Using Your Calculator - Two-Variable Statistics

With very little effort you can learn to use your calculator to calculate a number of important two-variable statistics. Instruction for using the Casio 300W are followed by instruction for using the Sharp EL-531R calculator. If you are using the Sharp, skip the instruction for the Casio. If you are using the Casio, skip the instructions for the Sharp.

Objectives:

By the end of this topic, you will be able to:

• Enter bivariate data points into your calculator.
• Enter frequency bivariate data into your calculator.
• Calculate the slope of a straight line.
• Calculate the intercept of a straight line.
• Calculate the Pearson Product Moment Correlation Coefficient for a set of data.
• Compute the linear regression line for a set of data.
• Compute the mean and standard deviation for each to the two variables.
• Predict the value of one variable given the value of the other variable.

Casio 300W

The statistical functions of your calculator can save you from doing days and days of monotonous, repetitious busy work.

Please have your calculator with you as you work through this lesson.

Modes

Press the [ON] button.

Press [MODE] once and you should see:

You calculator is always in one of these modes (COMP, SD, or REG).

• "REG" is the regression mode and is used for two-variable statistics (called "bivariate statistics"). To select "REG" from the first menu, you press 3.
Press [ON]
Press [MODE]
Press [3]
Then you are faced with a pair of secondary menus:

You move between the two secondary menus by using the two [REPLAY] buttons. Make sure the:

menu is showing and then Press [1]
(we will only do linear regressions, never logarithmic regressions, exponential regressions, power regressions, inverse regressions, or quadratic regressions (but it is nice to know we could if we wanted).

You should see a very small REG near the middle bottom of the screen - just above the [REPLAY] buttons.

Entering Data (Part 1)

This section assumes your calculator is in SD mode. Before we begin, I want you to locate the data entry key [M+] on your calculator. It is the button just above the read [AC] button. I also want you to note where the [,] button is. It is directly above the red [DEL] button. It is also important that you do not confuse the negative button [(-)] on the left side of your calculator with the subtraction button which is on the lower right side of your calculator. The negative button [(-)], just above the [STO] button, is used to enter negative numbers. The subtraction button [], just above the [=] button, is not used for data entry.

The basic unit in bivariate algebra, bivariate geometry, and bivariate statistics is a "point." A point is specified by two variables, an x-value and a y-value. Points are written in the form of (x, y). For example, (23, -6) is a point with an x-value of 23 and a y-value of -6. And, the point (35, 3) has an x-value of 35 and a y-value of 3. Let's turn on our calculator, put it in linear regression mode, clear the statistical memories, and enter these two points: (23, -6) and (35, 3).
Press [ON]
[MODE] [3] [1] to put your calculator in linear regression mode
[SHIFT] [AC] [=] to clear your statistical memory.
Your screen will show SCL for "Statistical Clear." Don't worry, this will disappear when we start entering data. Let's enter that data now:
[2] [3] [,] [(–)] [6] [M+] entered the first point.
[3] [5] [,] [3] [M+] entered the second point.

That's it!! Wasn't that easy? We only need to enter the data once, and as long as we do not clear our calculator, we can ask our calculator to compute a number of different statistics (or parameters) based on that data.

Calculating the Linear Equation

When given two different points, one, and only one, straight line can be drawn through those two points. Lines are specified with the equation y = a + bx where the value for a and/or b varies from line-to-line. Once a and b have been specified, the line has been specified.

Once your data has been entered, as ours has been, to obtain the value for a in the linear equation press [SHIFT] [7] [=] Your display should read -23.25 - note the small golden A above the [7] button

To obtain the b value, press [SHIFT] [8] [=] Your display should read 0.75. Note the small golden B above the [8] button.

The equation for the line that goes through the two points (23, -6) and (35, 3) is y = -23.25 + 0.75x.

Finding Other Points On The Line

The equation for our line is y = -23.25 + 0.75x. If you specify any x and then solve this equation for y, the point specified with this x, y pair fall on the line. For example, if we specify x as 1, we can solve for y: y = -23.25 + 0.75(1) = -23.25 + 0.75 = -22.5. Therefore, the point (1, -22.5) falls on this line.

This is far more easily done with our calculator. We have already entered our two points. To find the y value that goes with 1, we just press
[1] to specify the x value

[SHIFT] [] our display should read -22.5. Notice the golden (y-hat) written above the subtraction key. This symbol represents the estimated y value. I don't understand why we don't need to press [=] after the [SHIFT] [], but we don't.

Instead of specifying x and solving for y, we could specify y and solve for x. For example, we could specify the y in our linear equation of y = -23.25 + 0.75x to be 1 and solve for x:
1 = -23.25 + 0.75x which, by adding 23.25 to both sides becomes 24.25 = 0.75x which leads to . This means that (32.33, 1) also falls on our line.

Or, we could just use our calculator. We just press
[1] to specify the y value
[SHIFT] [+] our display should read 32.33333333. Notice the golden (x-hat) written above the addition key. This symbol represents the estimated x value. Again, we don't need to press [=] after the [SHIFT] [+].

Finding the Regression Line and the Correlation Coefficient

One of the things that we will learn in the Measurements of Association portion of this course is how to find the line that best fits numerous data points. And, then, to describe how well that line fits our data points. Let's enter the following data points into our calculator:

(-12, -4) (- 6, -3) (-2, -1) (1, 1) (3, 5) and (4, 7)

Press [ON]
[
MODE] [3] [1] to put your calculator in linear regression mode
[SHIFT] [AC] [=] to clear your statistical memory.
To enter the data, press:

 [(–)] [1] [2] [,] [(–)] [4] [M+] entered the first point. [(–)] [6] [,] [(–)] [3] [M+] entered the second point. [(–)] [2] [,] [(–)] [1] [M+] entered the third point. [1] [,] [1] [M+] entered the fourth point. [3] [,] [5] [M+] entered the fifth point, [4] [,] [7] [M+] entered the sixth point. x-value , y-value Enter General form

To find the linear equation for the best fitting line through this data, we must find a and b.
Press [SHIFT] [7] [=] Your display should read 2.14516129 which is the value of a.

To obtain the b value, press [SHIFT] [8] [=]. Your display should read 0.655913978

The linear equation for the best fitting line (called the regression line) to the data we entered is:

y = 2.15 + 0.66x

Another statistic that you will learn about, the Pearson Product Moment Correlation Coefficient, tells us how well the line fits the data. The symbol for the Pearson Product Moment Correlation Coefficient is r. To find the r for our data, press:
[SHIFT] [(] [=] your display should read 0.909056586. Again, please notice the golden r above the [9] button

Finding the Means and Standard Deviations

Again, let's look at the data points we entered: (-12, -4) (- 6, -3) (-2, -1) (1, 1) (3, 5) and (4, 7)

Notice that we have six points with six x-values (-12, - 6, -2, 1, 3, and 4) and six y-values (-4 , -3, -1, 1, 5, and 7).

We can find the mean and standard deviations for the x variable the same way we did when we had only one variable. To find the mean of the x-values press:
[SHIFT] [1] [=] your display should read -2

Before calculating the standard deviation and variance, you should determine if your data constitutes the entire population or just a sample of a population.

If you have the data for the entire population:

To find the population standard deviation for the x values, press

[SHIFT] [2] [=] your display should read 5.567764363

If you have the data for only a sample:

To find the sample standard deviation for the x values, press

[SHIFT] [3] [=] your display should read 6.099180273

Notice that these buttons [1], [2], and [3] are all in the same row. To find the mean and standard deviations for the y data, we move up a row to the [4], [5], and [6].

To find the mean of the y-values press:
[SHIFT] [4] [=] your display should read 0.833333333

To find the population standard deviation for the y values, press

[SHIFT] [5] [=] your display should read 4.017323598

To find the sample standard deviation for the y values, press

[SHIFT] [6] [=] your display should read 4.400757511.

If your display did not show the numbers above, you probably pushed the red [AC] button when you should not have. When you press the red [AC] button, you clear out your data and need to re-enter all the data again before computing another answer based on that data. Moral to the story: Do not press the clear button unless you are entering a new data set.

Practice Opportunities

1. What is linear equation for the line that goes through (3, -21) and (7, -15)?

Check answer by clicking here

2. What is the Pearson Product Moment Correlation Coefficient for the following data: (4, 7), (6, 9), (2, 6), (7, 7), (9, 8), (8, 8), (7, 4), and (10, 12)?

Check answer by clicking here

Entering Data (Part 2) - Entering the Same Points Repeatedly

 Point Frequency (7, -13) 3 (4, 6) 2 (8, -6) 6 (6, 7) 5 (2, 1) 0 (5, 7) 1

The general form for entering frequency data is to enter the point then [SHIFT] [,] then the frequency then [M+]. The frequency is always positive and nonzero. A point not followed by [SHIFT] [,] is treated as if the frequency is 1.

Before entering the data, make sure your calculator is in Linear Regression mode. Clear your statistical memories by pressing [SHIFT] [AC] [=].
Press [7] [,] [(-)] [1] [3] [SHIFT] [,] [3] [M+]

 Before pressing the [M+] your display will say: 7,-13;3 (the [SHIFT] [,] becomes a semicolon. When you press the [M+] your display will say 7 (the last x-value entered).

 [4] [,] [6] [SHIFT] [,] [2] [M+] 2nd point [8] [,] [(-)] [6] [SHIFT] [,] [6] [M+] 3rd point [6] [,] [7] [SHIFT] [,] [5] [M+] 4th point [6] [,] [1] [SHIFT] [,] [0] [M+] 5th point [5] [,] [7] [SHIFT] [,] [1] [M+] 6th point x-value , y-value ; Frequency Enter General Form

Press [SHIFT] [(] [=] and your display will tell you that the r is -0.724558575.

Practice Opportunities

3. What are the regression line, the correlation, and the mean of y for the following data?

 Point Frequency (1, 3) 12 (3, 4) 1 (5, 6) 7 (2, 3) 13 (3, 5) 0 (-7, -5) 6

Check answer by clicking here

That is it for the calculator.

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