Spotlight: Calculated Industries QR Calc

Spotlight: Calculated Industries QR Calc <Title>

Quick Facts

Model: 3375

Name: QR Calc

Company: Calculated Industries

Timeline: 1993

Type: Quality Control, Statistics

Operating System: Algebraic

Digits: 7

Memory: 1 general purpose memory register plus specific variable registers 

Power: 1 CR-2032 battery

When I purchased the QR Calc, there was no manual with it. Normally, it wouldn’t present any issues because I can often find manuals online. That is not the case with the QR Calc. If there is any manual online, please email me. I’ll have to be more careful next time.

I apologize that I will not be able to describe all the functions but hopefully I describe enough features to give the reader a general idea. Please check out the sources below.

Features

Let’s start with the mathematical functions included: powers and roots (y^x, x². √), natural logarithm (ln), exponential function (e^x), reciprocal (1/x), arithmetic (+, -, ×, ÷), and the standard percent key which operates they people expect it to (%). The order of operations is enforced. Like a lot of calculators, the change sign (+/-) key is a shifted function.

The operating range of the calculator is 7 digits: -9,999,999 to 9,999,999. Any calculator that has a result outside of this range causes the QR calculator to display an error message.

The store key for the QR Calc is labeled [ Set ].

The function →PPM changes any number into n parts per million.

Example: 0.0014 →PPM displays 1,400 PPM.

Here are some functions I was able to find out and figure out:

Normal Distribution

The normal distribution functions calculate areas of the standard normal distribution, assuming that μ = 0 and σ = 1.

n [ nZ ]: lower tail probability (from -∞ to x = n)

n [ 2^nd ] (ModZ): probability from x = 0 to x = n

One Variable Statistics

The QR handles a single set of statistics with the following key and key sequences:

[ Add ]: Add a data point (Σ+)

[ 2^nd ] [ Add ] (Subtract): Subtract a data point (Σ-)

[ 2^nd ] [ + ] (S←→P): Toggle between standard deviation and population deviation (σn indicator)

[ x-bar ]: Calculate the arithmetic mean.

[ R ]: Calculate the range of the data.

[ 2^nd ] [ x-bar ] (N): Calculate the number of data points.

[ 2^nd ] [ R ] (σ): Calculate the deviation, depending on the deviation mode set.

[ Low ]: Returns the minimum value of the data set entered.

[ 2^nd ] [ Low ] (High): Returns the maximum value of the date set entered.

[ Skew ]: Skew. I’m not sure what formula the QR Calc uses because I have not been able to match results with any formula I found yet.

[ 2^nd ] [ Skew ] (Kurt): Excess Kurtosis. Kurtosis measures how concentrated the data is with respect to the mean.

The formula used for Excess Kurtosis:

μ4 = 1/n * Σ((xi – mean)^4)

kurtosis = μ4/s^4 – 3

Process Capability Indices

If a process is mature, that is a process that is regular and has been executed for a period of time, we can measure the capability index.

There are two capability indices:

Cp: for a centered analysis

Cpk: for a non-centered analysis. This metric is used for potential future performance.

Generally, we want these indices to be at least 1. Capable processes have an Cp index of 1.33 or higher.

Key strokes:

Enter the mean: [ Set ] [ x-bar ]

Enter the deviation: [ Set ] [ 2^nd ] [ R ] ( σ )

Enter the lower specification limit (based on the normal distribution): [ Set ] [ LSL ]

Enter the upper specification limit: [ Set ] [ 2^nd ] [ LSL ] (USL)

Each variable entered will have an indicator.

Calculate Cp: [ 2^nd ] [ Cpk ] (Cp)

Calculate Cpk: [ Cpk ]

Example:

LSL = - 1, USL = 1, mean = 0.05, deviation = 0.27

[ 2^nd ] [ × ] (AC) to clear out the registers if needed

0.5 [ Set ] [ x-bar ] (x-bar indicator is on)

0.27 [ Set ] [ 2^nd ] [ R ] ( σ ) (σ indicator is on)

1 [ 2^nd ] [ - ] (+/-) [ Set ] [ LSL ] (L indicator is on)

1 [ Set ] [ 2^nd ] [ LSL ] (USL) (U indicator is on)

Results:

Cp: 1.234568

Cpk: 1.17284

That is a pretty good process.

Formulas Used:

Cp = (USL -LSL) / (6 * σ)

Cpx = min((x-bar – LSL) / (3 * σ), (USL – x-bar) / (3 * σ))

Control Charts

We get to the main feature of the QR Calc: Control Charts and Capability Limits. On the back of the calculator, the QR Calc has a list of handy formulas.

The heart of the QR Calc is the table of constants that are used in control charts and limit charts. Often the chart limits are built on many samples of n data points each, where x-bar is the average of the sample averages, and R is the average of the range samples. We can also build chart limits with one sample. The QR Calc can only handle sample sizes from 3 to 25 data points.

Mean Control Chart Limits:

Lower: LCL-mean = x-bar – n * A2 * R

Upper: UCL-mean = x-bar + n * A2 * R

R Control Chart Limits:

Lower: LCL-R = n * D3 * R

Upper: UCL-R = n * D4 * R

A2, D3, and D4 are constants used in calculating control chart limits. Accessing these constants takes one argument, which is the sample size.

The A2 constant for a sample size of 3: 3 [ nA2 ] returns 1.023.

Below is a short table of constants, as determined by the QR Calc.

Sample Size n	Constant A2	Constant D3	Constant D4
5	0.577	0	2.114
10	0.308	0.223	1.777
15	0.223	0.347	1.653
20	0.18	0.415	1.585
25	0.153	0.459	1.541

Standard deviation can be estimated by using the average range ( R ) and another constant d2:

σ ≈ R / d2

Sample Size n	Constant d2
5	2.326
10	3.078
15	3.472
20	3.735
25	3.931

A table of constants from n = 2 to 25 can be found here:

https://sixsigmastudyguide.com/x-bar-r-control-charts/

Example:

Construct mean and range charts from a sample (n = 5):

3.995

4.26

4.37

4.44

4.58

Keystrokes:

(after clearing data)

3.995 [ Add ] 4.26 [ Add ] 4.37 [ Add ] 4.44 [ Add ] 4.58 [ Add ]

X-bar chart:

LCL: [ x-bar ] - 5 [ nA2 ] [ × ] [ R ] [ = ] Result: 3.991455

UCL: [ x-bar ] + 5 [ nA2 ] [ × ] [ R ] [ = ] Result: 4.666545

R chart:

LCL: 5 [ 2^nd ] [ nD4 ] (nD3) [ × ] [ R ] [ = ] Result: 0

UCL: 5 [ nD4 ] [ × ] [ R ] [ = ] Result: 12.36669

The QC calc has contains the E2 constant.

Final Thoughts

The functions that I still do not know about or have figured out are: TRGa, TRGb, RS a, RS b, %Low, and %High.

This review is incomplete. I will keep searching for a manual, I may have to buy another QR Calc.

This calculator is a rarity, and one worth checking out.

Sources

Hessing, Ted. “Process Capability (Cp & Cpk)” 6σSTUDYGUIDE.COM (no specific date give, first comment on November 19, 2014) https://sixsigmastudyguide.com/process-capability-cp-cpk/. Accessed January 2025.

Hessing, Ted. “X Bar R Control Charts” 6σSTUDYGUIDE.COM (no specific date give, first comment on April 17, 2018) https://sixsigmastudyguide.com/x-bar-r-control-charts/. Accessed January 2025.

Hewlett Packard. HP-65 Stat Pac 2 Cupertino, CA. https://literature.hpcalc.org/items/975 1975

Next time, I’m going to see if a manual comes with it. Not everything has a manual online.

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

RPN with HP 15C & DM32: Expanding Linear Regression

RPN with HP 15C & DM32: Expanding Linear Regression

“Linearize” the Equation

The linear regression mode of the HP 15C and Swiss Micros DM32 can be expanded to fit different curves for bi-variate data.

Curves can be set up for linear regression if we can get the equation into the form:

y = b + m * x ⇒ f(y) = b + m * g(x)

where b is the y-intercept and x is the slope. The correlation is r and r^2 can be used to determine the relationship between x (independent) and y (dependent) variables. If r^2 is close to 1, the better the curve fits to the data.

Note: f(y) and g(x) are functions that contain only one term (sin y, cos y, e^y, 1/y, y^2 …. sin x, cos x, e^x, 1/x, x^2, ….)

The DM32 labels the slope as m, y-intercept as b. Each of the parameters can be recalled separately.

The HP 12C labels the slope as a, y-intercept as b. Both a and b are calculated by the key sequence [ f ] [ Σ+ ] (L.R.): slope is on the y-stack and the y-intercept is on the x-stack. To get the correlation, enter any (valid) number and press [ f ] [ . ] (y-hat, r) [ x<>y ].

Let’s illustrate this for with a couple of examples.

y = 1 / (b + m * e^(-x))

1/y = b + m * e^(-x)

We have the equation in the required form with the following adjustments: x’ = e^(-x), y’ = 1/x.

y = b * m^x

ln y = ln (b * m^x)

ln y = ln b + ln (m^x)

ln y = ln b + x * ln m

We have the equation in the required form with the following adjustments: x’ = x, y’ = ln y.

Now note that we have ln b and ln m. This will require an adjustment when the linear regression is calculated. To get the “true” slope and y-intercept in this case, we must calculate e^b and e^m.

To employ different curve fittings, I use (at least) two programs:

Entering data (DM32, HP15C):

LBL #

g(x)

x<>y

f(y)

x<>y

Σ+

R/S

GTO #

Entering data:

CLEAR Σ (CLΣ)

y1 ENTER x1 XEQ/GSB #

yn ENTER xn R/S

You can enter as many data points as you like.

Calculating the Parameters:

DM32:

LBL @

b

(adjustments if needed)

R/S

m

(adjustments if needed)

R/S

r

x^2

RTN

HP 15C:

LBL @

L.R.

(adjustment if needed)

R/S

x<>y

(adjustment if needed)

R/S

1

y-hat, r

x<>y

x^2

RTN

Example 1: y = 1 / (b + m * e^(-x))

Linearized: 1 / y = b + m * e^(-x)

Adjustments: x’ = e^(-x), y’ = 1 / y, no adjustment to b or m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 11	LBL A	LBL D
16	CHS	+/-
12	e^x	e^x
34	x<>y	x<>y
15	1/x	1/x
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 11	GTO A	GTO D

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 1	LBL 1	LBL R
42, 49	L.R.	b
31	R/S	R/S
34	x<>y	m
31	R/S	R/S
1	1	r
42, 48	y-hat, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
0.0	0.125
0.1	0.13
0.2	0.134
0.3	0.138
0.4	0.143
0.5	0.147

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	4.98312	4.98312
Slope (m/a)	3.01575	3.01575
r^2	0.99841	0.99841

Example 2: y = ln(b + m * e^(-x))

Linearized: e^y = b + m * e^(-x)

Adjustments: x’ = e^(-x), y’ = e^(y), no adjustment to b or m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 12	LBL B	LBL E
16	CHS	+/-
12	e^x	e^x
34	x<>y	x<>y
12	e^x	e^x
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 12	GTO B	GTO E

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 1	LBL 1	LBL R
42, 49	L.R.	b
31	R/S	R/S
34	x<>y	m
31	R/S	R/S
1	1	r
42, 48	y-hat, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
0.98	1.946
0.99	1.942
1.00	1.938
1.01	1.934
1.02	1.929
1.03	1.925

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	3.99987	3.99985
Slope (m/a)	8.00051	8.00056
r^2	0.99844	0.99844

*differences may be due to rounding error in the internal algorithms

Example 3: y = √(b + m * x^2)

Linearized: y^2 = b + m * x^2

Adjustments: x’ = x^2, y’ = y^2, no adjustment to b or m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 13	LBL C	LBL F
43, 11	x^2	x^2
34	x<>y	x<>y
43, 11	x^2	x^2
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 13	GTO C	GTO F

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 1	LBL 1	LBL R
42, 49	L.R.	b
31	R/S	R/S
34	x<>y	m
31	R/S	R/S
1	1	r
42, 48	y-hat, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
1.05	2.066
1.25	2.337
1.45	2.620
1.65	2.912
1.85	3.209
2.05	3.511

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	1.40009	1.40009
Slope (m/a)	2.59996	2.59996
r^2	1.00000	1.00000

Example 4: y = b * m^x

Linearized: ln y = ln b + x * ln m

Adjustments: x’ = x, y’ = ln y, result adjustments: e^b, e^m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 14	LBL D	LBL G
34	x<>y	x<>y
43, 12	LN	LN
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 14	GTO D	GTO G

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 2	LBL 2	LBL S
42, 49	L.R.	b
12	e^x	e^x
31	R/S	R/S
34	x<>y	m
12	e^x	e^x
31	R/S	R/S
1	1	r
42, 48	y-bar, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
0.84	2.358
0.87	2.363
0.90	2.369
0.93	2.374
0.96	2.379
0.99	2.385

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	2.21302	2.21302
Slope (m/a)	1.07843	1.07843
r^2	0.99916	0.99919

*differences may be due to rounding error in the internal algorithms

Expanding Linear Regression Table

Regression	X	Y	B = ITC	M = SLP
Logarithmic: y = b + m * ln x	ln x	y	b	m
Exponential: y = b * e^(m*x)	x	ln y	e^b	m
Inverse: y = b + m/x	1/x	y	b	m
Power: y = b * x^m	ln x	ln y	e^b	m
General Exponential: y = b * m^x	x	ln y	e^b	e^m
Simple Logistic: y = 1/(b + m * e^(-x))	e^(-x)	1/y	b	m
Square Root Linear: y = √(b + m * x)	x	y^2	b	m
Cosine: y = b + m*cos(ω(x – ϕ)) With ϕ = the point (x) nearest to zero where the trough or peak begins ω = (2*π)/period (radians) or ω = 360°/period (degrees)	cos(ω(x – ϕ))	y	b	m
Logarithmic-Linear-Exponential: y = ln(b + m * e^(-x))	e^(-x)	e^y	b	m
Square Root Quadratic: y = √(b + m * x^2)	x^2	y^2	b	m

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.