor of Biostatistics, Chennai, Tamilnadu, India
Corresponding Author: K. Balasubramanian, E-mail: balamanidhanam@gmail.com
Received: 21/11/2024; Revised: 16/12/2024; Accepted: 18/12/2024; Published: 29/01/2025
__________________________________________________________________________________________
Abstract
Scientific calculators could be used effectively in data analysis in biological research. Scientists often use
calculators to perform statistical calculations like mean, median, standard deviation and more complex
regression analysis to analyze experimental results. This paper outlines the necessary steps for using scientific
calculators, specifically in statistical calculations, in order to make the process more accessible and beneficial
for students.
Keywords: Scientific calculator, standard deviation, means, median, regression analysis.
__________________________________________________________________________________________
Introduction
It was found from my teaching experience that students not only from biological science stream but also from
mathematical science stream are not using scientific calculators effectively (Figure 1). The aim of this article is
to enlighten the best benefits of scientific calculators for students while solving statistical problems. [1,2]
Figure 1: Scientific calculator IS fx 991 MS
STEPS FOR USING A SCIENTIFIC CALCULATOR
1. ERASE MEMORY STORAGE
This is done by
Click “SHIFT KEY”. Then click “MODE / CLR” key
In display: you can see
1 2 3
Mc1 Mode All
Press “3”, then = and Press “AC” Key.
2. SETTING CALCULATOR FOR STATISTICAL CALCULATIONS
A. To calculate AM [or Mean] & SD [ Standard deviation]
This is done by setting “SD” Mode.
For this, Click “MODE” once. Check the display.
If SD is not seen, Click once more. Now you will see “SD”. You can also see a number below
“SD”.
Now Click that number. If the number is “2” Click number “2”.
If the number is “1” Click number “1” in the number key.
Now, in the display you will see “SD”.
Also, you will see “D”. Do not worry about this. Let it be there.
I. Entering data
As an example:
Find SD of 2, 3, 4, 5, 6
Click “2” in the number key and then click “M+” key.
Now the display will be “n = 1”
Then, click “3” and Click “M+”.
Now the display will be “n = 2”
Go on entering like that.
When you finish pressing “6” and “M+” key, the display will be “n = 5”. Now, data entry
over.
Now, you have entered all values in calculator memory.
II. Getting Σ x, Σ x2 for the calculations using the formulas.
Press “Shift” { note the color } key and then Number “1” key[ you can see “S-SUM”
above it { note the color }] key.
The display will be
Σ x2Σ x n
1 2 2
Now, press “1” and then “=” key to get Σ x2 which will be = 90 in the display.
Again, Press “Shift” and then Number “1” keys. Now, Press “2” and “=” to get Σ x which
will be = 20. Press “shift” and Number “3” to get “n” which will be = 5
III. Calculating AM and SD
Use the formula as follows
AM =
n
x
= x =
= 4
The SD formula used is from
SD =
n
xx
2
)(
( 1 )
S = SD =
2
2
xx
nn
=
² = = = 1.4142
If someone wish to use the formula for SD as
1
)( 2
n
xx
( 2 ),
Then calculate it using S *
{ ‘S’ calculated as per formula ( 1 ).
I would like to mention here that S *
is the formula for [ Estimate of
when the small sample is small {
sample size ‘n’ < 29 }. This is referred as sample SD in textbooks. But I would advise to
use calculate SD using formula ( 1 ).
The divisor in formula ( 2 ) is nothing but degrees of freedom.
Please note that Standard Error [ SE ] is used in applied statistics which will be same
using the SD formulas ( 1 ) Or ( 2 ).
IV. Verification of answers of AM and SD { The advantage in Scientific calculators }
For this,
Press “shift” and then “2” [ you can see “S-VAR” above it] keys
The display will be
x σn x σn-1
1 2 3
Now, press “1” and “=” keys to get which will be = 4
Note that the SD value we must take is xσn ( The result got using formula ( 1 ). If you
take x σn-1, it is the result of formula using (2).
In some calculators these options are a little different
σxSx
1 2 3
Here also take option ‘2’ for SD value.
Again, Press “Shift” and Number “2” keys. Now, Press “2” and “=” to get ‘S’
{xσn}which will be = 1.4142
THE ADVANTAGE IN USING SCIENTIFIC CALCULATOR IS GETTING THE Σ
x2Σ x and n VALUES QUICKLY AND VERIFYING THE ANSWERS
NOTE: THE STUDENTS ARE UNABLE TO GET ANY FORMULAE BY USING A
SCIENTIFIC CALCULATOR. THE CALCULATOR TYPE IS fx 991 MS
V. Calculating Coefficient of Variation [CV]
CV =
* 100 =
* 100 = 35.355 = 35.36
CV value cannot be verified since the formula is not built in, in the calculator.
B. To calculate Coefficient of correlation [ “r” value]
This is done by setting “REG” Mode.
For this, Click “MODE” once. Check the display.
If “REG” is not seen, Click once more. Now you will see “REG”. You can also see a number
below “REG”.
Now Click that number. If the number is “2” Click number “2”.
If the number is “1” Click number “1”.
There will again a display where you can see the following.
Lin Log Exp
1 2 3
Now, in the display you will have to select “Lin”.
If the number is “1” Click number “1”.
Now, in the display you can see “REG”
Also you will see “D”. Do not worry about this. Let it be there.
Now, the calculator is set for both “Correlation” and “Regression” calculations
I. Entering data
If the problem is
Find Correlation coefficient of
X: 2 3 4 6 8
Y: 5 7 8 9 10
Click “2”, then click “,“, then click “5” and Click “M+” key.
Now the display will be “n = 1”
Now, the first pair of values is entered.
Now, click “3”, then click “, “, then click “7” and Click “M+” key.
Now the display will be “n = 2”
Now, the second pair of values is entered.
Go on entering like that.
When you finish the last pair, the display will be “n = 5”. Now, data entry over.
II. Getting Σ x, Σ x2, Σ y, Σ y2, Σ xy and “n”for the calculations using the formulas.
For this, Press “Shift” and Number “1” [ you can see “S-SUM” above it] keys.
The display will be
Σ x2Σ x n
1 2 2
You can see a “REPLAY” button in the middle of the calculator. Also you can see an:
RIGHT ARROW [ like triangle]. Press on it and there will be display where you can see
Σ y2Σ y Σ xy
1 2 3
Now, pressing the corresponding numbers and
“=” keys, all the six values [Σ x, Σ x2, Σ y, Σ y2, Σ
xy and “n”] can be got.
Σ x2 = 129 Σ x = 23 n = 5
Σ y2 = 319 Σ y = 39 Σ xy = 197
THE ADVANTAGE IN USING SCIENTIFIC CALCULATOR IS GETTING Σ x, Σ x2,
Σ y, Σ y2, Σ xy and “n” VALUES QUICKLY AND VERIFYING THE ANSWERS -
THE CALCULATOR TYPE IS fx 991 MS OR SIMILAR
III. Calculating Correlation coefficient and Regression equation
The following formulas will be the best for the Correlation and Regression calculations.
Sxx =
n
x
x2
2)(
= 23.2
Syy =
n
y
y2
2)(
= 14.8
Sxy =
( )( )
() xy
xy n
= 17.6
Then, r =
)).(( SyySxx
Sxy
=
=
= 0.9498
Then the general form of the regression equation (linear) is given as y = b x + a
In this,
‘b’ is calculated as b =
Sxx
Sxy
=
= 0.7586
and ‘a’ is given by a = y – b.x
Also x and y, the Mean of ‘x’ and Mean of ‘y’ is to be obtained.
=
=
= 4.6
and =
=
= 7.8
a = 7.8 - 0.7586*4.6 = 4.3103
The Regression equation is
Y = 0.7586x + 4.3103
IV. Verification of answers of Correlation coefficient [r], Regression coefficient [b] and
Y-intercept [a]
For this
Press “shift” and “2” [ you can see “S-VAR” above it] keys
The display will be
x σn x σn-1
1 2 3
In the “REPLAY” button in the middle of the calculator Click RIGHT ARROW [
like triangle].
Now, there will be display where you can see
y σn y σn-1
1 2 3
Again, click in the “REPLAY” button in the middle of the calculator Click RIGHT
ARROW [ like triangle]. Now, there will be display where you can see
A B r
1 2 3
‘r, Correlation Coefficient
‘B’ [b] and ‘A’ [a] the coefficients in the Regression equation y = b x + a
Pressing the corresponding numbers and pressing “=”, the corresponding values can be
seen in the display.
Calculated answers can be verified.
C. ‘t’ test and paired ‘t’ also
For these calculations the above procedures can be used to get Mean and SD to apply in the
respective formulas.
The following formulas could possibly be advantageous in “t” test calculations.
Significance Tests – Small Samples (n < 30)
When samples are small, the test statistic follows Student’s ‘ t ‘ distribution
Tests based on Student’s ‘t’ distribution
I. Testing sample mean against population mean ( One Sample ‘t’ Test )
Null hypothesis is set as H0: = 0. Alternate hypothesis can be any one of the following.
(1) H1 : 0 (2) H1 : <0 (3) H1 : >0
‘’, the population SD is unknown, ‘t’ is calculated as
t = df = n-1
1
0
n
S
x
NOTE:
The SD is calculated as S = SD =
n
xx
2
)(
, The divisor is ‘n’ only.
NOTE:
If SD calculated using S = SD =
1
)( 2
n
xx
Then the formula used for the ‘t’ calculation will be t = ˳
Students please, do not use this formula.
II. Testing of Two Means Based on ‘t’ Test ( Two independent samples ‘t’ test )
The test statistic is set as
t =
df = n1 + n2 – 2
It will be easy to use the formula given correlation coefficient calculation to get n1S12 and
n2S22
Calculate and n2S22
Sxx = n1S12 =
n
x
x2
2)(
; (x)2
= (x) (x)
Financial support and sponsorship
Nil
Conflict of interest
There are no conflicts of interest.
References
1. Trouche L. Calculators in Mathematics Education: A Rapid Evolution of Tools, with Differential Effects.
In: The Didactical Challenge of Symbolic Calculators. Springer. 2005:36: 9-39.
2. Close S, Oldham E, Shiel G, Dooley T, Oleary M. Effects of calculators on mathematics achievement and
attitudes of ninth-grade students. J Educ Res. 2012;105(6):377–90.
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