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 confidence 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 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.269 • {CALC} ... {statistical calculation menu} P.276 • {TEST} ... {test menu} P.293 • {INTR} ... {confidence interval menu} P.303 • {DIST} ... {distribution menu} P.
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 k Inputting 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.
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. 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.
Paired-Variable Statistical Calculation Examples 18 - 2 • 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.
18 - 2 Paired-Variable Statistical Calculation Examples 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. u StatGraph (statistical graph specification) • {GPH1}/{GPH2}/{GPH3} ... graph {1}/{2}/{3} u Graph Type (graph type specification) • {Scat}/{ xy}/{NPP} ...
Paired-Variable Statistical Calculation Examples 18 - 2 u Graph Color (graph color specification) • {Blue}/{Orng}/{Grn} ... {blue}/{orange}/{green} u Outliers (outliers specification) • {On}/{Off} ... {display}/{non-display} k Drawing an xy Line Graph P.254 (Graph Type) (xy) Paired data items can be used to plot a scatter diagram. A scatter diagram where the points are linked is an xy line graph. Press J or !Q to return to the statistical data list. k Drawing a Normal Probability Plot P.
18 - 2 Paired-Variable Statistical Calculation Examples k Displaying Statistical Calculation Results Whenever you perform a regression calculation, the regression formula parameter (such as a and b in 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.
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.
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. k Mean-box Graph P.254 (Graph Type) (Box) This type of graph shows the distribution around the mean when there is a large number of data items.
Calculating and Graphing Single-Variable Statistical Data 18 - 3 k Line Graph P.254 (Graph Type) (Brkn) A line graph is formed by plotting the data in one list against the frequency 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), changing the settings to drawing of a line graph, and then drawing a graph creates a line graph.
18 - 3 Calculating and Graphing Single-Variable Statistical Data 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.
18-4 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 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 y-intercept ( y-coordinate when x = 0) of the line.
18 - 4 Calculating and Graphing Paired-Variable Statistical Data 6(DRAW) a ...... Med-Med graph slope b ...... Med-Med graph intercept 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. The formula that represents this is quadratic/cubic/quartic regression. Ex.
Calculating and Graphing Paired-Variable Statistical Data 18 - 4 k Logarithmic Regression Graph P.254 Logarithmic regression expresses y as 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) 1 2 3 4 5 6 6(DRAW) a ...... regression constant term (intercept) b ...... regression coefficient (slope) r ...... correlation coefficient r2 .....
18 - 4 Calculating and Graphing Paired-Variable Statistical Data k Power Regression Graph P.254 Exponential regression expresses y as 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 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) 1 2 3 4 5 6 6(DRAW) a ...... regression coefficient b ...... regression power r ......
Calculating and Graphing Paired-Variable Statistical Data 18 - 4 6(DRAW) 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.
- 4 Calculating and Graphing Paired-Variable Statistical Data Execute the calculation and produce sine regression analysis results. 1(CALC) 6 Display a sine regression graph based on the analysis results. 6(DRAW) k Residual Calculation Actual plot points (y-coordinates) and regression model distance can be calculated 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”.
Calculating and Graphing Paired-Variable Statistical Data 18 - 4 • Use c to scroll the list so you can view the items that run off the bottom of the screen. _ x ..................... mean of xList data Σ x ................... sum of xList data Σ x2 .................. 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 .....................
18 - 4 Calculating and Graphing Paired-Variable Statistical Data k Multiple Graphs P.252 You can draw more than one graph on the same display by using the procedure 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). After drawing the graphs, you can select which graph formula to use when performing single-variable statistic or regression calculations. 6(DRAW) P.
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 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. Display the statistical data and then press 2(CALC)6 (SET). The following is the meaning for each item.
18 - 5 Performing Statistical Calculations Now you can use the cursor keys to view the characteristics of the variables. P.259 For details on the meanings of these statistical values, see “Displaying SingleVariable Statistical Results”. k Paired-Variable Statistical Calculations In the previous examples from “Linear Regression Graph” to “Sine Regression Graph,” statistical calculation results were displayed after the graph was drawn.
Performing Statistical Calculations 18 - 5 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 x and y parameters. • Note that you cannot obtain estimated value for a Med-Med, quadratic regression, cubic regression, quartic regression, or sine regression graph.
18 - 5 Performing Statistical Calculations k Probability Distribution Calculation and Graphing You can calculate and graph probability distributions for single-variable statistics. uProbability distribution calculations Use the RUN Mode to perform 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. • {P(}/{Q(}/{R(} ...
Performing Statistical Calculations 18 - 5 2. Use the STAT Mode to perform the single-variable statistical calculations. 2(CALC)6(SET) 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). • You obtain the normalized variate immediately after performing singlevariable statistical calculations only. 4(t () bga.f)w (Normalized variate t for 160.5cm) Result: –1.633855948 ( –1.634) 4(t() bhf.
18 - 5 Performing Statistical Calculations k Probability Graphing You can graph a probability distribution with Graph Y = in the Sketch Mode. Example To graph 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 The following shows the View Window settings for the graph. Ymin ~ Ymax –0.1 0.45 Xmin ~ Xmax –3.2 3.
18-6 Tests The Z 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. Z testing is used for market research and public opinion research that need to be performed repeatedly. 1-Sample Z Test tests the population mean when the standard deviation is known.
18 - 6 Tests 2-Sample F 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 carcinogenic effects of multiple suspected factors such as tobacco use, alcohol, vitamin deficiency, high coffee intake, inactivity, poor living habits, etc.
Tests 18 - 6 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 0” specifies two-tail test, “< µ 0” specifies lower one-tail test, “> µ0 ” specifies upper one-tail test.) µ 0 .................... assumed population mean σ ..................... population standard deviation (σ > 0) List .................. list whose contents you want to use as data (List 1 to 6) Freq ......
18 - 6 Tests Perform the following key operation from the statistical result screen. J(To data input screen) cccccc(To Execute line) 6(DRAW) u 2-Sample Z Test This test is used when the sample standard deviations for two populations are known to test the hypothesis that the population means of the two populations are equal. The 2-Sample Z Test is applied to standard normal distribution.
Tests 18 - 6 The following shows the meaning of parameter data specification items that are different from list data specification. o1 .................... n1 .................... o2 .................... n2 .................... Example sample 1 mean sample 1 size (positive integer) sample 2 mean sample 2 size (positive integer) To perform a 2-Sample Z Test when two lists of data are input For this example, we will perform a µ 1 < µ2 test for the data List1 = {11.2, 10.9, 12.5, 11.3, 11.
18 - 6 Tests u 1-Prop Z Test This test is used to test whether data that satisfies certain criteria reaches a specific proportion. It tests the hypothesis when sample size and the number of data satisfying the criteria are specified. The 1-Prop Z Test is applied to standard normal distribution. x – p0 n Z= p0 (1– p0) n p0 : expected sample proportion n : sample size Perform the following key operation from the statistical data list. 3(TEST) 1(Z) 3(1-P) Prop ................
Tests 18 - 6 The following key operation can be used to draw a graph. J cccc 6(DRAW) u 2-Prop Z Test This test is used to compare the proportions of two samples that satisfy certain criteria. It tests the hypothesis that the size and the number of data of two samples that satisfy the criteria are as specified. The 2-Prop Z Test is applied to standard normal distribution.
18 - 6 Tests 3(>)c ccfw daaw cdaw daaw 1(CALC) p1>p2 ............... z ...................... p ..................... p̂1 .................... p̂2 .................... p̂ ..................... n1 .................... n2 .................... direction of test Z score p-value estimated proportion of population 1 estimated proportion of population 2 estimated sample proportion sample 1 size sample 2 size The following key operation can be used to draw a graph.
Tests 18 - 6 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 0” specifies twotail test, “< µ0 ” specifies lower one-tail test, “> µ0” specifies upper one-tail test.) µ0 .................... assumed population mean List .................. list whose contents you want to use as data Freq ................ frequency Execute ..........
18 - 6 Tests u 2-Sample t Test 2-Sample t Test uses the sample means, variance, and sample sizes when the sample standard deviations for two populations are unknown to test the hypothesis that the two samples were taken from the same population. The 2-Sample t Test is applied to standard normal distribution.
Tests 18 - 6 The following shows the meaning of each item in the case of list data specification. Data ................ data type µ 1 .................... sample mean value test conditions (“G µ2” specifies two-tail test, “< µ2 ” specifies one-tail test where sample 1 is smaller than sample 2, “> µ2” specifies one-tail test where sample 1 is greater than sample 2.) List1 ................ list whose contents you want to use as sample 1 data List2 ................
18 - 6 Tests µ1Gµ2 .............. direction of test t ...................... p ..................... df .................... o1 .................... o2 .................... x1σn -1 ............... x2σn -1 ............... n1 .................... n2 .................... t-value p-value degrees of freedom sample 1 mean sample 2 mean sample 1 standard deviation sample 2 standard deviation sample 1 size sample 2 size Perform the following key operation to display a graph.
Tests 18 - 6 The following shows the meaning of each item in the case of list data specification. 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 ............... list for x-axis data YList ............... list for y-axis data Freq ................ frequency Execute ..........
18 - 6 Tests k Other Tests u χ2 Test χ2 Test sets up a number of independent groups and tests hypotheses related to the proportion of the sample included in each group. The χ2 Test is applied to dichotomous variables (variable with two possible values, such as yes/no). expected counts n : all data values k Σ x ×Σ x = Σn ij Fij i =1 k χ2 = Σ Σ i =1 j =1 ij j =1 (xij – Fij)2 Fij For the above, data must already be input in a matrix using the MAT Mode.
Tests 18 - 6 χ2 .................... χ2 value p ..................... p-value df .................... degrees of freedom Expected ........ expected counts (Result is always stored in MatAns.) The following key operation can be used to display the graph. J c 6(DRAW) u 2-Sample F Test 2-Sample F 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 factors are removed. The F Test is applied to F distribution.
18 - 6 Tests The following shows the meaning of parameter data specification items that are different from list data specification. x1σn -1 ............... n1 .................... x2σn -1 ............... n2 ....................
Tests 18 - 6 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. F = M1 Me SS Fdf MS = SSe Edf MSe = k : number of populations oi : mean of each list xiσn-1 : standard deviation of each ni o list : size of each list : mean of all lists k SS = Σni (oi – o)2 i=1 k SSe = Σ(ni – 1)xiσn–12 i =1 Fdf = k – 1 k Edf = Σ(ni – 1) i =1 Perform the following key operation from the statistical data list.
18 - 6 Tests 2(3)c 1(List1)c 2(List2)c 3(List3)c 1(CALC) F ..................... F value p ..................... p-value xpσn -1 ............... pooled sample standard deviation Fdf .................. numerator degrees of freedom SS ................... factor sum of squares MS .................. factor mean squares Edf .................. denominator degrees of freedom SSe ................. error sum of squares MSe ................
18 - 8 18-7 Confidence Interval Confidence Interval A confidence interval is a range (interval) that includes the population mean value. A confidence interval that is too broad makes it difficult to get an idea of where the population value (true value) is located. A narrow confidence interval, on the other hand, limits the population value and makes it possible to obtain reliable results. The most commonly used confidence levels are 95% and 99%.
18 - 7 Confidence Interval k Z Confidence Interval You can use the following menu to select from the different types of Z confidence interval. • {1-S}/{2-S}/{1-P}/{2-P} ... {1-Sample}/{2-Sample}/{1-Prop}/{2-Prop} Z Interval u 1-Sample Z Interval 1-Sample Z Interval calculates the confidence interval when standard deviation is known. Z Interval is applied to normal distribution. The following is the confidence interval.
Confidence Interval Example 18 - 7 To calculate the 1-Sample Z Interval for one list of data For this example, we will obtain the Z Interval for the data {11.2, 10.9, 12.5, 11.3, 11.7}, when C-Level = 0.95 (95% confidence level) and σ = 3. 1(List)c a.jfw dw 1(List1)c1(1)c1(CALC) Left ................. interval lower limit (left edge) Right ............... interval upper limit (right edge) o ..................... sample mean xσn-1 ................ sample standard deviation n .....................
18 - 7 Confidence Interval σ1 .................... population standard deviation of sample 1 (σ1 > 0) σ2 .................... population standard deviation of sample 2 (σ2 > 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 .............. frequency of sample 1 Freq2 .............. frequency of sample 2 Execute ..........
Confidence Interval 18 - 7 u 1-Prop Z Interval 1-Prop Z Interval uses the number of data to calculate the confidence interval when the proportion is not known. The 1-Prop Z Interval is applied to standard normal distribution. The following is the confidence interval. x Left = n – Z α 2 x Right = n + Z α 2 1 x x n n 1– n n : sample size x : data 1 x x n n 1– n Perform the following key operation from the statistical data list. 4(INTR) 1(Z) 3(1-P) Data is specified using parameter specification.
18 - 7 Confidence Interval u 2-Prop Z Interval 2-Prop Z Interval calculates the confidence interval when the proportions of two samples are known. The 2-Prop Z Interval is applied to standard normal distribution. The following is the confidence interval. x x Left = n1 – n2 – Z α 1 2 2 x1 x2 x2 x1 n1 1– n1 n2 1– n2 + n1 n2 x x Right = n1 – n2 + Z α 1 2 2 n1, n2 : sample size x1, x2 : data x1 x2 x2 x1 n1 1– n1 n2 1– n2 + n1 n2 Perform the following key operation from the statistical data list.
Confidence Interval p̂1 .................... p̂2 .................... n1 .................... n2 .................... 18 - 7 expected p-value 1 expected p-value 2 sample 1 size sample 2 size k t Confidence Interval You can use the following menu to select from two types of t confidence interval. • {1-S}/{2-S} ... {1-Sample}/{2-Sample} t Interval u 1-Sample t Interval 1-Sample t Interval calculates the confidence interval when the mean value of the sample is known.
18 - 7 Confidence Interval Example To calculate the 1-Sample t Interval for one list of data For this example, we will obtain the 1-Sample t 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 ................. interval lower limit (left edge) Right ............... interval upper limit (right edge) o ..................... sample mean xσn-1 ................ sample standard deviation n .....................
Confidence Interval 18 - 7 Perform the following key operation from the statistical data list. 4(INTR) 2( t) 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 ................ list whose contents you want to use as sample 1 data List2 ................ list whose contents you want to use as sample 2 data Freq1 .............. frequency of sample 1 Freq2 ..............
18 - 7 Confidence Interval Example To calculate the 2-Sample t Interval when two lists of data are input For this example, we will obtain the 2-Sample t 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. 1(List)c a.jfw 1(List1)c2(List2)c1(1)c 1(1)c2(Off)c1(CALC) Left ................. interval lower limit (left edge) Right ............... interval upper limit (right edge) df .................... o1 ....................
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 occurrences of mean data (highest frequency), with the frequency decreasing as you move away from the center. Poisson distribution, geometric distribution, and various other distribution shapes are also used, depending on the data type.
18 - 8 Distribution 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 Normal probability density Normal probability density calculates the probability that data taken from a normal distribution is less than a specific value. Normal probability density is applied to standard normal distribution.
Distribution 18 - 8 Perform the following key operation to display a graph. J ccc 6(DRAW) u Normal distribution probability Normal distribution probability calculates the probability of normal distribution data falling between two specific values. p= 1 2πσ ∫ b e a – (x – µ µ) 2σ 2 2 dx a : lower boundary b : upper boundary Perform the following key operation from the statistical data list. 5(DIST) 1(NORM) 2(Ncd) Data is specified using parameter specification.
18 - 8 Distribution • This calculator performs the above calculation using the following: ∞ = 1E99, –∞ = –1E99 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 α=? Specify the probability and use this formula to obtain the integration interval. Perform the following key operation from the statistical data list.
Distribution 18 - 8 k Student-t Distribution You can use the following menu to select from the different types of Student-t distribution. • {tpd}/{tcd} ... {Student-t probability density}/{Student-t distribution probability} calculation u Student-t probability density Student-t probability density calculates whether data taken from a t distribution is less than a specific value. df + 1 1 + x2 df 2 f (x) = π df df Γ 2 Γ – df +1 2 Perform the following key operation from the statistical data list.
18 - 8 Distribution Perform the following key operation to display a graph. J cc 6(DRAW) u Student-t distribution probability Student-t distribution probability calculates the probability of t distribution data falling between two specific values. df + 1 2 p= df Γ 2 π df Γ ∫ b a 1 + x2 df – df +1 2 dx a : lower boundary b : upper boundary Perform the following key operation from the statistical data list. 5(DIST) 2( t) 2(tcd) Data is specified using parameter specification.
Distribution 18 - 8 k Chi-square Distribution You can use the following menu to select from the different types of chi-square distribution. • {Cpd}/{Ccd} ... {χ2 probability density}/{χ2 distribution probability} calculation u χ2 probability density χ2 probability density calculates whether data taken from a χ2 distribution is less than a specific value. f(x) = 1 df Γ 2 1 2 df 2 df –1 – x2 e x 2 (x > 0) Perform the following key operation from the statistical data list.
18 - 8 Distribution Perform the following key operation to display a graph. J cc 6(DRAW) u χ2 distribution probability χ2 distribution probability calculates the probability of χ2 distribution data falling between two specific values. p= 1 df Γ 2 1 2 df 2 ∫ b x df x –1 – 2 2 e dx a : lower boundary b : upper boundary a Perform the following key operation from the statistical data list. 5(DIST) 3(CHI) 2(Ccd) Data is specified using parameter specification.
Distribution 18 - 8 k F Distribution You can use the following menu to select from the different types of F distribution. • {Fpd}/{Fcd} ... {F probability density}/{F distribution probability} calculation u F probability density F probability density calculates whether data taken from a F distribution is less than a specific value. n+d 2 f (x) = n d Γ Γ 2 2 Γ n d n 2 x n –1 2 1 + nx d – n+d 2 ( x > 0) Perform the following key operation from the statistical data list.
18 - 8 Distribution u F distribution probability F distribution probability calculates the probability of F distribution data falling between two specific values. n+d 2 p= n d Γ Γ 2 2 Γ n d n 2 ∫ b x n –1 2 a 1 + nx d – a : lower boundary b : upper boundary n+d 2 dx Perform the following key operation from the statistical data list. 5(DIST) 4(F) 2(Fcd) Data is specified using parameter specification. The following shows the meaning of each item. Lower ............. lower boundary Upper ....
Distribution 18 - 8 u Binomial probability Binomial probability calculates whether data taken from a binomial distribution is less than a specific value. f (x) = n C x px (1–p) n – x (x = 0, 1, ·······, n) p : success probability (0 < p < 1) n : number of trials Perform the following key operation from the statistical data list. 5(DIST) 5(BINM) 1(Bpd) The following shows the meaning of each item when data is specified using list specification. Data ................ data type List ..................
18 - 8 Distribution u Binomial cumulative density Binomial cumulative density calculates the probability of binomial distribution data falling between two specific values. Perform the following key operation from the statistical data list. 5(DIST) 5(BINM) 2(Bcd) The following shows the meaning of each item when data is specified using list specification. Data ................ data type List .................. list whose contents you want to use as sample data Numtrial ..........
Distribution 18 - 8 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 Poisson probability Poisson probability calculates whether data taken from a Poisson distribution is less than a specific value. f (x) = e– µµ x x! ( x = 0, 1, 2, ···) µ : population mean (µ > 0) Perform the following key operation from the statistical data list.
18 - 8 Distribution u Poisson cumulative density Poisson cumulative density calculates the probability of Poisson distribution data falling between two specific values. Perform the following key operation from the statistical data list. 5(DIST) 6(g) 1(POISN) 2(Pcd) The following shows the meaning of each item when data is specified using list specification. Data ................ data type List .................. list whose contents you want to use as sample data µ .....................
Distribution 18 - 8 u Geometric probability Geometric probability calculates whether data taken from a geometric distribution is less than a specific value. f (x) = p(1– p) x – 1 ( x = 1, 2, 3, ···) Perform the following key operation from the statistical data list. 5(DIST) 6(g) 2(GEO) 1(Gpd) The following shows the meaning of each item when data is specified using list specification. Data ................ data type List .................. list whose contents you want to use as sample data p .........
18 - 8 Distribution u Geometric cumulative density Geometric cumulative density calculates the probability of geometric distribution data falling between two specific values. Perform the following key operation from the statistical data list. 5(DIST) 6(g) 2(GEO) 2(Gcd) The following shows the meaning of each item when data is specified using list specification. Data ................ data type List .................. list whose contents you want to use as sample data p .....................
