Casio ALGEBRA FX 2.0 User manual

19990401
Chapter
Statistical Graphs and
Calculations
This chapter describes how to input statistical data into lists, and
how to calculate the mean, maximum and other statistical values.
It also tells you how to perform regression calculations.
6-1 Before Performing Statistical Calculations
6-2 Calculating and Graphing Single-Variable Statistical
Data
6-3 Calculating and Graphing Paired-Variable Statistical
Data
6-4 Performing Statistical Calculations
6-5 Distribution
Important!
•This chapter contains a number of graph screen shots. In each case, new
data values were input in order to highlight the particular characteristics of
the graph being drawn. Note that when you try to draw a similar graph, the
unit uses data values that you have input using the List function. Because of
this, the graphs that appear on the screen when you perform a graphing
operation will probably differ somewhat from those shown in this manual.
6

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6-1 Before Performing Statistical Calculations
From the Main Menu, enter the STAT Mode and display the statistical data lists.
Use the statistical data lists to input data and to perform statistical calculations.
Use f, c, d and e to move
the highlighting around the lists.
Once you input data, you can use it to produce a graph and check for tendencies. You can
also use a variety of different regression calculations to analyze the data.
kInputting Data into Lists
○○○○○
Example To input the following two data groups
0.5, 1.2, 2.4, 4.0, 5.2
–2.1, 0.3, 1.5, 2.0, 2.4
a.fwb.cw
c.ewewf.cw
e
-c.bwa.dw
b.fwcwc.ew
Once data is input, you can use it for graphing and statistical calculations.
6-1-1
Before Performing Statistical Calculations
# Except for complex numbers, calculation
results can be input as statistical data.
#You can use the f, c, dand ekeys
to move the highlighting to any cell in the lists
for data input.

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kChanging Graph Parameters
Use the following procedures to specify the graph draw/non-draw status, the graph type, and
other general settings for each of the graphs in the graph menu (GPH1, GPH2, GPH3).
While the statistical data list is on the display, press 1(GRPH) to display the graph menu,
which contains the following items.
•{S-Gph1}/{S-Gph2}/{S-Gph3} ... graph {1}/{2}/{3} drawing*1
•{Select} ... {simultaneous graph (GPH1, GPH2, GPH3) selection} (You can specify the
multiple graphs.)
•{Set} ... {graph settings (graph type, list assignments)}
1. General graph settings [GRPH]-[Set]
This section describes how to use the general graph settings screen to make the following
settings for each graph (GPH1, GPH2, GPH3).
• Graph Type
The initial default graph type setting for all the graphs is scatter graph.You can select one of
a variety of other statistical graph types for each graph.
• List
The initial default statistical data is List 1 for single-variable data, and List 1 and List 2 for
paired-variable data. You can specify which statistical data list you want to use for x-data and
y-data.
• Frequency
Normally, each data item or data pair in the statistical data list is represented on a graph as a
point. When you are working with a large number of data items however, this can cause
problems because of the number of plot points on the graph. When this happens, you can
specify a frequency list that contains values indicating the number of instances (the
frequency) of the data items in the corresponding cells of the lists you are using for x-data
and y-data. Once you do this, only one point is plotted for the multiple data items, which
makes the graph easier to read.
6-1-2
Before Performing Statistical Calculations
*1 The initial default graph type setting for all the
graphs (Graph 1 through Graph 3) is scatter
diagram, but you can change to one of a
number of other graph types.
#You can specify the graph draw/non-draw
status, the graph type, and other general
settings for each of the graphs in the graph
menu (GPH1, GPH2, GPH3).

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• Mark Type
This setting lets you specify the shape of the plot points on the graph.
uTo display the general graph settings screen [GRPH]-[Set]
Pressing 1(GRPH)f(Set) displays the general graph settings screen.
•The settings shown here are examples only. The settings on your general graph settings
screen may differ.
• StatGraph (statistical graph specification)
•{GPH1}/{GPH2}/{GPH3} ... graph {1}/{2}/{3}
• Graph Type (graph type specification)
•{Scat}/{xy}/{NPP} ... {scatter diagram}/{xy line graph}/{normal probability plot}
•{Hist}/{Box}/{ModB}/{N·Dis}/{Brkn} ... {histogram}/{med-box graph}/{modified-box
graph}/{normal distribution curve}/{line graph}
•{X}/{Med}/{X^2}/{X^3}/{X^4} ... {linear regression graph}/{Med-Med graph}/{quadratic
regression graph}/{cubic regression graph}/{quartic regression graph}
•{Log}/{Exp}/{Pwr}/{Sin}/{Lgst} ... {logarithmic regression graph}/{exponential regression
graph}/{power regression graph}/{sinusoidal regression graph}/{logistic regression graph}
• XList (x-axis data list)
•{LIST} ... {List 1 to 20}
• YList (y-axis data list)
•{LIST} ... {List 1 to 20}
• Frequency (number of times a value occurs)
•{1} ... {1-to-1 plot}
•{LIST} ... contents of these lists indicate the frequency of XList and YList data
• Mark Type (plot mark type)
•{ }/{×}/{•} ... scatter diagram plot points
6-1-3
Before Performing Statistical Calculations

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2. Graph draw/non-draw status [GRPH]-[Select]
The following procedure can be used to specify the draw (On)/non-draw (Off) status of each of
the graphs in the graph menu.
uTo specify the draw/non-draw status of a graph
1. Pressing 1(GRPH) e(Select) displays the graph On/Off screen.
•Note that the StatGraph1 setting is for Graph 1 (GPH1 of the graph menu), StatGraph2
is for Graph 2, and StatGraph3 is for Graph 3.
2. Use the cursor keys to move the highlighting to the graph whose status you want to
change, and press the applicable function key to change the status.
•{On}/{Off} ... {On (draw)}/{Off (non-draw)}
•{DRAW} ... {draws all On graphs}
3. To return to the graph menu, press i.
6-1-4
Before Performing Statistical Calculations
#ViewWindow parameters are normally set
automatically for statistical graphing. If you
want to set View Window parameters
manually, you must change the Stat Wind item
to “Manual”.
While the statistical data list is on the display,
perform the following procedure.
u3(SET UP)2(Man)
i(Returns to previous menu.)
#The default setting automatically uses List 1
data as x-axis (horizontal) values and List 2
data as y-axis (vertical) values. Each set of x/y
data is a point on the scatter diagram.
# Pressinguadoes not hide the menu while a
statistical graph is on the display.

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6-2 Calculating and Graphing Single-Variable
Statistical Data
Single-variable data is data with only a single variable. If you are calculating the average
height of the members of a class for example, there is only one variable (height).
Single-variable statistics include distribution and sum. The following types of graphs are
available for single-variable statistics.
You can also use the procedures under “Changing Graph Parameters” on page 6-1-2 to make
the settings you want before drawing each graph.
kk
kk
kNormal Probability Plot (NPP)
This plot compares the data accumulated ratio with a normal distribution accumulated ratio.
XList specifies the list where data is input, and Mark Type is used to select from among the
marks { / ×/ •}you want to plot.
Press ior !i(QUIT) to return to the statistical data list.
kk
kk
kHistogram (Bar Graph) (Hist)
XList specifies the list where the data is input, while Freq specifies the list where the data
frequency is input. 1 is specified for Freq when frequency is not specified.
6-2-1
Calculating and Graphing Single-Variable Statistical Data
⇒
w(Draw)
The display screen appears as shown above before the graph is drawn. At this point, you
can change the Start and pitch values.

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kMed-box Graph (Box)
This type of graph lets you see how a large number of data items are grouped within specific
ranges. A box encloses all the data in an area from the 25th percentile to the 75th percentile,
with a line drawn at the 50th percentile. Lines (called whiskers) extend from either end of the
box up to the minimum and maximum of the data.
XList specifies the list where the data is input, while Freq specifies the list where the data
frequency is input. 1 is specified for Freq when frequency is not specified.
kModified Box Graph (ModB)
The modified box graph omits everything in the range past 1.5 ×IQR (IQR = Q3 – Q1,
Q3: 3rd quartile, Q1: 1st quartile) from the med-box 4th quartile and draws whiskers.
Outliers are displayed as plot points.
XList specifies the list where the data is input, while Freq specifies the list where the data
frequency is input. 1 is specified for Freq when frequency is not specified.
6-2-2
Calculating and Graphing Single-Variable Statistical Data
#Input a positive integer for frequency data.
Other types of values (decimals, etc.) cause
an error.

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kNormal Distribution Curve (N •Dis)
The normal distribution curve is graphed using the following normal distribution function.
y
= 1
(2 π)xσ n
e
–2xσ
n
2
(x–x)
2
XList specifies the list where the data is input, while Freq specifies the list where the data
frequency is input. 1 is specified for Freq when frequency is not specified.
kLine Graph (Brkn)
Lines connect center points of a histogram bar.
XList specifies the list where the data is input, while Freq specifies the list where the data
frequency is input. 1 is specified for Freq when frequency is not specified.
6-2-3
Calculating and Graphing Single-Variable Statistical Data
⇒
w(Draw)
The display screen appears as shown above before the graph is drawn. At this point, you
can change the Start and pitch values.

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kDisplaying the Calculation Results of a Drawn Single-Variable Graph
Single-variable statistics can be expressed as both graphs and parameter values. When
these graphs are displayed, the single-variable calculation results appear as shown below
when you press 4(CALC)b(1VAR).
•Use cto scroll the list so you can view the items that run off the bottom of the screen.
The following describes the meaning of each of the parameters.
o............. mean
Σx........... sum
Σx2.......... sum of squares
xσn.......... population standard deviation
xσn–1 ........ sample standard deviation
n............. number of data items
minX ....... minimum
Q1 .......... first quartile
Med ........ median
Q3 .......... third quartile
maxX ...... maximum
Mod ........ mode
Mod : n ... number of data mode items
Mod : F ... data mode frequency
•Press 6(DRAW) to return to the original single-variable statistical graph.
6-2-4
Calculating and Graphing Single-Variable Statistical Data
#When Mod has multiple solutions, they are all
displayed.

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6-3-1
Calculating and Graphing Paired-Variable Statistical Data
6-3 Calculating and Graphing Paired-Variable
Statistical Data
kDrawing a Scatter Diagram and xy Line Graph
Description
The following procedure plots a scatter diagram and connects the dots to produce an xy line
graph.
Set Up
1. From the Main Menu, enter the STAT Mode.
Execution
2. Input the data into a list.
3. Specify Scat (scatter diagram) or xy (xy line graph) as the graph type, and then
execute the graph operation.
Press ior !i(QUIT) to return to the statistical data list.

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○○○○○
Example Input the two sets of data shown below. Next, plot the data on a
scatter diagram and connect the dots to produce an xy line graph.
0.5, 1.2, 2.4, 4.0, 5.2,
–2.1, 0.3, 1.5, 2.0, 2.4
Procedure
1mSTAT
2a.fwb.cw
c.ewewf.cw
e
-c.bwa.dw
b.fwcwc.ew
3(Scatter diagram)1(GRPH)f(Set)c1(Scat)i
1(GRPH)b(S-Gph1)
3(xy line graph)1(GRPH)f(Set)c2(xy)i
1(GRPH)b(S-Gph1)
Result Screen
(Scatter diagram)
(xy line graph)
6-3-2
Calculating and Graphing Paired-Variable Statistical Data

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kDrawing a Regression Graph
Description
Use the following procedure to input paired-variable statistical data, perform a regression
calculation using the data, and then graph the results.
Set Up
1. From the Main Menu, enter the STAT Mode.
Execution
2. Input the data into a list, and plot the scatter diagram.
3. Select the regression type, execute the calculation, and display the regression
parameters.
4. Draw the regression graph.
6-3-3
Calculating and Graphing Paired-Variable Statistical Data
#You can perform trace on a regression graph.
You cannot perform trace scroll.

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○○○○○
Example Input the two sets of data shown below and plot the data on a scatter
diagram. Next, perform logarithmic regression on the data to display
the regression parameters, and then draws the corresponding
regression graph.
0.5, 1.2, 2.4, 4.0, 5.2,
–2.1, 0.3, 1.5, 2.0, 2.4
Procedure
1mSTAT
2a.fwb.cw
c.ewewf.cw
e
-c.bwa.dw
b.fwcwc.ew
1(GRPH)f(Set)c1(Scat)i
1(GRPH)b(S-Gph1)
34(CALC)h(Log)
46(DRAW)
Result Screen
6-3-4
Calculating and Graphing Paired-Variable Statistical Data

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kk
kk
kSelecting the Regression Type
After you graph paired-variable statistical data, press 4(CALC). Then you can use the
function menu at the bottom of the display to select from a variety of different types of
regression.
•{2VAR} ... {paired-variable statistical results}
•{Linear}/{MedMed}/{Quad}/{Cubic}/{Quart}/{Log}/{Exp}/{Power}/{Sin}/{Lgstic}
... {linear regression}/{Med-Med}/{quadratic regression}/{cubic regression}/{quartic
regression}/{logarithmic regression}/{exponential regression}/{power regression}/
{sinusoidal regression}/{logistic regression} calculation and graphing
kk
kk
kDisplaying Statistical Calculation Results
Whenever you perform a regression calculation, the regression formula parameter (such as a
and bin the linear regression y= ax + b) calculation results appear on the display. You can
use these to obtain statistical calculation results.
Regression parameters are calculated as soon as you press a function key to select a
regression type while a graph is on the display.
kk
kk
kGraphing Statistical Calculation Results
While the parameter calculation result is on the display, you can graph the displayed
regression formula by pressing 6(DRAW).
6-3-5
Calculating and Graphing Paired-Variable Statistical Data

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kk
kk
kLinear Regression Graph
Linear regression uses the method of least squares to plot a straight line that passes close to
as many data points as possible, and returns values for the slope and y - intercept
(y -coordinate when x= 0) of the line.
The graphic representation of this relationship is a linear regression graph.
4(CALC)c(Linear)
6(DRAW)
The following is the linear regression model formula.
y= ax + b
a............. regression coefficient (slope)
b............. regression constant term (intercept)
r............. correlation coefficient
r2............ coefficient of determination
kk
kk
kMed-Med Graph
When it is suspected that there are a number of extreme values, a Med-Med graph can be
used in place of the least squares method. This is similar to linear regression, but it
minimizes the effects of extreme values.
4(CALC)d(MedMed)
6(DRAW)
The following is the Med-Med graph model formula.
y= ax + b
a............. Med-Med graph slope
b............. Med-Med graph intercept
6-3-6
Calculating and Graphing Paired-Variable Statistical Data
#Input a positive integer for frequency data.
Other types of values (decimals, etc.) cause
an error.

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kk
kk
kQuadratic/Cubic/Quartic Regression Graph
A quadratic/cubic/quartic regression graph represents connection of the data points of a
scatter diagram. It uses the method of least squares to draw a curve that passes close to as
many data points as possible. The formula that represents this is quadratic/cubic/quartic
regression.
Ex. Quadratic regression
4(CALC)e(Quad)
6(DRAW)
Quadratic regression
Model formula ..... y= ax2+ bx + c
a............. regression second coefficient
b............. regression first coefficient
c............. regression constant term (intercept)
Cubic regression
Model formula ..... y= ax3+ bx2+ cx + d
a............. regression third coefficient
b............. regression second coefficient
c............. regression first coefficient
d............. regression constant term (intercept)
Quartic regression
Model formula ..... y= ax4+ bx3+ cx2+ dx + e
a............. regression fourth coefficient
b............. regression third coefficient
c............. regression second coefficient
d............. regression first coefficient
e............. regression constant term (intercept)
6-3-7
Calculating and Graphing Paired-Variable Statistical Data

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kLogarithmic Regression Graph
Logarithmic regression expresses yas a logarithmic function of x. The standard logarithmic
regression formula is y= a+ b×In x, so if we say that X = In x, the formula corresponds to
linear regression formula y= a+ bX.
4(CALC)h(Log)
6(DRAW)
The following is the logarithmic regression model formula.
y= a+ b•ln x
a............. regression constant term
b............. regression coefficient
r.............. correlation coefficient
r2............ coefficient of determination
kExponential Regression Graph
Exponential regression expresses yas a proportion of the exponential function of x. The
standard exponential regression formula is y= a×ebx, so if we take the logarithms of both
sides we get In y= In a + bx. Next, if we say Y = In y, and a= In a, the formula corresponds
to linear regression formula Y = a+ bx.
4(CALC)i(Exp)
6(DRAW)
The following is the exponential regression model formula.
y= a•ebx
a............. regression coefficient
b............. regression constant term
r.............. correlation coefficient
r2............ coefficient of determination
6-3-8
Calculating and Graphing Paired-Variable Statistical Data

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kk
kk
kPower Regression Graph
Power regression expresses yas a proportion of the power of x. The standard power
regression formula is y= a×xb, so if we take the logarithms of both sides we get In y= In a+
b×In x. Next, if we say X = In x, Y = In y, and a= In a, the formula corresponds to linear
regression formula Y = a+ bX.
4(CALC)j(Power)
6(DRAW)
The following is the power regression model formula.
y= a•xb
a............. regression coefficient
b............. regression power
r.............. correlation coefficient
r2............. coefficient of determination
kk
kk
kSinusoidal Regression Graph
Sinusoidal regression is best applied for cyclical data.
The following is the sinusoidal regression model formula.
y= a·sin(bx + c) + d
While the statistical data list is on the display, perform the following key operation.
4(CALC)v(Sin)
6(DRAW)
Drawing a sinusoidal regression graph causes the angle unit setting of the calculator to
automatically change to Rad (radians). The angle unit does not change when you perform a
sinusoidal regression calculation without drawing a graph.
• Certain types of data may take a long time to calculate. This does not indicate malfunction.
6-3-9
Calculating and Graphing Paired-Variable Statistical Data

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kk
kk
kLogistic Regression Graph
Logistic regression is best applied for time-based phenomena in which there is a continual
increase until a saturation point is reached.
The following is the logistic regression model formula.
y=c
1 + ae–bx
4(CALC)l(Lgstic)
6(DRAW)
• Certain types of data may take a long time to calculate. This does not indicate malfunction.
kk
kk
kResidual Calculation
Actual plot points (y-coordinates) and regression model distance can be calculated during
regression calculations.
While the statistical data list is on the display, recall the SET UP screen to specify a LIST
(“List 1” through “List 20”) for “Resid List”. Calculated residual data is stored in the specified
list.
The vertical distance from the plots to the regression model will be stored in the list.
Plots that are higher than the regression model are positive, while those that are lower are
negative.
Residual calculation can be performed and saved for all regression models.
6-3-10
Calculating and Graphing Paired-Variable Statistical Data
#Any data already existing in the selected list is
cleared. The residual of each plot is stored in
the same precedence as the data used as the
model.

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kk
kk
kDisplaying the Calculation Results of a Drawn Paired-Variable Graph
Paired-variable statistics can be expressed as both graphs and parameter values. When
these graphs are displayed, the paired-variable calculation results appear as shown below
when you press 4(CALC)b(2VAR).
•Use cto scroll the list so you can view the items that run off the bottom of the screen.
o............... mean of data stored in xList
Σx............. sum of data stored in xList
Σx2........... sum of squares of data
stored in xList
xσn............ population standard
deviation of data stored in
xList
xσn-1 .......... sample standard deviation
of data stored in xList
n............... number of data
p............... mean of data stored in yList
Σy............. sum of data stored in yList
kk
kk
kCopying a Regression Graph Formula to the GRPH
•
TBL Mode
You can copy regression formula calculation results to the GRPH •TBL Mode graph formula
area, and store and compare.
1. Press 5(COPY) to copy the regression formula that produced the displayed data to
the GRPH •TBL Mode graph formula area*1.
2. Press wto save the copied graph formula and return to the previous regression
calculation result display.
6-3-11
Calculating and Graphing Paired-Variable Statistical Data
Σy2...... sum of squares of data stored in yList
yσn...... population standard deviation of data
stored in yList
yσn-1 .... sample standard deviation of data
stored in yList
Σxy ..... sum of data stored in xList and yList
minX ... minimum of data stored in xList
maxX .. maximum of data stored in xList
minY ... minimum of data stored in yList
maxY .. maximum of data stored in yList
*1You cannot edit regression formulas for graph
formulas in the GRPH
•
TBL Mode.