**Statistics on the HP 15C**

The HP 15C performs one and two variable statistics. In statistical calculation, data is entered into Y and X registers. When the key [ ∑+ ] is pressed the following variables are updated:

R2 = n = number of data points

R3 = ∑x = sum of x data

R4 = ∑x^2 = sum of x-squared data

R5 = ∑y = sum of y data

R6 = ∑y^2 = sum of y-squared data

R7 = ∑xy = sum of the products of x-data and y-data

You can recall these values.

Additional statistical functions are available:

Arithmetic Mean: [ g ] [ 0 ]. The arithmetic mean of x-data is displayed. Press [x<>y] to display the arithmetic mean of y-data is displayed.

Standard Deviation: [ g ] [ . ]. The standard deviation of x-data is displayed. Press [x<>y] to display the standard deviation of y-data is displayed.

Clearing the Statistics Registers

[ f ] [GSB] (CLEAR ∑) clears memory registers R2, R3, R4, R5, R6, and R7 to 0.

Linear Regression

The HP 15C includes linear regression functions to fit data to the line:

y = a x + b

where a = slope and b = y-intercept with r = correlation coefficient. If r is near 1 or -1, the data has a good fit to the line. If r is near 0, the data does not have a good fit to a line.

To find a, b, and r:

b: [ f ] [∑+] (L.R.)

a: [ f ] [∑+] (L.R.) [x<>y]

r: a valid number [ f ] [ . ] (y-hat, r) [x<>y]

Entering Data

One Variable Data:

1. Enter x

2. Press [∑+]

Two Variable Data:

1. Enter y

2. Press [ENTER]

3. Enter x

4. Press [∑+]

The key sequence [ g ] [∑+] (∑-) can be used to remove data.

A Example of Linear Regression

Find a linear fit to the data:

x y

1 3

2 6

4 11

8 15

16 36

Caution: In two variable data, enter the

*y*data first.

Key press:

[ f ] [GSB] (CLEAR ∑)

3 [ENTER] 1 [∑+]

6 [ENTER] 2 [∑+]

11 [ENTER] 4 [∑+]

15 [ENTER] 8 [∑+]

36 [ENTER] 16 [∑+]

First the slope and y-intercept:

[ f ] [∑+] (L.R.) (** 1.0833 is displayed)

[x<>y] (** 2.1156 is displayed)

1 [ f ] [ . ] (y-hat, r) [x<>y] (** 0.9905 is displayed)

Results: The linear fit is approximately is y ≈ 2.1156 + 1.0833x with r ≈ 0.9905 (a very good fit!)

In the next program, we will use the statistical features to fit data to a logarithmic fit.

**Logarithmic Regression**

This program fits the data to the equation:

y = b + a ln x

Note that this is similar to the linear equation y = b + a x, except that we are working with "ln x" instead of "x".

This program adjusts the x-data to ln(x-data).

Labels used:

Label A: Initialize data; clear registers R2 - R7

Label B: Adjusts the data and enters the data

Label C: Calculates a, b, and r.

Program Listing

Key Codes Key

001 42 21 22 LBL A * Initialization

002 42 32 CLR ∑

003 43 32 RTN

004 42 21 12 LBL B * Data Entry

005 43 12 LN

006 49 ∑+

007 43 32 RTN

008 42 21 13 LBL C * Calculate b, a, r

009 42 49 L.R.

010 31 R/S * Display b

011 34 x<>y

012 31 R/S * Display a

013 1 1

014 42 48 y-hat, r

015 34 x<>y * Display r

016 43 32 RTN

Example

Data is presented regarding the growth of a human male:

X = Age (in years) Y = Height (in feet)

1 0.88

2 1.44

3 2.06

4 2.18

5 2.45

10 5.02

15 5.76

20 6.09

Use the program to find a logarithmic fit.

Key Presses:

[ f ] [ √ ] (A)

0.88 [ENTER] 1 [ f ] [e^x] (B)

1.44 [ENTER] 2 [ f ] [e^x] (B)

2.06 [ENTER] 3 [ f ] [e^x] (B)

2.18 [ENTER] 4 [ f ] [e^x] (B)

2.45 [ENTER] 5 [ f ] [e^x] (B)

5.02 [ENTER] 10 [ f ] [e^x] (B)

5.76 [ENTER] 15 [ f ] [e^x] (B)

6.09 [ENTER] 20 [ f ] [e^x] (B)

[ f ] [10^x] (C) (* Display: 0.1363)

[R/S] (* Display: 1.9376)

[R/S] (* Display: 0.9656)

Results:

y ≈ 0.1363 + 1.9376 ln x with r ≈ 0.9656 (very good fit!)

Regressions

The following table shows various fits that can be used using L.R. and y-hat,r functions. Use the necessary adjustments to get the correct data.

For example, for logarithmic fit, take the natural logarithm of x before pressing [∑+]. For exponential fit, take the natural logarithm of y before pressing [∑+]; when pressing [L.R.], take the exponential of both a and b to get the true slope and y-intercept.

Fit Equation x y a b

Linear y = ax+b x y a b

Logarithmic y = b + a ln x y a b

Power y = b * x^a ln x ln y a b

Exponential y = b * a^x x ln y e^a e^b

Inverse y = b + a/x 1/x y a b

Logistic y = L/(1+ae^(bx)) x ln(L/y - 1) e^a b

L > 2y_max

Sine y = b + a sin(px) sin(px) y a b

p = (x_max - x_min)/(2π)

Hope you find this table useful.

Until then, happy programming,

Eddie

*This tutorial is property of Edward Shore. © 2011*