Difference between revisions of "Applications-of-GeoGebra/C3/Statistics-using-GeoGebra/English"
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− | | | '''Visual Cue''' | + | || '''Visual Cue''' |
− | | | '''Narration''' | + | || '''Narration''' |
|- | |- | ||
− | | | '''Slide Number 1''' | + | || '''Slide Number 1''' |
'''Title Slide''' | '''Title Slide''' | ||
− | | | Welcome to this '''tutorial''' on '''Statistics using GeoGebra''' | + | || Welcome to this '''tutorial''' on '''Statistics using GeoGebra''' |
|- | |- | ||
− | | | '''Slide Number 2''' | + | || '''Slide Number 2''' |
'''Learning Objectives''' | '''Learning Objectives''' | ||
− | | | In this '''tutorial''', we will learn how to use '''GeoGebra''' to perform: | + | || In this '''tutorial''', we will learn how to use '''GeoGebra''' to perform: |
'''One Variable Analysis''' to calculate different statistical parameters | '''One Variable Analysis''' to calculate different statistical parameters | ||
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'''Multiple Variable Analysis''' to calculate different statistical parameters | '''Multiple Variable Analysis''' to calculate different statistical parameters | ||
|- | |- | ||
− | | | '''Slide Number 3''' | + | || '''Slide Number 3''' |
'''System Requirement''' | '''System Requirement''' | ||
− | | | Here I am using: | + | || Here I am using: |
− | '''Ubuntu Linux ''' | + | '''Ubuntu Linux '''Operating System version 16.04 |
'''GeoGebra''' 5.0.481.0-d | '''GeoGebra''' 5.0.481.0-d | ||
|- | |- | ||
− | | | '''Slide Number 4''' | + | || '''Slide Number 4''' |
'''Pre-requisites''' | '''Pre-requisites''' | ||
− | | | To follow this '''tutorial''', you should be familiar with: | + | || To follow this '''tutorial''', you should be familiar with: |
'''GeoGebra''' interface | '''GeoGebra''' interface | ||
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Statistics | Statistics | ||
|- | |- | ||
− | | | '''Slide Number 5''' | + | || '''Slide Number 5''' |
'''Statistics''' | '''Statistics''' | ||
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'''Additional material ''' | '''Additional material ''' | ||
− | | | | + | ||Statistics deals with |
− | + | ||
− | + | ||
− | + | ||
− | Statistics deals with | + | |
Data analysis and interpretation | Data analysis and interpretation | ||
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Please refer to '''additional material''' provided along with this '''tutorial'''. | Please refer to '''additional material''' provided along with this '''tutorial'''. | ||
|- | |- | ||
− | | | '''Slide Number 6''' | + | || '''Slide Number 6''' |
'''Fish Feed ''' | '''Fish Feed ''' | ||
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Girth (mm) | Girth (mm) | ||
− | | | '''Fish Feed ''' | + | || '''Fish Feed ''' |
Let us look at an example. | Let us look at an example. | ||
− | A fishery is testing four types of feed formulations on its fish: '''A, B, C''' and '''D'''. | + | A fishery is testing four types of feed formulations on its fish: '''A''', '''B''', '''C''' and '''D'''. |
Data to be collected after feeding the fish for 6 months are: | Data to be collected after feeding the fish for 6 months are: | ||
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Let us look at some of these data. | Let us look at some of these data. | ||
|- | |- | ||
− | | | '''Slide Number 7''' | + | || '''Slide Number 7''' |
'''Fish Feed Data''' | '''Fish Feed Data''' | ||
− | + | || We will use these data for our '''analyses'''. | |
− | + | ||
− | | | We will use these data for our '''analyses'''. | + | |
Please download the '''code file''', '''Fishery-data''', provided along with this '''tutorial'''. | Please download the '''code file''', '''Fishery-data''', provided along with this '''tutorial'''. | ||
|- | |- | ||
− | | | Show the '''GeoGebra''' window. | + | || Show the '''GeoGebra''' window. |
− | | | I have opened the '''GeoGebra''' interface. | + | || I have opened the '''GeoGebra''' interface. |
|- | |- | ||
− | | | Click on '''View''' tool | + | || Click on '''View''' tool >> select '''Spreadsheet'''. |
− | | | Click on '''View''' tool and select '''Spreadsheet'''. | + | || Click on '''View''' tool and select '''Spreadsheet'''. |
|- | |- | ||
− | | | Click on '''X''' at top right corner of '''Graphics''' | + | || Click on '''X''' at top right corner of '''Graphics''', '''Algebra''' views. |
− | | | Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views. | + | || Click on '''X''' at top right corner of '''Graphics''' and '''Algebra''' views. |
This will close these views. | This will close these views. | ||
|- | |- | ||
− | | | In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''. | + | || In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''. |
Show data in '''columns H''' and '''I'''. | Show data in '''columns H''' and '''I'''. | ||
Hold '''Ctrl''' key down and press '''C'''. | Hold '''Ctrl''' key down and press '''C'''. | ||
− | | | In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''. | + | || In the '''code file''', drag '''mouse''' to highlight length and weight data from '''columns H''' and '''I'''. |
These are data for fish that have been fed formulation '''C'''. | These are data for fish that have been fed formulation '''C'''. | ||
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Hold '''Control''' key down and press '''C'''. | Hold '''Control''' key down and press '''C'''. | ||
|- | |- | ||
− | | | Click on '''Spreadsheet''' view in '''GeoGebra'''. | + | || Click on '''Spreadsheet''' view in '''GeoGebra'''. |
− | | | Click in the top of the '''Spreadsheet''' in '''GeoGebra'''. | + | || Click in the top of the '''Spreadsheet''' in '''GeoGebra'''. |
|- | |- | ||
− | | | Press '''Ctrl''' key down | + | || Press '''Ctrl''' key down, press '''V'''. |
− | | | Hold '''Control''' key down and press '''V'''. | + | || Hold '''Control''' key down and press '''V'''. |
This will copy and paste the highlighted data from the '''code file''' into '''GeoGebra'''. | This will copy and paste the highlighted data from the '''code file''' into '''GeoGebra'''. | ||
|- | |- | ||
− | | | Place the '''cursor''' on the first column header in '''Spreadsheet''' view. | + | || Place the '''cursor''' on the first column header in '''Spreadsheet''' view. |
− | Drag and adjust '''column A'''' | + | Drag and adjust '''column A's''' width. |
− | | | Place the '''cursor''' on the first column header in '''Spreadsheet''' view. | + | || Place the '''cursor''' on the first column header in '''Spreadsheet''' view. |
− | Drag and adjust '''column A'''' | + | Drag and adjust '''column A's''' width. |
|- | |- | ||
− | | | Right-click on '''column A''' heading of '''Length (mm)'''. | + | || Right-click on '''column A''' heading of '''Length (mm)'''. |
Select '''Object Properties'''. | Select '''Object Properties'''. | ||
Point to '''dialog box'''. | Point to '''dialog box'''. | ||
− | | | Right-click on '''column A''' heading of '''Length millimetres'''. | + | || Right-click on '''column A''' heading of '''Length millimetres'''. |
Select '''Object Properties'''. | Select '''Object Properties'''. | ||
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A '''dialog box''' opens. | A '''dialog box''' opens. | ||
|- | |- | ||
− | | | Click on '''Text tab''' and change the name to '''Length (mm)-C'''. | + | || Click on '''Text tab''' and change the name to '''Length (mm)-C'''. |
Close the '''dialog box'''. | Close the '''dialog box'''. | ||
Similarly, add '''–C''' to '''Weight (lbs)'''. | Similarly, add '''–C''' to '''Weight (lbs)'''. | ||
− | | | Click on '''Text tab''' and change the name to '''Length millimetres | + | || Click on '''Text tab''' and change the name to '''Length millimetres hyphen '''C'''. |
Close the '''dialog box'''. | Close the '''dialog box'''. | ||
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Similarly, add '''hyphen C''' to '''Weight pounds'''. | Similarly, add '''hyphen C''' to '''Weight pounds'''. | ||
|- | |- | ||
− | | | Adjust '''column B''' width. | + | || Adjust '''column B''' width. |
− | | | Adjust '''column B''' width. | + | || Adjust '''column B''' width. |
|- | |- | ||
− | | | Use '''mouse''' to drag and highlight first '''column A'''’s length data and label in '''GeoGebra'''. | + | || Use '''mouse''' to drag and highlight first '''column A'''’s length data and label in '''GeoGebra'''. |
− | | | Click on '''column A''' heading of '''Length millimetres C'''. | + | || Click on '''column A''' heading of '''Length millimetres C'''. |
Drag to highlight length data in '''Spreadsheet''' view. | Drag to highlight length data in '''Spreadsheet''' view. | ||
|- | |- | ||
− | | | Below the '''menubar''', click on '''One Variable Analysis''' tool. | + | || Below the '''menubar''', click on '''One Variable Analysis''' tool. |
Point to '''Data Source''' popup window. | Point to '''Data Source''' popup window. | ||
Click on '''Analyze button'''. | Click on '''Analyze button'''. | ||
− | | | Below the '''menubar''', click on '''One Variable Analysis''' tool. | + | || Below the '''menubar''', click on '''One Variable Analysis''' tool. |
A '''Data Source''' popup window appears. | A '''Data Source''' popup window appears. | ||
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Click on '''Analyze button'''. | Click on '''Analyze button'''. | ||
|- | |- | ||
− | | | Point to '''Data Analysis''' window and '''histogram'''. | + | || Point to '''Data Analysis''' window and '''histogram'''. |
− | | | A '''Data Analysis''' window appears. | + | || A '''Data Analysis''' window appears. |
By default, a '''histogram''' is plotted. | By default, a '''histogram''' is plotted. | ||
|- | |- | ||
− | | | Drag the boundary to see the graph properly. | + | || Drag the boundary to see the graph properly. |
− | | | Drag the boundary to see the graph properly. | + | || Drag the boundary to see the graph properly. |
|- | |- | ||
− | | | Point to length on the '''x-axis''' and frequency on the '''y-axis'''. | + | || Point to length on the '''x-axis''' and frequency on the '''y-axis'''. |
− | | | The length is plotted on the '''x-axis'''. | + | || The length is plotted on the '''x-axis'''. |
The number of fish that are of a particular length, the '''frequency''', is plotted on the '''y-axis'''. | The number of fish that are of a particular length, the '''frequency''', is plotted on the '''y-axis'''. | ||
|- | |- | ||
− | | | Point to the '''display box''' above the graph containing the word '''Histogram'''. | + | || Point to the '''display box''' above the graph containing the word '''Histogram'''. |
− | | | Note the '''display box''' above the graph containing the word '''Histogram'''. | + | || Note the '''display box''' above the graph containing the word '''Histogram'''. |
|- | |- | ||
− | | | In the '''display box''', click on the '''dropdown menu button'''. | + | || In the '''display box''', click on the '''dropdown menu button'''. |
− | | | In the '''display box''', click on the '''dropdown menu button''' to display the list of plots. | + | || In the '''display box''', click on the '''dropdown menu button''' to display the list of plots. |
|- | |- | ||
− | | | Select '''Histogram'''. | + | || Select '''Histogram'''. |
Point to '''slider''' to the right of the display. | Point to '''slider''' to the right of the display. | ||
− | | | We will stay with the '''histogram''' option. | + | || We will stay with the '''histogram''' option. |
To the right of the '''dropdown menu''' is a '''slider'''. | To the right of the '''dropdown menu''' is a '''slider'''. | ||
|- | |- | ||
− | | | Drag the '''slider''' from left to right to go to 20. | + | || Drag the '''slider''' from left to right to go to 20. |
− | | | Drag the '''slider''' from left to right to go to 20. | + | || Drag the '''slider''' from left to right to go to 20. |
|- | |- | ||
− | | | Point to rectangles between minimum and maximum values of data. | + | || Point to rectangles between minimum and maximum values of data. |
− | | | The '''slider''' changes the number of rectangles between the minimum and maximum values of data. | + | || The '''slider''' changes the number of rectangles between the minimum and maximum values of data. |
|- | |- | ||
− | | | Click on '''Options''' button to the right of the '''slider'''. | + | || Click on '''Options''' button to the right of the '''slider'''. |
− | | | Click on '''Options''' button to the right of the '''slider'''. | + | || Click on '''Options''' button to the right of the '''slider'''. |
|- | |- | ||
− | | | Under '''Classes''', check '''Set Classes Manually'''. | + | || Under '''Classes''', check '''Set Classes Manually'''. |
− | | | Under '''Classes''', check '''Set Classes Manually check box'''. | + | || Under '''Classes''', check '''Set Classes Manually check box'''. |
This displays '''Start''' and '''Width text-boxes''' to the left of the '''Options button'''. | This displays '''Start''' and '''Width text-boxes''' to the left of the '''Options button'''. | ||
|- | |- | ||
− | | | Type 800 in the '''Start text-box''' and press '''Enter'''. | + | || Type 800 in the '''Start text-box''' and press '''Enter'''. |
Show the value of 5 in the '''Width text-box'''. | Show the value of 5 in the '''Width text-box'''. | ||
− | | | As all the fish are over 800 mm long, type 800 in the '''Start text-box''' and press '''Enter'''. | + | || As all the fish are over 800 mm long, type 800 in the '''Start text-box''' and press '''Enter'''. |
We will stay with the default value of 5 for rectangle '''width'''. | We will stay with the default value of 5 for rectangle '''width'''. | ||
|- | |- | ||
− | | | Uncheck '''Set Classes Manually'''. | + | || Uncheck '''Set Classes Manually'''. |
− | | | Uncheck '''Set Classes Manually''' check box. | + | || Uncheck '''Set Classes Manually''' check box. |
|- | |- | ||
− | | | Under '''Show''', uncheck '''Histogram''' check box. | + | || Under '''Show''', uncheck '''Histogram''' check box. |
− | | | Under '''Show''', uncheck '''Histogram''' check box to make it disappear. | + | || Under '''Show''', uncheck '''Histogram''' check box to make it disappear. |
|- | |- | ||
− | | | Scroll down and check '''Frequency Polygon''' to show it. | + | || Scroll down and check '''Frequency Polygon''' to show it. |
− | | | Scroll down and check '''Frequency Polygon''' to show it. | + | || Scroll down and check '''Frequency Polygon''' to show it. |
|- | |- | ||
− | | | Check '''Cumulative''' option as the '''Frequency Type'''. | + | || Check '''Cumulative''' option as the '''Frequency Type'''. |
− | | | Under '''Frequency Type''', check '''Cumulative''' option. | + | || Under '''Frequency Type''', check '''Cumulative''' option. |
|- | |- | ||
− | | | Point to default '''Count''' selection. | + | || Point to default '''Count''' selection. |
Point to the '''cumulative frequency count'''. | Point to the '''cumulative frequency count'''. | ||
− | | | The default '''Count''' selection shows the '''cumulative frequency count''' for the data. | + | || The default '''Count''' selection shows the '''cumulative frequency count''' for the data. |
|- | |- | ||
− | | | Drag '''slider''', bring it back to 20. | + | || Drag '''slider''', bring it back to 20. |
− | | | Drag the '''slider''' and note the effects on smoothness of the '''cumulative frequency count curve'''. | + | || Drag the '''slider''' and note the effects on smoothness of the '''cumulative frequency count curve'''. |
We will drag the '''slider''' back to 20. | We will drag the '''slider''' back to 20. | ||
|- | |- | ||
− | | | Under '''Frequency Type''', uncheck '''Cumulative''' and under '''Show''', uncheck '''Frequency Polygon'''. | + | || Under '''Frequency Type''', uncheck '''Cumulative''' and under '''Show''', uncheck '''Frequency Polygon'''. |
− | | | Under '''Frequency Type''', uncheck '''Cumulative''' and under '''Show''', uncheck '''Frequency Polygon'''. | + | || Under '''Frequency Type''', uncheck '''Cumulative''' and under '''Show''', uncheck '''Frequency Polygon'''. |
|- | |- | ||
− | | | Under '''Show''', check '''Histogram''' and uncheck '''Frequency Polygon''' > > click on '''Options''' button again to hide the window. | + | || Under '''Show''', check '''Histogram''' and uncheck '''Frequency Polygon''' > > click on '''Options''' button again to hide the window. |
− | | | Under '''Show''', check '''Histogram | + | || Under '''Show''', check '''Histogram''' . |
And click on '''Options''' button again to hide the window. | And click on '''Options''' button again to hide the window. | ||
|- | |- | ||
− | | | Click on '''Show Data''' tool >> point to data highlighted in the '''Spreadsheet'''. | + | || Click on '''Show Data''' tool >> point to data highlighted in the '''Spreadsheet'''. |
− | | | Above the '''Histogram text-box''', click on the third '''Show Data tool button'''. | + | || Above the '''Histogram text-box''', click on the third '''Show Data tool button'''. |
This displays all the data highlighted in the '''Spreadsheet'''. | This displays all the data highlighted in the '''Spreadsheet'''. | ||
|- | |- | ||
− | | | Drag the boundary to see the data properly. | + | || Drag the boundary to see the data properly. |
− | | | Drag the boundary to see the data properly. | + | || Drag the boundary to see the data properly. |
|- | |- | ||
− | | | Click on '''Show Data''' tool again to hide the list. | + | || Click on '''Show Data''' tool again to hide the list. |
− | | | Click on the '''Show Data''' tool again to hide the list. | + | || Click on the '''Show Data''' tool again to hide the list. |
|- | |- | ||
− | | | Click on '''Show 2<sup>nd</sup> Plot''' tool. | + | || Click on '''Show 2<sup>nd</sup> Plot''' tool. |
− | | | Above the '''Histogram text-box''', click on the last '''Show 2<sup>nd</sup> Plot tool button'''. | + | || Above the '''Histogram text-box''', click on the last '''Show 2<sup>nd</sup> Plot tool button'''. |
|- | |- | ||
− | | | Select '''histogram''' for top plot and '''box plot''' for bottom plot. | + | || Select '''histogram''' for top plot and '''box plot''' for bottom plot. |
− | | | The same data are graphed in two vertically placed plots. | + | || The same data are graphed in two vertically placed plots. |
You can select plot types from the '''dropdown menu button''' above each plot. | You can select plot types from the '''dropdown menu button''' above each plot. | ||
|- | |- | ||
− | | | Click on '''Show Statistics''' tool. | + | || Click on '''Show Statistics''' tool. |
Point to '''Statistics''' for both plots. | Point to '''Statistics''' for both plots. | ||
− | | | Above the '''Histogram text-box''', click on the second '''Show Statistics tool button'''. | + | || Above the '''Histogram text-box''', click on the second '''Show Statistics tool button'''. |
'''Statistics''' for the plot appears as a '''panel''' in the middle. | '''Statistics''' for the plot appears as a '''panel''' in the middle. | ||
|- | |- | ||
− | | | Drag the boundary to see it properly. | + | || Drag the boundary to see it properly. |
− | | | Drag the boundary to see it properly. | + | || Drag the boundary to see it properly. |
|- | |- | ||
− | | | '''Slide Number 8''' | + | || '''Slide Number 8''' |
'''Box Plot''' | '''Box Plot''' | ||
− | + | || '''Box Plot''' | |
− | | | '''Box Plot''' | + | |
'''Box plot''' is a standardized way of showing data, based on the '''five number summary'''. | '''Box plot''' is a standardized way of showing data, based on the '''five number summary'''. | ||
|- | |- | ||
− | | | Click and point to '''Median''', '''Min''', '''Max''', '''Q<sub>1</sub>''' and '''Q<sub>3'''</sub> values in the '''box plot'''. | + | || Click and point to '''Median''', '''Min''', '''Max''', '''Q<sub>1</sub>''' and '''Q<sub>3'''</sub> values in the '''box plot'''. |
− | | | Let us compare '''histogram''' and '''box plot'''. | + | || Let us compare '''histogram''' and '''box plot'''. |
In the '''box plot''', locate the '''Median''', '''Min''', '''Max''', '''Q<sub>1</sub>''' and '''Q<sub>3'''</sub> values. | In the '''box plot''', locate the '''Median''', '''Min''', '''Max''', '''Q<sub>1</sub>''' and '''Q<sub>3'''</sub> values. | ||
|- | |- | ||
− | | | Click on the button next to '''Options button''' above the plot. | + | || Click on the button next to '''Options button''' above the plot. |
− | | | Above each plot, in the upper right corner, click on the '''button''' next to '''Options'''. | + | || Above each plot, in the upper right corner, click on the '''button''' next to '''Options'''. |
A '''dropdown menu''' appears with which you can copy each plot to '''Clipboard''' or '''export''' it as an '''image'''. | A '''dropdown menu''' appears with which you can copy each plot to '''Clipboard''' or '''export''' it as an '''image'''. | ||
|- | |- | ||
− | | | Click on '''Show Statistics tool button''' to hide the data. | + | || Click on '''Show Statistics tool button''' to hide the data. |
− | | | Click on '''Show Statistics tool button''' to hide the data. | + | || Click on '''Show Statistics tool button''' to hide the data. |
|- | |- | ||
− | | | Close the '''Data Analysis''' window. | + | || Close the '''Data Analysis''' window. |
− | | | Close the '''Data Analysis''' window. | + | || Close the '''Data Analysis''' window. |
|- | |- | ||
− | | | '''Slide Number 9''' | + | || '''Slide Number 9''' |
'''Least Squares Linear Regression (LSLR)''' | '''Least Squares Linear Regression (LSLR)''' | ||
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'''Coefficient of determination R<sup>2</sup>''' | '''Coefficient of determination R<sup>2</sup>''' | ||
− | | | '''Least Squares Linear Regression (LSLR)''' | + | || '''Least Squares Linear Regression (LSLR)''' |
Changing an independent variable '''x''' changes the dependent variable '''y'''. | Changing an independent variable '''x''' changes the dependent variable '''y'''. | ||
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The closer '''R squared''' is to 1, the better is the prediction of variance in '''y''' from '''x'''. | The closer '''R squared''' is to 1, the better is the prediction of variance in '''y''' from '''x'''. | ||
|- | |- | ||
− | | | Show length and weight data in the '''Spreadsheet''' in the '''GeoGebra'''. | + | || Show length and weight data in the '''Spreadsheet''' in the '''GeoGebra'''. |
− | | | Let us go back to the length and weight data in the '''Spreadsheet''' view in '''GeoGebra'''. | + | || Let us go back to the length and weight data in the '''Spreadsheet''' view in '''GeoGebra'''. |
|- | |- | ||
− | | | Drag '''mouse''' to highlight all labels and data in the two '''columns'''. | + | || Drag '''mouse''' to highlight all labels and data in the two '''columns'''. |
− | | | Drag and select all the data in both '''columns'''. | + | || Drag and select all the data in both '''columns'''. |
|- | |- | ||
− | | | Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool. | + | || Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool. |
− | | | Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool. | + | || Under '''One Variable Analysis''', click on '''Two Variable Regression Analysis''' tool. |
|- | |- | ||
− | | | Click '''Analyze button''' in the '''Data Source window''' that pops up. | + | || Click '''Analyze button''' in the '''Data Source window''' that pops up. |
− | | | In the '''Data Source window''' that pops up, click '''Analyze button'''. | + | || In the '''Data Source window''' that pops up, click '''Analyze button'''. |
|- | |- | ||
− | | | '''Data Analysis''' window appears. | + | || '''Data Analysis''' window appears. |
− | | | A '''Data Analysis''' window appears with two plots. | + | || A '''Data Analysis''' window appears with two plots. |
|- | |- | ||
− | | | Show both plots. | + | || Show both plots. |
− | | | By default, the upper plot is a '''Scatterplot''' and the lower a '''Residual plot'''. | + | || By default, the upper plot is a '''Scatterplot''' and the lower a '''Residual plot'''. |
|- | |- | ||
− | | | Click on '''Show Statistics''' tool to see '''Statistics'''. | + | || Click on '''Show Statistics''' tool to see '''Statistics'''. |
− | | | Click on '''Show Statistics''' tool to see the '''Statistics'''. | + | || Click on '''Show Statistics''' tool to see the '''Statistics'''. |
|- | |- | ||
− | | | Drag the boundary to see them properly. | + | || Drag the boundary to see them properly. |
− | | | Drag the boundary to see them properly. | + | || Drag the boundary to see them properly. |
|- | |- | ||
− | | | Below '''Statistics window''', click on the '''Regression Model menu button''' | + | || Below '''Statistics window''', click on the '''Regression Model menu button''' >> select '''Linear'''. |
− | | | Below the '''Statistics window''', click on the '''Regression Model menu button''' and select '''Linear'''. | + | || Below the '''Statistics window''', click on the '''Regression Model menu button''' and select '''Linear'''. |
|- | |- | ||
− | | | Point to the red line that is drawn through some points. | + | || Point to the red line that is drawn through some points. |
− | | | Note the red line in the '''Scatterplot'''. | + | || Note the red line in the '''Scatterplot'''. |
|- | |- | ||
− | | | Point to equation is given in red, '''y= 0.08x-48.39'''. | + | || Point to equation is given in red, '''y= 0.08x-48.39'''. |
− | | | This is the '''best fit line''' that passes through as many points as possible. | + | || This is the '''best fit line''' that passes through as many points as possible. |
Its equation is given in red at the bottom. | Its equation is given in red at the bottom. | ||
|- | |- | ||
− | | | Point to '''R<sup>2'''</sup> value of 0.7722. | + | || Point to '''R<sup>2'''</sup> value of 0.7722. |
− | | | This ''' R<sup> </sup>squared''' value indicates good fit between the model and the actual data. | + | || This ''' R<sup> </sup>squared''' value indicates good fit between the model and the actual data. |
|- | |- | ||
− | | | Select other '''regression models''' to see effects on '''R<sup>2'''</sup>. | + | || Select other '''regression models''' to see effects on '''R<sup>2'''</sup>. |
− | | | Select other '''regression models''' to see effects on the '''R<sup> </sup>squared''' value. | + | || Select other '''regression models''' to see effects on the '''R<sup> </sup>squared''' value. |
|- | |- | ||
− | | | Point to the lower '''Residual Plot'''. | + | || Point to the lower '''Residual Plot'''. |
− | | | The lower plot is the '''Residual Plot'''. | + | || The lower plot is the '''Residual Plot'''. |
'''Residuals '''are the differences between observed and predicted values of all points. | '''Residuals '''are the differences between observed and predicted values of all points. | ||
|- | |- | ||
− | | | Click on '''Switch Axes button'''. | + | || Click on '''Switch Axes button'''. |
− | | | Above the '''Statistics window''', click on the last '''Switch Axes button'''. | + | || Above the '''Statistics window''', click on the last '''Switch Axes button'''. |
|- | |- | ||
− | | | Point to length now plotted on '''y-axis''' and weight on '''x-axis'''. | + | || Point to length now plotted on '''y-axis''' and weight on '''x-axis'''. |
− | | | For the '''scatterplot''', length is now plotted along '''y-axis''' and weight along '''x-axis'''. | + | || For the '''scatterplot''', length is now plotted along '''y-axis''' and weight along '''x-axis'''. |
|- | |- | ||
− | | | Point to the '''best fit line''' and '''statistics'''. | + | || Point to the '''best fit line''' and '''statistics'''. |
Point to equation '''y= 9.91x + 684.3'''. | Point to equation '''y= 9.91x + 684.3'''. | ||
− | | | Observe that the '''best fit line''' and many '''statistics''' change. | + | || Observe that the '''best fit line''' and many '''statistics''' change. |
Its equation is now '''y= 9.91x + 684.3'''. | Its equation is now '''y= 9.91x + 684.3'''. | ||
|- | |- | ||
− | | | Point to '''r''', '''R<sup>2</sup> '''and '''rho (ρ)'''. | + | || Point to '''r''', '''R<sup>2</sup> '''and '''rho (ρ)'''. |
− | | | The only statistics that remain the same are '''r''', '''R<sup> </sup>squared '''and '''rho (ρ)'''. | + | || The only statistics that remain the same are '''r''', '''R<sup> </sup>squared '''and '''rho (ρ)'''. |
Note that '''r''' and '''rho''' are greater than 0.8, indicating positive '''correlation'''. | Note that '''r''' and '''rho''' are greater than 0.8, indicating positive '''correlation'''. | ||
Line 413: | Line 406: | ||
The relationship is strong and well predicted by the '''best fit lines'''. | The relationship is strong and well predicted by the '''best fit lines'''. | ||
|- | |- | ||
− | | | | + | || Click on '''Switch Axes button'''. |
− | | | Again, click on '''Switch Axes button'''. | + | || Again, click on '''Switch Axes button'''. |
|- | |- | ||
− | | | Point to '''Symbolic Evaluation''' at the bottom. | + | || Point to '''Symbolic Evaluation''' at the bottom. |
− | | | At the bottom, in '''Symbolic Evaluation''', you can enter a value for '''x''' to get a prediction for '''y'''. | + | || At the bottom, in '''Symbolic Evaluation''', you can enter a value for '''x''' to get a prediction for '''y'''. |
|- | |- | ||
− | | | Point at the line in the '''Scatterplot'''. | + | || Point at the line in the '''Scatterplot'''. |
− | | | To get logical predictions, we will enter '''x''' values above the '''x-intercept'''. | + | || To get logical predictions, we will enter '''x''' values above the '''x-intercept'''. |
|- | |- | ||
− | | | In '''Symbolic Evaluation''', type in a value for '''x''' | + | || In '''Symbolic Evaluation''', type in a value for '''x''' >> press '''Enter'''. |
− | | | In '''Symbolic Evaluation''', in the text-box for '''x''', type 800 and press '''Enter'''. | + | || In '''Symbolic Evaluation''', in the text-box for '''x''', type 800 and press '''Enter'''. |
|- | |- | ||
− | | | Point to '''y''' value appearing next to the '''display box'''. | + | || Point to '''y''' value appearing next to the '''display box'''. |
− | | | Note that a '''y''' value appears next to the '''display box'''. | + | || Note that a '''y''' value appears next to the '''display box'''. |
The '''x''' value was substituted in the '''best fit line''' equation to get the '''y''' value. | The '''x''' value was substituted in the '''best fit line''' equation to get the '''y''' value. | ||
|- | |- | ||
− | | | | + | || Click on '''Show Statistics tool button'''. |
− | | | Again, click on '''Show Statistics tool button'''. | + | || Again, click on '''Show Statistics tool button'''. |
|- | |- | ||
− | | | Close the '''Data Analysis window'''. | + | || Close the '''Data Analysis window'''. |
− | | | Close the '''Data Analysis window'''. | + | || Close the '''Data Analysis window'''. |
|- | |- | ||
− | | | Point to length and weight data in the '''Spreadsheet'''. | + | || Point to length and weight data in the '''Spreadsheet'''. |
− | | | Let’s go back to the length and weight data in the '''Spreadsheet'''. | + | || Let’s go back to the length and weight data in the '''Spreadsheet'''. |
|- | |- | ||
− | | | Drag '''mouse''' to highlight all labels and data in the two '''columns'''. | + | || Drag '''mouse''' to highlight all labels and data in the two '''columns'''. |
− | | | In the '''Spreadsheet''', select all the data in both '''columns'''. | + | || In the '''Spreadsheet''', select all the data in both '''columns'''. |
|- | |- | ||
− | | | Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool. | + | || Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool. |
− | | | Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool. | + | || Under '''One Variable Analysis''', click on '''Multiple Variable Analysis''' tool. |
|- | |- | ||
− | | | Click '''Analyze button''' in the '''Data Source window''' that pops up. | + | || Click '''Analyze button''' in the '''Data Source window''' that pops up. |
− | | | In the '''Data Source window''' that pops up, click '''Analyze button'''. | + | || In the '''Data Source window''' that pops up, click '''Analyze button'''. |
|- | |- | ||
− | | | Point to '''Box Plots''' in the window and to the cell numbers in each row. | + | || Point to '''Box Plots''' in the window and to the cell numbers in each row. |
− | | | '''Box Plots''' appear in the window. | + | || '''Box Plots''' appear in the window. |
They are for length and weight data. | They are for length and weight data. | ||
|- | |- | ||
− | | | Click on '''Show Statistics''' tool. | + | || Click on '''Show Statistics''' tool. |
Point to '''Statistics''' for both plots. | Point to '''Statistics''' for both plots. | ||
− | | | Above the plot, click on the second '''Show Statistics''' tool. | + | || Above the plot, click on the second '''Show Statistics''' tool. |
'''Statistics''' for both plots appear below. | '''Statistics''' for both plots appear below. | ||
|- | |- | ||
− | | | Place the '''cursor''' on the boundary between the plot and statistics. | + | || Place the '''cursor''' on the boundary between the plot and statistics. |
When the arrow appears, drag the boundary to resize the windows. | When the arrow appears, drag the boundary to resize the windows. | ||
− | | | Place the '''cursor''' on the boundary between the plot and statistics. | + | || Place the '''cursor''' on the boundary between the plot and statistics. |
When the '''arrow''' appears, drag the boundary to resize the windows. | When the '''arrow''' appears, drag the boundary to resize the windows. | ||
|- | |- | ||
− | | | | + | || |
− | | | Let us summarize. | + | || Let us summarize. |
|- | |- | ||
− | | | '''Slide Number 10''' | + | || '''Slide Number 10''' |
'''Summary''' | '''Summary''' | ||
− | | | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to perform: | + | || In this '''tutorial''', we have learnt how to use '''GeoGebra''' to perform: |
'''One Variable Analysis''' to calculate different statistical parameters | '''One Variable Analysis''' to calculate different statistical parameters | ||
Line 481: | Line 474: | ||
'''Multiple Variable Analysis''' to calculate different statistical parameters | '''Multiple Variable Analysis''' to calculate different statistical parameters | ||
|- | |- | ||
− | | | '''Slide Number 11''' | + | || '''Slide Number 11''' |
'''Assignment''' | '''Assignment''' | ||
Line 489: | Line 482: | ||
Is any of the oils absorbed more than the others? | Is any of the oils absorbed more than the others? | ||
− | + | || '''Assignment''' | |
− | + | ||
− | | | '''Assignment''' | + | |
Perform statistical analyses for weight and girth data given in this '''tutorial''' | Perform statistical analyses for weight and girth data given in this '''tutorial''' | ||
Line 497: | Line 488: | ||
Four oils were used to deep fry chips. Amount of absorbed fat was measured for 6 chips fried in 4 oils. Is any of the oils absorbed more than the others? | Four oils were used to deep fry chips. Amount of absorbed fat was measured for 6 chips fried in 4 oils. Is any of the oils absorbed more than the others? | ||
|- | |- | ||
− | | | '''Slide Number 12''' | + | || '''Slide Number 12''' |
'''About Spoken Tutorial project''' | '''About Spoken Tutorial project''' | ||
− | | | The video at the following link summarizes the '''Spoken Tutorial project'''. | + | || The video at the following link summarizes the '''Spoken Tutorial project'''. |
Please download and watch it. | Please download and watch it. | ||
|- | |- | ||
− | | | '''Slide Number 13''' | + | || '''Slide Number 13''' |
'''Spoken Tutorial workshops''' | '''Spoken Tutorial workshops''' | ||
− | | | The '''Spoken Tutorial Project''' team conducts workshops and gives certificates. | + | || The '''Spoken Tutorial Project''' team conducts workshops and gives certificates. |
For more details, please write to us. | For more details, please write to us. | ||
|- | |- | ||
− | | | '''Slide Number 14''' | + | || '''Slide Number 14''' |
'''Forum for specific questions:''' | '''Forum for specific questions:''' | ||
Line 524: | Line 515: | ||
Someone from our team will answer them | Someone from our team will answer them | ||
− | | | Please post your timed queries on this forum. | + | || Please post your timed queries on this forum. |
|- | |- | ||
− | | | '''Slide Number 15''' | + | || '''Slide Number 15''' |
'''Acknowledgement''' | '''Acknowledgement''' | ||
− | | | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India. | + | || '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India. |
More information on this mission is available at this link. | More information on this mission is available at this link. | ||
|- | |- | ||
− | | | | + | || |
− | | | This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off. | + | || This is '''Vidhya Iyer''' from '''IIT Bombay''', signing off. |
Thank you for joining. | Thank you for joining. | ||
|- | |- | ||
|} | |} |
Revision as of 12:18, 16 January 2019
Visual Cue | Narration |
Slide Number 1
Title Slide |
Welcome to this tutorial on Statistics using GeoGebra |
Slide Number 2
Learning Objectives |
In this tutorial, we will learn how to use GeoGebra to perform:
One Variable Analysis to calculate different statistical parameters Two Variable Regression Analysis to estimate best fit line Multiple Variable Analysis to calculate different statistical parameters |
Slide Number 3
System Requirement |
Here I am using:
Ubuntu Linux Operating System version 16.04 GeoGebra 5.0.481.0-d |
Slide Number 4
Pre-requisites |
To follow this tutorial, you should be familiar with:
GeoGebra interface Statistics |
Slide Number 5
Statistics Data analysis and interpretation Measures of central tendency Measures of Dispersion Comparing variability of data series Additional material |
Statistics deals with
Data analysis and interpretation Measures of central tendency Measures of Dispersion Comparing variability of data series Please refer to additional material provided along with this tutorial. |
Slide Number 6
Fish Feed A fishery is testing four feed formulations on its fish: A, B, C and D Length (mm) Weight (lbs) Girth (mm) |
Fish Feed
Let us look at an example. A fishery is testing four types of feed formulations on its fish: A, B, C and D. Data to be collected after feeding the fish for 6 months are: Length in millimeters Weight in pounds Girth in millimeters Let us look at some of these data. |
Slide Number 7
Fish Feed Data |
We will use these data for our analyses.
Please download the code file, Fishery-data, provided along with this tutorial. |
Show the GeoGebra window. | I have opened the GeoGebra interface. |
Click on View tool >> select Spreadsheet. | Click on View tool and select Spreadsheet. |
Click on X at top right corner of Graphics, Algebra views. | Click on X at top right corner of Graphics and Algebra views.
This will close these views. |
In the code file, drag mouse to highlight length and weight data from columns H and I.
Show data in columns H and I. Hold Ctrl key down and press C. |
In the code file, drag mouse to highlight length and weight data from columns H and I.
These are data for fish that have been fed formulation C. Hold Control key down and press C. |
Click on Spreadsheet view in GeoGebra. | Click in the top of the Spreadsheet in GeoGebra. |
Press Ctrl key down, press V. | Hold Control key down and press V.
This will copy and paste the highlighted data from the code file into GeoGebra. |
Place the cursor on the first column header in Spreadsheet view.
Drag and adjust column A's width. |
Place the cursor on the first column header in Spreadsheet view.
Drag and adjust column A's width. |
Right-click on column A heading of Length (mm).
Select Object Properties. Point to dialog box. |
Right-click on column A heading of Length millimetres.
Select Object Properties. A dialog box opens. |
Click on Text tab and change the name to Length (mm)-C.
Close the dialog box. Similarly, add –C to Weight (lbs). |
Click on Text tab and change the name to Length millimetres hyphen C.
Close the dialog box. Similarly, add hyphen C to Weight pounds. |
Adjust column B width. | Adjust column B width. |
Use mouse to drag and highlight first column A’s length data and label in GeoGebra. | Click on column A heading of Length millimetres C.
Drag to highlight length data in Spreadsheet view. |
Below the menubar, click on One Variable Analysis tool.
Point to Data Source popup window. Click on Analyze button. |
Below the menubar, click on One Variable Analysis tool.
A Data Source popup window appears. Click on Analyze button. |
Point to Data Analysis window and histogram. | A Data Analysis window appears.
By default, a histogram is plotted. |
Drag the boundary to see the graph properly. | Drag the boundary to see the graph properly. |
Point to length on the x-axis and frequency on the y-axis. | The length is plotted on the x-axis.
The number of fish that are of a particular length, the frequency, is plotted on the y-axis. |
Point to the display box above the graph containing the word Histogram. | Note the display box above the graph containing the word Histogram. |
In the display box, click on the dropdown menu button. | In the display box, click on the dropdown menu button to display the list of plots. |
Select Histogram.
Point to slider to the right of the display. |
We will stay with the histogram option.
To the right of the dropdown menu is a slider. |
Drag the slider from left to right to go to 20. | Drag the slider from left to right to go to 20. |
Point to rectangles between minimum and maximum values of data. | The slider changes the number of rectangles between the minimum and maximum values of data. |
Click on Options button to the right of the slider. | Click on Options button to the right of the slider. |
Under Classes, check Set Classes Manually. | Under Classes, check Set Classes Manually check box.
This displays Start and Width text-boxes to the left of the Options button. |
Type 800 in the Start text-box and press Enter.
Show the value of 5 in the Width text-box. |
As all the fish are over 800 mm long, type 800 in the Start text-box and press Enter.
We will stay with the default value of 5 for rectangle width. |
Uncheck Set Classes Manually. | Uncheck Set Classes Manually check box. |
Under Show, uncheck Histogram check box. | Under Show, uncheck Histogram check box to make it disappear. |
Scroll down and check Frequency Polygon to show it. | Scroll down and check Frequency Polygon to show it. |
Check Cumulative option as the Frequency Type. | Under Frequency Type, check Cumulative option. |
Point to default Count selection.
Point to the cumulative frequency count. |
The default Count selection shows the cumulative frequency count for the data. |
Drag slider, bring it back to 20. | Drag the slider and note the effects on smoothness of the cumulative frequency count curve.
We will drag the slider back to 20. |
Under Frequency Type, uncheck Cumulative and under Show, uncheck Frequency Polygon. | Under Frequency Type, uncheck Cumulative and under Show, uncheck Frequency Polygon. |
Under Show, check Histogram and uncheck Frequency Polygon > > click on Options button again to hide the window. | Under Show, check Histogram .
And click on Options button again to hide the window. |
Click on Show Data tool >> point to data highlighted in the Spreadsheet. | Above the Histogram text-box, click on the third Show Data tool button.
This displays all the data highlighted in the Spreadsheet. |
Drag the boundary to see the data properly. | Drag the boundary to see the data properly. |
Click on Show Data tool again to hide the list. | Click on the Show Data tool again to hide the list. |
Click on Show 2^{nd} Plot tool. | Above the Histogram text-box, click on the last Show 2^{nd} Plot tool button. |
Select histogram for top plot and box plot for bottom plot. | The same data are graphed in two vertically placed plots.
You can select plot types from the dropdown menu button above each plot. |
Click on Show Statistics tool.
Point to Statistics for both plots. |
Above the Histogram text-box, click on the second Show Statistics tool button.
Statistics for the plot appears as a panel in the middle. |
Drag the boundary to see it properly. | Drag the boundary to see it properly. |
Slide Number 8
Box Plot |
Box Plot
Box plot is a standardized way of showing data, based on the five number summary. |
Click and point to Median, Min, Max, Q_{1} and Q_{3} values in the box plot. | Let us compare histogram and box plot.
In the box plot, locate the Median, Min, Max, Q_{1} and Q_{3} values. |
Click on the button next to Options button above the plot. | Above each plot, in the upper right corner, click on the button next to Options.
A dropdown menu appears with which you can copy each plot to Clipboard or export it as an image. |
Click on Show Statistics tool button to hide the data. | Click on Show Statistics tool button to hide the data. |
Close the Data Analysis window. | Close the Data Analysis window. |
Slide Number 9
Least Squares Linear Regression (LSLR) Changing an independent variable x changes the dependent variable y. LSLR predicts y based on x value. LSRL (best fit line) y = b_{0} + b_{1}x Coefficient of determination R^{2} |
Least Squares Linear Regression (LSLR)
Changing an independent variable x changes the dependent variable y. LSLR predicts y based on x value. Least Squares Regression Line (LSRL) is also called the best fit line. It is given by y = b_{0} + b_{1}x. b_{1}, the slope, is the regression coefficient. Coefficient of determination R^{ }squared R^{ }squared ranges from 0 to 1. The closer R squared is to 1, the better is the prediction of variance in y from x. |
Show length and weight data in the Spreadsheet in the GeoGebra. | Let us go back to the length and weight data in the Spreadsheet view in GeoGebra. |
Drag mouse to highlight all labels and data in the two columns. | Drag and select all the data in both columns. |
Under One Variable Analysis, click on Two Variable Regression Analysis tool. | Under One Variable Analysis, click on Two Variable Regression Analysis tool. |
Click Analyze button in the Data Source window that pops up. | In the Data Source window that pops up, click Analyze button. |
Data Analysis window appears. | A Data Analysis window appears with two plots. |
Show both plots. | By default, the upper plot is a Scatterplot and the lower a Residual plot. |
Click on Show Statistics tool to see Statistics. | Click on Show Statistics tool to see the Statistics. |
Drag the boundary to see them properly. | Drag the boundary to see them properly. |
Below Statistics window, click on the Regression Model menu button >> select Linear. | Below the Statistics window, click on the Regression Model menu button and select Linear. |
Point to the red line that is drawn through some points. | Note the red line in the Scatterplot. |
Point to equation is given in red, y= 0.08x-48.39. | This is the best fit line that passes through as many points as possible.
Its equation is given in red at the bottom. |
Point to R^{2} value of 0.7722. | This R^{ }squared value indicates good fit between the model and the actual data. |
Select other regression models to see effects on R^{2}. | Select other regression models to see effects on the R^{ }squared value. |
Point to the lower Residual Plot. | The lower plot is the Residual Plot.
Residuals are the differences between observed and predicted values of all points. |
Click on Switch Axes button. | Above the Statistics window, click on the last Switch Axes button. |
Point to length now plotted on y-axis and weight on x-axis. | For the scatterplot, length is now plotted along y-axis and weight along x-axis. |
Point to the best fit line and statistics.
Point to equation y= 9.91x + 684.3. |
Observe that the best fit line and many statistics change.
Its equation is now y= 9.91x + 684.3. |
Point to r, R^{2} and rho (ρ). | The only statistics that remain the same are r, R^{ }squared and rho (ρ).
Note that r and rho are greater than 0.8, indicating positive correlation. Weight increases as length increases for fish given feed C. The relationship is strong and well predicted by the best fit lines. |
Click on Switch Axes button. | Again, click on Switch Axes button. |
Point to Symbolic Evaluation at the bottom. | At the bottom, in Symbolic Evaluation, you can enter a value for x to get a prediction for y. |
Point at the line in the Scatterplot. | To get logical predictions, we will enter x values above the x-intercept. |
In Symbolic Evaluation, type in a value for x >> press Enter. | In Symbolic Evaluation, in the text-box for x, type 800 and press Enter. |
Point to y value appearing next to the display box. | Note that a y value appears next to the display box.
The x value was substituted in the best fit line equation to get the y value. |
Click on Show Statistics tool button. | Again, click on Show Statistics tool button. |
Close the Data Analysis window. | Close the Data Analysis window. |
Point to length and weight data in the Spreadsheet. | Let’s go back to the length and weight data in the Spreadsheet. |
Drag mouse to highlight all labels and data in the two columns. | In the Spreadsheet, select all the data in both columns. |
Under One Variable Analysis, click on Multiple Variable Analysis tool. | Under One Variable Analysis, click on Multiple Variable Analysis tool. |
Click Analyze button in the Data Source window that pops up. | In the Data Source window that pops up, click Analyze button. |
Point to Box Plots in the window and to the cell numbers in each row. | Box Plots appear in the window.
They are for length and weight data. |
Click on Show Statistics tool.
Point to Statistics for both plots. |
Above the plot, click on the second Show Statistics tool.
Statistics for both plots appear below. |
Place the cursor on the boundary between the plot and statistics.
When the arrow appears, drag the boundary to resize the windows. |
Place the cursor on the boundary between the plot and statistics.
When the arrow appears, drag the boundary to resize the windows. |
Let us summarize. | |
Slide Number 10
Summary |
In this tutorial, we have learnt how to use GeoGebra to perform:
One Variable Analysis to calculate different statistical parameters Two Variable Regression Analysis to estimate best fit line Multiple Variable Analysis to calculate different statistical parameters |
Slide Number 11
Assignment Perform statistical analyses for weight and girth data Is any of the oils absorbed more than the others? |
Assignment
Perform statistical analyses for weight and girth data given in this tutorial Four oils were used to deep fry chips. Amount of absorbed fat was measured for 6 chips fried in 4 oils. Is any of the oils absorbed more than the others? |
Slide Number 12
About Spoken Tutorial project |
The video at the following link summarizes the Spoken Tutorial project.
Please download and watch it. |
Slide Number 13
Spoken Tutorial workshops |
The Spoken Tutorial Project team conducts workshops and gives certificates.
For more details, please write to us. |
Slide Number 14
Forum for specific questions: Do you have questions in THIS Spoken Tutorial? Please visit this site Choose the minute and second where you have the question Explain your question briefly Someone from our team will answer them |
Please post your timed queries on this forum. |
Slide Number 15
Acknowledgement |
Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link. |
This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |