Casio CFX-9850G PLUS User manual

Chapter
18
Statistical Graphs and
Calculations
This chapter describes how to input statistical data into lists, how
to calculate the mean, maximum and other statistical values, how
to perform various statistical tests, how to determine the confi-
dence interval, and how to produce a distribution of statistical
data. It also tells you how to perform regression calculations.
18-1 Before Performing Statistical Calculations
18-2 Paired-Variable Statistical Calculation Examples
18-3 Calculating and Graphing Single-Variable Statistical
Data
18-4 Calculating and Graphing Paired-Variable Statistical
Data
18-5 Performing Statistical Calculations
18-6 Tests
18-7 Confidence Interval
18-8 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 appears on the screen when you perform a graphing
operation will probably differ somewhat from those shown in this manual.

250
18-1 Before Performing Statistical Calculations
In the Main Menu, select the STAT icon to 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.
P. 251 •{GRPH} ... {graph menu}
P. 270 •{CALC} ... {statistical calculation menu}
P. 277 •{TEST} ... {test menu}
P. 294 •{INTR} ... {confidence interval menu}
P. 304 •{DIST} ... {distribution menu}
P. 234 •{SRT·A}/{SRT·D} ... {ascending}/{descending} sort
P. 233 •{DEL}/{DEL·A} ... deletes {highlighted data}/{all data}
P. 234 •{INS} ... {inserts new cell at highlighted cell}
•The procedures you should use for data editing are identical to those you use
P. 229 with the list function. For details, see “17. List Function”.

251
18-2 Paired-Variable Statistical Calculation
Examples
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.
Example To input the following two data groups and perform statistical
calculations
{0.5, 1.2, 2.4, 4.0, 5.2}
{–2.1, 0.3, 1.5, 2.0, 2.4}
kk
kk
kInputting Data into Lists
Input the two groups of data into List 1 and List 2.
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.
•Input values can be up to 10 digits long.
•You can use the f, c, dand ekeys to move the highlighting to any cell
in the lists for data input.
kk
kk
kPlotting a Scatter Diagram
Use the data input above to plot a scatter diagram.
1(GRPH)1(GPH1)
•To return to the statistical data list, press Jor !Q.
•View Window 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”.
Note that View Window parameters are set automatically for the following
types of graphs regardless of whether or not the Stat Wind item is set to
“Manual”.
1-Sample ZTest, 2-Sample ZTest, 1-Prop ZTest, 2-Prop ZTest, 1-Sample
tTest, 2-Sample tTest, χ2Test, 2-Sample FTest (x-axis only disregarded).

252
While the statistical data list is on the display, perform the following procedure.
!Z2(Man)
J(Returns to previous menu.)
•It is often difficult to spot the relationship between two sets of data (such as
height and shoe size) by simply looking at the numbers. Such relationship
become clear, however, when we plot the data on a graph, using one set of
values as x-data and the other set as y-data.
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/ydata is a point on the scatter
diagram.
kk
kk
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.
•{GPH1}/{GPH2}/{GPH3} ... only one graph {1}/{2}/{3} drawing
•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.
P. 252 •{SEL} ... {simultaneous graph (GPH1, GPH2, GPH3) selection}
P. 254 •{SET} ... {graph settings (graph type, list assignments)}
•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).
•You can press any function key (1,2,3) to draw a graph regardless of
the current location of the highlighting in the statistical data list.
1. Graph draw/non-draw status [GRPH]-[SEL]
The following procedure can be used to specify the draw (On)/non-draw (Off)
status of each of the graphs in the graph menu.
uu
uu
uTo specify the draw/non-draw status of a graph
1. Pressing 4(SEL) displays the graph On/Off screen.
18 - 2 Paired-Variable Statistical Calculation Examples

253
•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} ... setting {On (draw)}/{Off (non-draw)}
• {DRAW} ... {draws all On graphs}
3. To return to the graph menu, press J.
uu
uu
uTo draw a graph
Example To draw a scatter diagram of Graph 3 only
1(GRPH)4(SEL) 2(Off)
cc1(On)
6(DRAW)
2. 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.
• Mark Type
This setting lets you specify the shape of the plot points on the graph.
Paired-Variable Statistical Calculation Examples 18 - 2

254
uu
uu
uTo display the general graph settings screen [GRPH]-[SET]
Pressing 6(SET) displays the general graph settings screen.
•The settings shown here are examples only. The settings on your general graph
settings screen may differ.
uu
uu
uStatGraph (statistical graph specification)
•{GPH1}/{GPH2}/{GPH3} ... graph {1}/{2}/{3}
uu
uu
uGraph Type (graph type specification)
•{Scat}/{xy}/{NPP} ... {scatter diagram}/{xy line graph}/{normal probability plot}
–––
•{Hist}/{Box}/{Box}/{N·Dis}/{Brkn} ... {histogram}/{med-box graph}/{mean-box
graph}/{normal distribution curve}/{broken 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}/{sine regression graph}/
{logistic regression graph}
uu
uu
uXList (x-axis data list)
•{List1}/{List2}/{List3}/{List4}/{List5}/{List6} ... {List 1}/{List 2}/{List 3}/{List 4}/
{List 5}/{List 6}
uu
uu
uYList (y-axis data list)
•{List1}/{List2}/{List3}/{List4}/{List5}/{List6} ... {List 1}/{List 2}/{List 3}/{List 4}/
{List 5}/{List 6}
uu
uu
uFrequency (number of data items)
•{1} ... {1-to-1 plot}
•{List1}/{List2}/{List3}/{List4}/{List5}/{List6} ... frequency data in {List 1}/
{List 2}/{List 3}/{List 4}/{List 5}/{List 6}
uu
uu
uMark Type (plot mark type)
•{ }/{×}/{•} ... plot points: { }/{×}/{•}
18 - 2 Paired-Variable Statistical Calculation Examples

255
uu
uu
uGraph Color (graph color specification)
•{Blue}/{Orng}/{Grn} ... {blue}/{orange}/{green}
uu
uu
uOutliers (outliers specification)
•{On}/{Off} ... {display}/{do not display} Med-Box outliers
kk
kk
kDrawing an xy Line Graph
P. 254 Paired data items can be used to plot a scatter diagram. A scatter diagram where
(Graph Type) the points are linked is an xy line graph.
(xy)
Press Jor !Q to return to the statistical data list.
kk
kk
kDrawing a Normal Probability Plot
P. 254 Normal probability plot contrasts the cumulative proportion of variables with the
(Graph Type) cumulative proportion of a normal distribution and plots the result. The expected
(NPP) values of the normal distribution are used as the vertical axis, while the observed
values of the variable being tested are on the horizontal axis.
Press Jor !Q to return to the statistical data list.
kk
kk
kSelecting the Regression Type
After you graph paired-variable statistical data, you can use the function menu at
the bottom of the display to select from a variety of different types of regression.
•{X}/{Med}/{X^2}/{X^3}/{X^4}/{Log}/{Exp}/{Pwr}/{Sin}/{Lgst} ... {linear regres-
sion}/{Med-Med}/{quadratic regression}/{cubic regression}/{quartic
regression}/{logarithmic regression}/{exponential regression}/{power
regression}/{sine regression}/{logistic regression} calculation and graphing
•{2VAR} ... {paired-variable statistical results}
Paired-Variable Statistical Calculation Examples 18 - 2
CFX

256
kk
kk
kDisplaying Statistical Calculation Results
Whenever you perform a regression calculation, the regression formula parameter
(such as aand 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.
Example To display logarithmic regression parameter calculation results
while a scatter diagram is on the display
6(g)1(Log)
kk
kk
kGraphing Statistical Calculation Results
You can use the parameter calculation result menu to graph the displayed
regression formula.
P. 268 • {COPY} ... {stores the displayed regression formula as a graph function}
• {DRAW} ... {graphs the displayed regression formula}
Example To graph a logarithmic regression
While logarithmic regression parameter calculation results are on the display,
press 6(DRAW).
P. 255 For details on the meanings of function menu items at the bottom of the display, see
“Selecting the Regression Type”.
18 - 2 Paired-Variable Statistical Calculation Examples

257
Calculating and Graphing Single-Variable Statistical Data 18 - 3
18-3 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.
kk
kk
kDrawing a Histogram (Bar Graph)
From the statistical data list, press 1(GRPH) to display the graph menu, press
6(SET), and then change the graph type of the graph you want to use (GPH1,
GPH2, GPH3) to histogram (bar graph).
Data should already be input in the statistical data list (see “Inputting Data into
Lists”). Draw the graph using the procedure described under “Changing Graph
Parameters”.
⇒
6(DRAW)
The display screen appears as shown above before the graph is drawn. At this
point, you can change the Start and pitch values.
kk
kk
kMed-box Graph (Med-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 first quartile
(Q1) to the third quartile (Q3), with a line drawn at the median (Med). Lines (called
whiskers) extend from either end of the box up to the minimum and maximum of
the data.
From the statistical data list, press 1(GRPH) to display the graph menu, press
6(SET), and then change the graph type of the graph you want to use (GPH1,
GPH2, GPH3) to med-box graph.
P. 251
P. 252
P. 254
(Graph Type)
(Hist)
6
P. 254
(Graph Type)
(Box)
Q1 Med Q3 maxX
minX

258
To plot the data that falls outside the box, first specify “MedBox” as the graph
type. Then, on the same screen you use to specify the graph type, turn the outliers
item “On”, and draw the graph.
kk
kk
kMean-box Graph
This type of graph shows the distribution around the mean when there is a large
number of data items. A line is drawn at the point where the mean is located, and
then a box is drawn so that it extends below the mean up to the population
standard deviation (o– x σ n) and above the mean up to the population standard
deviation (o+ x σ n). Lines (called whiskers) extend from either end of the box up to
the minimum (minX) and maximum (maxX) of the data.
From the statistical data list, press 1(GRPH) to display the graph menu, press
6(SET), and then change the graph type of the graph you want to use (GPH1,
GPH2, GPH3) to mean-box graph.
kk
kk
kNormal Distribution Curve
P. 254 The normal distribution curve is graphed using the following normal distribution
(Graph Type) function.
(N·Dis)
y
= 1
(2 π)xσ n
e
–2xσ
n
2
(x–x)2
The distribution of characteristics of items manufactured according to some fixed
standard (such as component length) fall within normal distribution. The more data
items there are, the closer the distribution is to normal distribution.
From the statistical data list, press 1(GRPH) to display the graph menu, press
6(SET), and then change the graph type of the graph you want to use (GPH1,
GPH2, GPH3) to normal distribution.
18 - 3 Calculating and Graphing Single-Variable Statistical Data
P. 254
(Graph Type)
(Box)
Note :
This function is not usually used in
the classrooms in U.S. Please use
Med-box Graph, instead.
o– x σ no o + x σ n
minX
maxX

259
Calculating and Graphing Single-Variable Statistical Data 18 - 3
kk
kk
kBroken Line Graph
P. 254 A broken line graph is formed by plotting the data in one list against the frequency
(Graph Type) of each data item in another list and connecting the points with straight lines.
(Brkn) Calling up the graph menu from the statistical data list, pressing 6(SET),
changing the settings to drawing of a broken line graph, and then drawing a graph
creates a broken line graph.
⇒
6(DRAW)
6
The display screen appears as shown above before the graph is drawn. At this
point, you can change the Start and pitch values.
kk
kk
kDisplaying Single-Variable Statistical Results
Single-variable statistics can be expressed as both graphs and parameter values.
When these graphs are displayed, the menu at the bottom of the screen appears
as below.
• {1VAR} ... {single-variable calculation result menu}
Pressing 1(1VAR) displays the following screen.
•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.
_
x..................... mean of data
Σx................... sum of data
Σx2.................. sum of squares
xσn.................. population standard deviation
xσn-1 ................ sample standard deviation
n..................... number of data items

260
minX ............... minimum
Q1 .................. first quartile
Med ................ median
Q3 .................. third quartile
_
x–xσn............ data mean – population standard deviation
_
x+xσn............ data mean + population standard deviation
maxX .............. maximum
Mod ................ mode
•Press 6(DRAW) to return to the original single-variable statistical graph.
18 - 3 Calculating and Graphing Single-Variable Statistical Data

261
18-4 Calculating and Graphing Paired-Variable
Statistical Data
Under “Plotting a Scatter Diagram,” we displayed a scatter diagram and then
performed a logarithmic regression calculation. Let’s use the same procedure to
look at the various regression functions.
kk
kk
kLinear Regression Graph
P. 254 Linear regression plots 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.
(Graph Type) !Q1(GRPH)6(SET)c
(Scatter) 1(Scat)
(GPH1) !Q1(GRPH)1(GPH1)
(X) 1(X)
123456
6(DRAW)
a...... regression coefficient (slope)
b...... regression constant term (y-intercept)
r....... correlation coefficient
r2...... coefficient of determination
kk
kk
kMed-Med Graph
P. 254 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 also a type of
linear regression, but it minimizes the effects of extreme values. It is especially
useful in producing highly reliable linear regression from data that includes
irregular fluctuations, such as seasonal surveys.
2(Med)
123456

262
6(DRAW)
a...... Med-Med graph slope
b...... Med-Med graph y-intercept
kk
kk
kQuadratic/Cubic/Quartic Regression Graph
P. 254 A quadratic/cubic/quartic regression graph represents connection of the data
points of a scatter diagram. It actually is a scattering of so many points that are
close enough together to be connected. The formula that represents this is
quadratic/cubic/quartic regression.
Ex. Quadratic regression
3(X^ 2)
123456
6(DRAW)
Quadratic regression
a...... regression second coefficient
b...... regression first coefficient
c...... regression constant term (y-intercept)
Cubic regression
a...... regression third coefficient
b...... regression second coefficient
c...... regression first coefficient
d...... regression constant term (y-intercept)
Quartic regression
a...... regression fourth coefficient
b...... regression third coefficient
c...... regression second coefficient
d...... regression first coefficient
e...... regression constant term (y-intercept)
18 - 4 Calculating and Graphing Paired-Variable Statistical Data

263
Calculating and Graphing Paired-Variable Statistical Data 18 - 4
kk
kk
kLogarithmic Regression Graph
P. 254 Logarithmic regression expresses yas a logarithmic function of x. The standard
logarithmic regression formula is y= a+ b×Inx, so if we say that X = Inx, the
formula corresponds to linear regression formula y= a+ bX.
6(g)1(Log)
123456
6(DRAW)
a...... regression constant term
b...... regression coefficient
r...... correlation coefficient
r2..... coefficient of determination
kk
kk
kExponential Regression Graph
P. 254 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 Iny= Ina+ bx. Next, if we say Y = Iny, and A = Ina,
the formula corresponds to linear regression formula Y = A + bx.
6(g)2(Exp)
123456
6(DRAW)
a...... regression coefficient
b...... regression constant term
r...... correlation coefficient
r2..... coefficient of determination

264
18 - 4 Calculating and Graphing Paired-Variable Statistical Data
kk
kk
kPower Regression Graph
P.254 Exponential regression expresses yas a proportion of the power of x. The
standard power regression formula is y= a×xb, so if we take the logarithm of both
sides we get Iny= Ina+ b×Inx. Next, if we say X = Inx, Y = Iny, and A = Ina, the
formula corresponds to linear regression formula Y = A + bX.
6(g)3(Pwr)
123456
6(DRAW)
a...... regression coefficient
b...... regression power
r...... correlation coefficient
r2..... coefficient of determination
kk
kk
kSine Regression Graph
Sine regression is best applied for phenomena that repeats within a specific
range, such as tidal movements.
y= a·sin(bx + c) + d
While the statistical data list is on the display, perform the following key operation.
6(g)5(Sin)
6(DRAW)
Drawing a sine 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 sine regression calculation without drawing a graph.
P. 254
6

265
Calculating and Graphing Paired-Variable Statistical Data 18 - 4
Gas bills, for example, tend to be higher during the winter when heater use is
more frequent. Periodic data, such as gas usage, is suitable for application of sine
regression.
Example To perform sine regression using the gas usage data shown
below
List 1 (Month Data)
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48}
List 2 (Gas Usage Meter Reading)
{130, 171, 159, 144, 66, 46, 40, 32, 32, 39, 44, 112, 116, 152, 157,
109, 130, 59, 40, 42, 33, 32, 40, 71, 138, 203, 162, 154, 136, 39,
32, 35, 32, 31, 35, 80, 134, 184, 219, 87, 38, 36, 33, 40, 30, 36, 55,
94}
Input the above data and plot a scatter diagram.
1(GRPH)1(GPH1)
Execute the calculation and produce sine regression analysis results.
6(g)5(Sin)
Display a sine regression graph based on the analysis results.
6(DRAW)
kk
kk
kLogistic Regression Graph
Logistic regression is best applied for phenomena in which there is a continual
increase in one factor as another factor increases until a saturation point is
reached. Possible applications would be the relationship between medicinal
dosage and effectiveness, advertising budget and sales, etc.
6
P. 254

266
y=C
1 + ae
–bx
6(g)6(g)1(Lgst)
6(DRAW)
Example Imagine a country that started out with a television diffusion
rate of 0.3% in 1966, which grew rapidly until diffusion reached
virtual saturation in 1980. Use the paired statistical data shown
below, which tracks the annual change in the diffusion rate, to
perform logistic regression.
List1(Year Data)
{66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83}
List2(Diffusion Rate)
{0.3, 1.6, 5.4, 13.9, 26.3, 42.3, 61.1, 75.8, 85.9, 90.3, 93.7, 95.4, 97.8, 97.8,
98.2, 98.5, 98.9, 98.8}
1(GRPH)1(GPH1)
Perform the calculation, and the logistic regression analysis values appear on the
display.
6(g)6(g)1(Lgst)
18 - 4 Calculating and Graphing Paired-Variable Statistical Data
6
6

267
Draw a logistic regression graph based on the parameters obtained from the
analytical results.
6(DRAW)
kk
kk
kResidual Calculation
Actual plot points (y-coordinates) and regression model distance can be calcu-
lated during regression calculations.
P. 6 While the statistical data list is on the display, recall the set up screen to specify a
list (“List 1” through “List 6”) 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.
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.
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.
kk
kk
kDisplaying Paired-Variable Statistical Results
Paired-variable statistics can be expressed as both graphs and parameter values.
When these graphs are displayed, the menu at the bottom of the screen appears
as below.
•{2VAR} ... {paired-variable calculation result menu}
Pressing 4(2VAR) displays the following screen.
Calculating and Graphing Paired-Variable Statistical Data 18 - 4

268
•Use cto scroll the list so you can view the items that run off the bottom of the
screen.
_
x..................... mean of xList data
Σx................... sum of xList data
Σx2.................. sum of squares of xList data
xσn.................. population standard deviation of xList data
xσn-1 ................ sample standard deviation of xList data
n..................... number of xList data items
_
y..................... mean of yList data
Σy................... sum of yList data
Σy2.................. sum of squares of yList data
yσn.................. population standard deviation of yList data
yσn-1 ................ sample standard deviation of yList data
Σxy .................. sum of the product of data stored in xList and yList
minX ............... minimum of xList data
maxX .............. maximum of xList data
minY ............... minimum of yList data
maxY .............. maximum of yList data
kk
kk
kCopying a Regression Graph Formula to the Graph Mode
After you perform a regression calculation, you can copy its formula to the
GRAPH Mode.
The following are the functions that are available in the function menu at the
bottom of the display while regression calculation results are on the screen.
• {COPY} ... {stores the displayed regression formula to the GRAPH Mode}
• {DRAW} ... {graphs the displayed regression formula}
1. Press 5(COPY) to copy the regression formula that produced the displayed
data to the GRAPH Mode.
Note that you cannot edit regression formulas for graph formulas in the GRAPH
Mode.
2. Press wto save the copied graph formula and return to the previous
regression calculation result display.
18 - 4 Calculating and Graphing Paired-Variable Statistical Data