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Casio CFX-9850G PLUS Calculations Manual
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Casio CFX-9850G PLUS Calculations Manual

18. statistical graphs and calculations
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Download this manual See also: Programming Manual, Function Manual
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.
Chapter
18

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Summary of Contents for Casio CFX-9850G PLUS

  • Page 1 Chapter 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.

  • Page 2: Before Performing Statistical Calculations

    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.
  • Page 3 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}...
  • Page 4 18 - 2 Paired-Variable Statistical Calculation Examples 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.
  • Page 5 18 - 2 Paired-Variable Statistical Calculation Examples • 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.
  • Page 6 18 - 2 Paired-Variable Statistical Calculation Examples u u u u u To 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.
  • Page 7 18 - 2 Paired-Variable Statistical Calculation Examples u u u u u Graph Color (graph color specification) • {Blue}/{Orng}/{Grn} ... {blue}/{orange}/{green} u u u u u Outliers (outliers specification) • {On}/{Off} ... {display}/{do not display} Med-Box outliers k k k k k Drawing an Line Graph P.254 Paired data items can be used to plot a scatter diagram.
  • Page 8 18 - 2 Paired-Variable Statistical Calculation Examples k k k k k Displaying Statistical Calculation Results Whenever you perform a regression calculation, the regression formula parameter (such as in the linear regression ) 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.

  • Page 9: Minx

    18 - 3 Calculating and Graphing Single-Variable Statistical Data 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).
  • Page 10 18 - 3 Calculating and Graphing Single-Variable Statistical Data 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.
  • Page 11 18 - 3 Calculating and Graphing Single-Variable Statistical Data k k k k k Broken 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. Calling up the graph menu from the statistical data list, pressing 6 (SET), (Brkn) changing the settings to drawing of a broken line graph, and then drawing a graph...
  • Page 12 18 - 3 Calculating and Graphing Single-Variable Statistical Data minX ....minimum Q1 ....first quartile Med ....median Q3 ....third quartile σ .... – data mean – population standard deviation σ .... data mean + population standard deviation maxX ....

  • Page 13: Med

    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. k k k k k Linear 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...
  • Page 14 18 - 4 Calculating and Graphing Paired-Variable Statistical Data 6(DRAW) a ..Med-Med graph slope b ..Med-Med graph -intercept k k k k k Quadratic/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.
  • Page 15 18 - 4 Calculating and Graphing Paired-Variable Statistical Data k k k k k Logarithmic Regression Graph P.254 Logarithmic regression expresses as a logarithmic function of . The standard × In logarithmic regression formula is , so if we say that X = In , the formula corresponds to linear regression formula 6(g)1(Log)
  • Page 16 18 - 4 Calculating and Graphing Paired-Variable Statistical Data k k k k k Power Regression Graph P.254 Exponential regression expresses as a proportion of the power of . The × standard power regression formula is , so if we take the logarithm of both ×...
  • Page 17 18 - 4 Calculating and Graphing Paired-Variable Statistical Data 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)
  • Page 18 18 - 4 Calculating and Graphing Paired-Variable Statistical Data 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.
  • Page 19 18 - 4 Calculating and Graphing Paired-Variable Statistical Data Draw a logistic regression graph based on the parameters obtained from the analytical results. 6(DRAW) k k k k k Residual Calculation Actual plot points ( -coordinates) and regression model distance can be calcu- lated during regression calculations.

  • Page 20: Maxx

    18 - 4 Calculating and Graphing Paired-Variable Statistical Data • Use c to scroll the list so you can view the items that run off the bottom of the screen...... mean of List data Σ ....sum of List data Σ...
  • Page 21 18 - 4 Calculating and Graphing Paired-Variable Statistical Data k k k k k Multiple Graphs You can draw more than one graph on the same display by using the procedure P.252 under “Changing Graph Parameters” to set the graph draw (On)/non-draw (Off) status of two or all three of the graphs to draw “On”, and then pressing 6 (DRAW).

  • Page 22: Performing Statistical Calculations

    18 - 5 Performing Statistical Calculations 18-5 Performing Statistical Calculations All of the statistical calculations up to this point were performed after displaying a graph. The following procedures can be used to perform statistical calculations alone. u u u u u To specify statistical calculation data lists You have to input the statistical data for the calculation you want to perform and specify where it is located before you start a calculation.
  • Page 23 18 - 5 Performing Statistical Calculations Now you can use the cursor keys to view the characteristics of the variables. For details on the meanings of these statistical values, see “Displaying Single- P.259 Variable Statistical Results”. k k k k k Paired-Variable Statistical Calculations In the previous examples from “Linear Regression Graph”...
  • Page 24 18 - 5 Performing Statistical Calculations k k k k k Estimated Value Calculation ( , ) After drawing a regression graph with the STAT Mode, you can use the RUN Mode to calculate estimated values for the regression graph's parameters.
  • Page 25 18 - 5 Performing Statistical Calculations k k k k k Normal Probability Distribution Calculation and Graphing You can calculate and graph normal probability distributions for single-variable statistics. u u u u u Normal probability distribution calculations Use the RUN Mode to perform normal probability distribution calculations. Press K in the RUN Mode to display the option number and then press 6 (g) 3 (PROB) 6 (g) to display a function menu, which contains the following items.
  • Page 26 18 - 5 Performing Statistical Calculations 2. Use the STAT Mode to perform the single-variable statistical calculations. 2(CALC)6(SET) 1(List1)c3(List2)J1(1VAR) 3. Press m to display the Main Menu, and then enter the RUN Mode. Next, press K to display the option menu and then 6 (g) 3 (PROB) 6 (g). •...
  • Page 27 18 - 5 Performing Statistical Calculations k k k k k Normal Probability Graphing You can graph a normal probability distribution with Graph Y = in the Sketch Mode. Example To graph normal probability P(0.5) Perform the following operation in the RUN Mode. !4(Sketch)1(Cls)w 5(GRPH)1(Y=)K6(g)3(PROB) 6(g)1(P()a.f)w...
  • Page 28 18-6 Tests Test provides a variety of different standardization-based tests. They make it possible to test whether or not a sample accurately represents the population when the standard deviation of a population (such as the entire population of a country) is known from previous tests. testing is used for market research and public opinion research, that need to be performed repeatedly.
  • Page 29 18 - 6 Tests 2-Sample Test tests the hypothesis that there will be no change in the result for a population when a result of a sample is composed of multiple factors and one or more of the factors is removed. It could be used, for example, to test the carcino- genic effects of multiple suspected factors such as tobacco use, alcohol, vitamin deficiency, high coffee intake, inactivity, poor living habits, etc.
  • Page 30 18 - 6 Tests The following shows the meaning of each item in the case of list data specifica- tion. Data ....data type µ ..... population mean value test conditions (“G µ ” specifies two-tail test, “< µ ” specifies lower one-tail test, “> µ ”...
  • Page 31 18 - 6 Tests Perform the following key operations from the statistical result screen. J(To data input screen) cccccc(To Execute line) 6(DRAW) u u u u u 2-Sample Z Test This test is used when the sample standard deviations for two populations are known to test the hypothesis.
  • Page 32 18 - 6 Tests The following shows the meaning of parameter data specification items that are different from list data specification..... sample 1 mean ....sample 1 size (positive integer) ....sample 2 mean ....sample 2 size (positive integer) Example To perform a 2-Sample Test when two lists of data are input...
  • Page 33 18 - 6 Tests u u u u u 1-Prop Z Test This test is used to test for an unknown proportion of successes. The 1-Prop Test is applied to the normal distribution. : expected sample proportion – p : sample size (1–...
  • Page 34 18 - 6 Tests The following key operations can be used to draw a graph. cccc 6(DRAW) u u u u u 2-Prop Z Test This test is used to compare the proportion of successes. The 2-Prop Test is applied to the normal distribution. : sample 1 data value –...
  • Page 35 18 - 6 Tests 3(>)c ccfw daaw cdaw daaw 1(CALC) > ....direction of test ...... value ..... p-value ˆ p ....estimated proportion of population 1 ˆ p ....estimated proportion of population 2 ˆ p ..... estimated sample proportion ....
  • Page 36 18 - 6 Tests The following shows the meaning of each item in the case of list data specification. Data ....data type µ ..... population mean value test conditions (“G µ ” specifies two- tail test, “< µ ” specifies lower one-tail test, “> µ ”...
  • Page 37 18 - 6 Tests u u u u u 2-Sample t Test 2-Sample Test compares the population means when the population standard deviations are unknown. The 2-Sample Test is applied to -distribution. The following applies when pooling is in effect. –...
  • Page 38 18 - 6 Tests The following shows the meaning of each item in the case of list data specifica- tion. Data ....data type µ ....sample mean value test conditions (“G µ ” specifies two-tail test, “< µ ” specifies one-tail test where sample 1 is smaller than sample 2, “>...
  • Page 39 18 - 6 Tests µ G µ ....direction of test ...... value ..... p-value ....degrees of freedom ....sample 1 mean ....sample 2 mean σ ....sample 1 standard deviation σ ....sample 2 standard deviation ....sample 1 size ....
  • Page 40 18 - 6 Tests The following shows the meaning of each item in the case of list data specifica- tion. β & ρ ....p-value test conditions (“G 0” specifies two-tail test, “< 0” specifies lower one-tail test, “> 0” specifies upper one-tail test.) XList ....
  • Page 41 18 - 6 Tests k k k k k Other Tests u u u u u χ Test χ Test sets up a number of independent groups and tests hypotheses related to the proportion of the sample included in each group. The χ Test is applied to dichotomous variables (variable with two possible values, such as yes/no).
  • Page 42 18 - 6 Tests χ ....χ value ..... p-value ....degrees of freedom Expected ..expected counts (Result is always stored in MatAns.) The following key operations can be used to display the graph. 6(DRAW) u u u u u 2-Sample F Test 2-Sample Test tests the hypothesis that when a sample result is composed of multiple factors, the population result will be unchanged when one or some of the...
  • Page 43 18 - 6 Tests The following shows the meaning of parameter data specification items that are different from list data specification. σ σ ....sample 1 standard deviation ( > 0) ....sample 1 size (positive integer) σ σ ....sample 2 standard deviation ( >...
  • Page 44 18 - 6 Tests u u u u u Analysis of Variance (ANOVA) ANOVA tests the hypothesis that when there are multiple samples, the means of the populations of the samples are all equal. : number of populations : mean of each list σ...
  • Page 45 18 - 6 Tests 2(3)c 1(List1)c 2(List2)c 3(List3)c 1(CALC) ..... value ..... p-value σ ....pooled sample standard deviation ....factor degrees of freedom ....factor sum of squares ....factor mean squares ....error degrees of freedom ....error sum of squares ....

  • Page 46: Confidence Interval

    18 - 8 Confidence Interval 18-7 Confidence Interval A confidence interval is a range (interval) that includes a statistical value, usually the population mean. A confidence interval that is too broad makes it difficult to get an idea of where the population value (true value) is located.
  • Page 47 18 - 7 Confidence Interval k k k k k Z Confidence Interval You can use the following menu to select from the different types of confidence interval. • {1-S}/{2-S}/{1-P}/{2-P} ... {1-Sample}/{2-Sample}/{1-Prop}/{2-Prop} Interval u u u u u 1-Sample Z Interval 1-Sample Interval calculates the confidence interval for an unknown population mean when the population standard deviation is known.
  • Page 48 18 - 7 Confidence Interval Example To calculate the 1-Sample Interval for one list of data For this example, we will obtain the Interval for the data {11.2, 10.9, 12.5, 11.3, 11.7}, when C-Level = 0.95 (95% confi- dence level) and = 3.
  • Page 49 18 - 7 Confidence Interval ....population standard deviation of sample 1 ( > 0) ....population standard deviation of sample 2 ( > 0) List1 ....list whose contents you want to use as sample 1 data List2 ....list whose contents you want to use as sample 2 data Freq1 ....
  • Page 50 18 - 7 Confidence Interval u u u u u 1-Prop Z Interval 1-Prop Interval uses the number of data to calculate the confidence interval for an unknown proportion of successes. The following is the confidence interval. The value 100 (1 – ) % is the confidence level.
  • Page 51 18 - 7 Confidence Interval u u u u u 2-Prop Z Interval 2-Prop Z Interval uses the number of data items to calculate the confidence interval for the defference between the proportion of successes in two populations. The following is the confidence interval. The value 100 (1 – ) % is the confidence level.
  • Page 52 18 - 7 Confidence Interval ˆ p ....estimated sample propotion for sample 1 ˆ p ....estimated sample propotion for sample 2 ....sample 1 size ....sample 2 size k k k k k t Confidence Interval You can use the following menu to select from two types of confidence interval.
  • Page 53 18 - 7 Confidence Interval Example To calculate the 1-Sample Interval for one list of data For this example, we will obtain the 1-Sample Interval for data = {11.2, 10.9, 12.5, 11.3, 11.7} when C-Level = 0.95. 1(List)c a.jfw 1(List1)c 1(1)c 1(CALC) Left ....
  • Page 54 18 - 7 Confidence Interval Perform the following key operations from the statistical data list. 4(INTR) 2(2-S) The following shows the meaning of each item in the case of list data specification. Data ....data type C-Level ... confidence level (0 < C-Level < 1) List1 ....
  • Page 55 18 - 7 Confidence Interval Example To calculate the 2-Sample Interval when two lists of data are input For this example, we will obtain the 2-Sample Interval for data 1 = {55, 54, 51, 55, 53, 53, 54, 53} and data 2 = {55.5, 52.3, 51.8, 57.2, 56.5} without pooling when C-Level = 0.95.
  • Page 56 18-8 Distribution There is a variety of different types of distribution, but the most well-known is “normal distribution,” which is essential for performing statistical calculations. Normal distribution is a symmetrical distribution centered on the greatest occur- rences of mean data (highest frequency), with the frequency decreasing as you move away from the center.
  • Page 57 18 - 8 Distribution k k k k k Normal Distribution You can use the following menu to select from the different types of calculation. • {Npd}/{Ncd}/{InvN} ... {normal probability density}/{normal distribution probability}/{inverse cumulative normal distribution} calculation u u u u u Normal probability density Normal probability density calculates the probability density of normal distribution from a specified value.
  • Page 58 18 - 8 Distribution Perform the following key operations to display a graph. 6(DRAW) u u u u u Normal distribution probability Normal distribution probability calculates the probability of normal distribution data falling between two specific values. : lower boundary (x –...
  • Page 59 18 - 8 Distribution • This calculator performs the above calculation using the following: = 1E99, – = –1E99 u u u u u Inverse cumulative normal distribution Inverse cumulative normal distribution calculates a value that represents the location within a normal distribution for a specific cumulative probability. f (x)dx = p Upper boundary of integration interval...
  • Page 60 18 - 8 Distribution k k k k k Student-t Distribution You can use the following menu to select from the different types of Student- distribution. • {tpd}/{tcd} ... {Student- probability density}/{Student- distribution probability} calculation u u u u u Student-t probability density Student- probability density calculates probability density from a specified...
  • Page 61 18 - 8 Distribution Perform the following key operation to display a graph. 6(DRAW) u u u u u Student-t distribution probability Student- distribution probability calculates the probability of distribution data falling between two specific values. : lower boundary df + 1 df+1 –...
  • Page 62 18 - 8 Distribution k k k k k Chi-square Distribution You can use the following menu to select from the different types of chi-square distribution. • {Cpd}/{Ccd} ... { probability density}/{ distribution probability} calculation u u u u u probability density probability density calculates the probability density function for the distribution at a specified...
  • Page 63 18 - 8 Distribution Perform the following key operations to display a graph. 6(DRAW) u u u u u distribution probability distribution probability calculates the probability of distribution data falling between two specific values. : lower boundary –1 – : upper boundary Perform the following key operations from the statistical data list.
  • Page 64 18 - 8 Distribution k k k k k F Distribution You can use the following menu to select from the different types of distribution. • {Fpd}/{Fcd} ... { probability density}/{ distribution probability} calculation u u u u u F probability density probability density calculates the probability density function for the F distribution at a specified value.
  • Page 65 18 - 8 Distribution u u u u u F distribution probability distribution probability calculates the probability of distribution data falling between two specific values. : lower boundary n + d n + d : upper boundary – –1 Perform the following key operations from the statistical data list. 5(DIST) 4(F) 2(Fcd)
  • Page 66 18 - 8 Distribution u u u u u Binomial probability Binomial probability calculates a probability at specified value for the discrete binomial distribution with the specified number of trials and probability of success on each trial. n) p = 0, 1, ·······, : success probability n –...
  • Page 67 18 - 8 Distribution u u u u u Binomial cumulative density Binomial cumulative density calculates a cumulative probability at specified value for the discrete binomial distribution with the specified number of trials and probability of success on each trial. Perform the following key operations from the statistical data list.
  • Page 68 18 - 8 Distribution k k k k k Poisson Distribution You can use the following menu to select from the different types of Poisson distribution. • {Ppd}/{Pcd} ... {Poisson probability}/{Poisson cumulative density} calculation u u u u u Poisson probability Poisson probability calculates a probability at specified value for the discrete Poisson distribution with the specified mean.
  • Page 69 18 - 8 Distribution u u u u u Poisson cumulative density Poisson cumulative density calculates a cumulative probability at specified value for the discrete Poisson distribution with the specified mean. Perform the following key operations from the statistical data list. 5(DIST) 6(g) 1(POISN)
  • Page 70 18 - 8 Distribution u u u u u Geometric probability Geometric probability calculates a probability at specified value, the number of the trial on which the first success occurs, for the discrete geometric distribution with the specified probability of success. x –...
  • Page 71 18 - 8 Distribution u u u u u Geometric cumulative density Geometric cumulative density calculates a cumulative probability at specified value, the number of the trial on which the first success occurs, for the discrete geometric distribution with the specified probability of success. Perform the following key operations from the statistical data list.

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