1. Descriptive Statistics

Transcription

1 1.1 Descriptive statistics 1. Descriptive Statistics A Data management Before starting any statistics analysis with a graphics calculator, you need to enter the data. We will illustrate the process by entering some data collected on the air pollution level (mg/m 3 ) in six cities. The data is shown below. City number Pollution level (mg/m 3 ) Data is stored in the graphics calculator in columns (called Lists) in what looks like a minispreadsheet, called the stat list editor. Before entering the data into the list, it is good statistical practice to give the list a name to help us keep in mind the quantity we are analysing. We will follow this practice throughout this book. A.1 Entering data into a named list Task: Enter the city numbers into a list named CITY. Step 1. Call up the stat list editor To start the process, press STAT and choose EDIT by pressing ENTER. The screen should then appear as shown*. * If there is already data in list (L1 ), either move to an empty list or clear the list (move the cursor to cover L1 and press CLEAR ENTER ). Step 2. Create and name the list CITY (1) To give the list a name, first move the cursor using the arrow keys (}) so that it sits on L1 (or any other empty column). (2) Press 2nd [INS] ( DEL ) and the screen should appear as shown. A new list has been created which you can now name. Peter Jones and Chris Barling, 2001

2 Descriptive statistics 1.2 (3) Type in the word CITY* and press ENTER and your new list will be named CITY. Press the down arrow key ( ) to move the cursor down into the list. *At this point the cursor is set to A(LPHA) mode, so you type by finding the green letters above the keys. Do not press the green ALPHA key as this will take you out of alphabetic mode. Step 3. Enter the data Type in the data values, pressing ENTER after each value. Note how the cursor moves down the list, and the entries are successively numbered at the foot of the screen. A.2 Recalling and displaying the contents of a data list on the Home Screen Tasks: Recall and display the contents of the data stored in the list named CITY on the Home Screen. Note: This sequence assumes that you have already entered data into a list called CITY, see section A.1. Step 1. Move to Home Screen If you are not on the Home Screen, return to the Home Screen by pressing 2nd [QUIT] ( MODE ). Clear the screen if necessary using the CLEAR key. Step 2. Locate the list named CITY in the LIST menu (1) Go to the LIST menu by pressing 2nd [LIST] ( STAT ) and you will see the names of all the lists stored in the calculator displayed on the screen starting with the lists L1 to L6. If there are more than seven lists, an arrow at the bottom of the screen shows that the list continues. (2) Use the down arrow key ( ) to move down the menu until it sits opposite CITY. Step 3. Paste the list named CITY onto the Home Screen and display its contents. (1) With the cursor opposite CITY in the LIST [NAMES] menu, press ENTER to paste the list name onto the Home Screen. *Note: When pasting onto the Home Screen, the calculator inserts an L before the name to indicate that it is a list name. (2) Display the list's contents by pressing ENTER. Peter Jones and Chris, 2001

3 1.3 Descriptive statistics Exercises 1. Enter the pollution levels in a list named POLL (an abbreviation for pollution). Keep the pollution values in order, to match the correct city numbers. If you have done it correctly, your calculator screen should now look something like that shown opposite. 2. Recall the list named POLL to the Home Screen and display its contents. If you have done it correctly, your calculator screen should now look something like that shown opposite*. *Note: The dots at the right indicate that the display is incomplete. To see the rest of the list, scroll to the right with the right arrow key. A.3 Clearing and deleting lists displayed in the stat list editor Tasks: 1. Clear the contents of a list displayed in the stat list editor. 2. Delete a list from the stat list editor. Note: This sequence assumes that you have already entered data into a list called CITY and a list called POLL, see section A.1. Task 1: Clear the contents of a list called CITY displayed in the stat list editor. Task 2: Delete a list called POLL displayed in the stat list editor. If you are not already in the stat list editor, press STAT ENTER ENTER. With the arrow keys, move the cursor so that it sits on the list name CITY. Press CLEAR ENTER to clear the contents of the list. If you are not already in the stat list editor, press STAT ENTER ENTER. With the arrow keys, move the cursor so that it sits on the list name POLL. Press DEL to delete the list from the Data Editor. Peter Jones and Chris Barling, 2001

4 Descriptive statistics 1.4 A.4 Deleting a named list from the memory Task: Delete the list named POLL from memory. Note: This sequence assumes that you have already entered data into a list called POLL, see section A.1. Even though you may have deleted a list from the stat list editor (as in A.3.1), the list will still be stored in memory for later use. If we no longer wish to keep the list, it can be deleted as follows. Step 1. Call up the MEMORY menu by pressing 2nd [MEM] ( + ) and use the down arrow key ( ) to move the cursor to option 2:Mem Mgmt/Del... Step 2. Press ENTER and use the down arrow key ( ) to move the cursor to option 4:List... Step 3. Press ENTER and use the down arrow key ( ) to move the cursor to POLL. Step 4. Press DEL to delete the list from the memory. B Displaying univariate data 1: Histograms The following set of data shows the marks, out of 30, for 27 students in a statistics test Tasks: 1. Enter the data into a list named MARK. 2. Display the students' marks in the form of a histogram. 3. Interrogate the histogram to determine key facts. 4. Investigate how changing the class intervals affects the shape of the histogram. Peter Jones and Chris, 2001

5 1.5 Descriptive statistics B.1 Constructing a histogram Step 1. Enter the data Enter the data into a list named MARK (refer back to section A.1 if in doubt). If you have done it correctly, your calculator screen should now look something like that shown opposite. Step 2. Call up the STAT PLOT menu Graphical displays of data are created by calling up the STAT PLOT menu ( 2nd [STAT PLOT] ( Y= )). Your screen should look something like that opposite. Step 3. Set up the plot Select Plot 1 by pressing ENTER. To set up the plot you need to (1) Turn the plot on. Move the cursor (if necessary) so that it covers On and press ENTER. (2) Move the cursor down to Type: (of display) and move the cursor across to cover the histogram. Select by pressing ENTER. (3) Move the cursor down to Xlist: The data we wish to display is stored in a list named MARK. You can either type in the word MARK (use of the ALPHA key is not necessary because of the "A" cursor). or go to the LIST menu ( 2nd [LIST] ( STAT )), move the cursor down the NAMES column until it sits opposite MARK and press ENTER. This will paste the word MARK onto STATPLOT screen opposite Xlist. (4) Leave the Freq:uency set at 1. If you have followed these steps correctly, your screen should look like the one opposite. Peter Jones and Chris Barling, 2001

6 Descriptive statistics 1.6 Step 4. Set up the viewing window Before plotting the histogram we need to set an appropriate viewing window. To do this press WINDOW and enter the settings as shown opposite. Explanations for the settings are as follows: Xmin = 0 a convenient lower value for the marks Xmax = 30 a convenient upper value for the marks Xscl = 5 class interval (with these setting marks will be grouped as 0-4, 5-9, etc.) Ymin = 0 minimum frequency Ymax = 15 maximum frequency ( an intelligent guess) Yscl = 1 scale on vertical axis Xres = 1 leave as 1; plots every pixel* * If you change to 2, it will plot every second pixel and so on. Step 5. Plotting the histogram Plot the histogram by pressing GRAPH. If you did everything as set out above, your histogram should be similar to that shown opposite.* *If you find that you have some other bits and pieces of graphs on your plot it is probably because (a) you have some function graphs stored under the Y= menu. Go back to this menu and either CLEAR these functions or turn them off. Press GRAPH to return to your plot. (b) another statistical plot you do not want is turned on. Check by going to the STAT PLOT. Turn any unwanted plots off. Press GRAPH to return to your plot. B. 2 Obtaining information from a histogram To find the boundaries of each class interval and the number of data values that fall into the interval (the frequency) press TRACE. This places the cursor in the middle of the first class interval and gives the required information. From there you can move to any other class interval using the horizontal arrow keys. The screen opposite shows the cursor in the third interval which starts at 10 and includes all values up to, but not including, 15. We can also see that there are 9 (n = 9) data values in this interval. Peter Jones and Chris, 2001

7 1.7 Descriptive statistics B.3 Changing the width of the class intervals in a histogram In constructing the present histogram, the data has been grouped into intervals 5 units wide. What would the histogram look like if data was grouped into intervals 10 units wide? Changing the interval width to 10. To change the class interval width to 10, press the WINDOW key and make Xscl = 10. You will also need to increase Ymax as having made the intervals wider, they will capture more data values. We will set Ymax = 30. Changing the interval width to 2. To change the class interval width to 2, press the WINDOW key and make Xscl = 2. You will also need to decrease Ymax as having made the intervals narrower, they will capture less data values. We will set Ymax = 8. Pressing GRAPH will then redraw the histogram with class intervals of 10 as shown below. Pressing GRAPH will then redraw the histogram with class intervals of 2 as shown below. Again the information about each class interval can be found by pressing TRACE and moving the cursor to the desired interval. Again the information about each class interval can be found by pressing TRACE and moving the cursor to the desired interval. One of the key issues when constructing histograms is choosing an appropriate width for the class intervals. Too wide and you will tend to lose the detail, as in the case when we used a class interval width of 10. Too small and the overall pattern in the data will be lost in the detail, as is the case when we use an interval width of 2. In this case an interval width of 5 (our first choice) seems to be the best of the three intervals investigated here and would seem to be an appropriate choice. Peter Jones and Chris Barling, 2001

8 Descriptive statistics 1.8 Exercises 1. The life expectancies (in years) of 25 countries are listed below Using your graphics calculator, construct a histogram with the first class interval starting at 35 and the last class interval ending at 80 using a class interval width of: (1) 2 (2) 5 (3) The data below gives the wrist circumference (in cm) of 15 men Using your graphics calculator, construct a histogram with the first class interval starting at 16.5 and the last class interval ending at 20.5 using a class interval width of: (1) 0.5 (2) 1.0 (3) 2.0 C Summarising univariate data: measures of centre and variability C.1 Working with raw data Ten households were surveyed and the number of people normally living there recorded with the following results: Tasks: For this data set, determine the values of the: 1. mean and standard deviation 2. unbiased estimate of the population variance 3. median and semi-interquartile range On the TI-83 Plus/ TI-84 Plus, these statistics are calculated by using the 1-VarStats command. Peter Jones and Chris, 2001

9 1.9 Descriptive statistics Step 1. Enter the data Call up the stat list editor and enter the data into list named NUM. If you have forgotten how to name a list see section A.1. Step 2. Set up and execute the 1-Var Stats command (1) Return to the Home Screen by pressing 2nd [QUIT] ( MODE ) and press CLEAR to clear the screen. (2) Press STAT and move the cursor to CALC(ulation). The 1-VarStats command should be automatically highlighted. (3) Press ENTER and the 1-VarStats command will be 'pasted' onto the Home Screen. (4) To complete the 1-VarStats command we need to tell the calculator where to find the data. In this case, it is stored in the list named NUM. Find NUM in the [LIST ] menu and paste it into the 1-VarStats command. Peter Jones and Chris Barling, 2001

10 Descriptive statistics 1.10 (5) Press ENTER and the required statistics will be generated. Note that the results are spread over two screens. We can scroll up and down with the up and down arrow keys (}, ). Screen 1 Screen 2 From this calculation we see that, for the households surveyed: A-level (Singapore / UK) the mean number of residents is x =3.300 with a standard deviation of Sx = the median number of residents is Med =3.50 or IQR =Q 3 Q 1 =2 the mean number of residents is x =3.300 with a standard deviation of σx = the unbiased estimate of the population variance is Sx 2 = = 1.79 Sx 2 = n n -1 σx2 = 10 9 x1.272 = 1.79 semi-iqr = 1 2 (Q 3 Q 1 ) = 1 (4 2) = 1 2 Exercise Ten households were surveyed and the weekly amount spent on food recorded with the following results: $170 $123 $87 $98 $112 $150 $98 $134 $106 $114 enter the data into a list called SPENT and show that, to the nearest dollar: A-level (Singapore / UK) the mean amount spent on food each the mean amount spent on food each week is x =$119 week is x =$119 with a standard deviation of σx = $24 with a standard deviation of Sx = $26 the unbiased estimate of the population variance is the median amount spent on food each Sx 2 = = 661 week is Med =$113 or IQR =Q 3 Q 1 =$36 Sx 2 n = n -1 σx2 = 10 9 x =661 the median amount spent on food each week is Med =$113 semi-iqr = 1 2 (Q 3 Q 1 ) = 1 (134-98) = $18 2 Peter Jones and Chris, 2001

11 1.11 Descriptive statistics C.2 Working with grouped data One hundred households were surveyed and the number of people normally in residence recorded in tabular form as follows: Number of residents Frequency Total 100 Tasks: For this data set, determine the values of the: 1. mean and standard deviation 2. unbiased estimate of the population variance 3. median and semi-interquartile range On the TI-83 Plus, these statistics are calculated by using the 1-VarStats command. Step 1. Enter the data With grouped data, we need to enter both the data values and the frequencies into the calculator. Call up the data entry screen and enter the number of residents into a list named NRES (an abbreviation for number of residents) and the corresponding frequencies into a list named FREQ (an abbreviation for frequency). If you have forgotten how to name a list, see section A.1. Step 2. Set up and execute the 1-VarStats command. (1) Return to the Home Screen by pressing 2nd [QUIT] ( MODE ) and press CLEAR to clear screen. (2) Press STAT and move the cursor to CAL C(ulation). The 1-VarStats command should be automatically highlighted. Peter Jones and Chris Barling, 2001

12 Descriptive statistics 1.12 (3) Press ENTER and the 1-VarStats command will be 'pasted' onto the Home Screen. (4) To complete the 1-VarStats command we need to tell the calculator where to find the data. In this case, the data values are stored in the list named NRES and the frequencies in the list named FREQ. Find NRES in the [LIST] menu and paste it into the 1-VarStats command. Next enter a comma (, ) from the keyboard. Finally, find FREQ in the [LIST] menu and paste it in after the comma. (5) Press ENTER and the required statistics will be generated. Note: The results are spread over two screens. Screen 1 Screen 2 From this calculation we see that, for the households surveyed: A-level (Singapore / UK) the mean number of residents is the mean number of residents is x =4.01 x =4.01 with a standard deviation of σx = 1.47 the standard deviation is the unbiased estimate of the population variance is Sx = 1.48 Sx 2 = = 2.2 the median number of residents is or Med =4 Sx IQR =Q 3 Q 1 =2 2 n = n -1 σx2 = x1.472 = 2.2 the median number of residents is Med =4 semi-iqr = 1 2 (Q 3 Q 1 ) = 1 2 (3-1 ) = 1 Peter Jones and Chris, 2001

13 1.13 Descriptive statistics Exercise One hundred households were surveyed and the number of children normally in residence recorded in tabular form as follows: Number of children Frequency Total Enter the data values into a list called NCHIL and the frequencies into a list called FREQ. If you already have a list called FREQ, clear the list by moving the cursor onto the name at the top of the column and press CLEAR ENTER. 2. Show that, correct to one decimal place: the mean number of children is x =2.0 the standard deviation is Sx = 1.3 the median number of children is Med =2 IQR =Q 3 Q 1 =2 A-level (Singapore / UK) the mean number of children is x =2.0 with a standard deviation of σx = 1.3 the unbiased estimate of the population variance is Sx 2 = = 1.8 or Sx 2 n = n -1 σx2 = x1.332 = 1.8 the median number of children is Med =2 semi-iqr = 1 2 (Q 3 Q 1 ) = 1 2 (3-1 ) = 1 D Displaying univariate data 2: Boxplots D.1 Constructing a boxplot The data shows the weights (in kg) carried by 14 horses in a race. 50, 54, 53, 53, 50, 52, 51, 53, 54, 50, 59, 51, 52, 53 Tasks: 1. Enter the data into a list called WT (an abbreviation for weight). 2. Generate a boxplot showing outliers. Step 1. Call up stat list editor To start the process, press STAT and choose EDIT by pressing ENTER. The screen should then appear as shown. Peter Jones and Chris Barling, 2001

14 Descriptive statistics 1.14 Step 2. Create and name the list WT (an abbreviation for 'weight') (1) To give the list a name, first move the cursor (using the arrow keys), so that it sits on L1 (or any other column name). (2) Press 2nd [INS] ( DEL ) and the screen should appear as shown. A new list has been created which you can now name. (3) Type in WT* and press ENTER and your new list will be named WT. *At this point the cursor is set to A(LPHA), so you type by finding the green letters above the keys; there is no need to use ALPHA. Step 3. Enter the data Move the cursor down into the list and type in the data values, pressing ENTER after each value. Note how the cursor moves down the list, and the entries are successively numbered at the foot of the screen. Step 4. Call up STAT PLOT Call up STAT PLOT( 2nd [STAT PLOT] ( Y= )) and select Plot 1 by pressing ENTER. Your screen should look something like that shown opposite. Peter Jones and Chris, 2001

15 1.15 Descriptive statistics Step 5. Set up Plot 1 To set up the plot you need to: (1) Turn the plot on. Move the cursor so that it covers On and press ENTER. (2) Move the cursor down to Type: (of display) and then move the cursor sideways to cover the boxplot with outliers. Select by pressing ENTER. (3) Next, move the cursor down to the Xlist: line and type in the letters WT (there is no need to press ALPHA as the cursor is set on "A"). Press ENTER *. (4) Leave the Freq:uency set at 1 and Mark: as a 'box'. * Alternatively, the name can be retrieved from the LIST menu and pasted in. See section A.2. Step 6. Plot To generate the boxplot, we can scale and plot at the same time by using the following key sequence: ZOOM 9 Your plot should be as shown*. * If you find that you have bits and pieces of graphs on your plot, see section B.1, Step 5. Step 7. Determine key values With boxplots you can determine key values in the plot by pressing TRACE. The cursor will jump to a value on the box plot, in this case, the median value (Med=52.5). The horizontal arrow keys then allow you to move the cursor to key points of the plot. For example, pressing the right arrow key moves the cursor to the third quartile and we find that Q3=53. Moving the horizontal arrow keys to move along the plot we find that the: minimum value minx = 50 first quartile Q 1 = 51 the median Med = 52.5 third quartile Q 3 = 53 maximum value x = 54 (excluding outlier) outlier maxx = 59 Peter Jones and Chris Barling, 2001

16 Descriptive statistics 1.16 Note 1: Note 2: The calculator calls any value that lies more than 1.5xIQR above the third quartile (Q3) or 1.5xIQR below the first quartile (Q1) an outlier. This is a commonly used convention. If you choose the second boxplot option in the STAT PLOT menu, the resulting boxplot will not show extreme values as outliers but simply use the maximum and minimum values in the data set as the end points for the whiskers (see below). Exercise 1. The life (in hours) of 11 batteries is 30, 31, 38, 35, 36, 60, 40, 31, 33, 62, 43 (a) (b) Enter the data into a list called BAT. Generate a boxplot and write down the key values. minimum value minx = 30 first quartile Q 1 = 31 the median Med = 36 third quartile Q 3 = 43 maximum value x = 60 (excluding outlier) outlier maxx = 62. E Displaying bivariate data The table shows the life expectancy (in years) and birthrate (per 1000) of people living in 10 countries. life expectancy (years) birthrate (per 1000) E.1 Constructing a scatterplot Task: Construct a scatterplot with life expectancy on the vertical axis (y-axis) and birthrate on the horizontal axis (x-axis). Peter Jones and Chris, 2001

17 1.17 Descriptive statistics Step 1. Enter data into two lists named LIFE (expectancy) and BIRTH(rate)*. *If you are unsure how to do this, see section A.1. Step 2. Call up the STAT PLOT menu Graphical displays of data are created by calling up the STAT PLOT menu ( 2nd [STAT PLOT] ( Y= )). Your screen should look something like that shown opposite. Step 3. Set up scatterplot Select Plot 1 by pressing ENTER. To set up the plot you need to: (1) Turn the plot ON. Move the cursor so that it covers On and press ENTER. (2) Move the cursor down to Type: (of display) and then move the cursor to cover the scatterplot. Select by pressing ENTER. (3) Next, move the cursor down to Xlist: and type in the word BIRTH (or you can 'paste' this list name from the LIST menu*). *See section A.2 for details. (4) Next, move the cursor down to Ylist: and type in the word BIRTH (or you can 'paste' this list name from the LIST menu). Press ENTER. (5) Leave Mark: as a 'box'. Your screen should appear as shown. Step 4. Plot To generate the scatterplot, scale and plot at the same time by using the key sequence ZOOM 9. Your plot should be as shown*. * If you find that you have bits and pieces of graphs on your plot, see section B.1, Step 5. Step 5. Reading values off the scatterplot Individual values can be read off the scatterplot by pressing TRACE and moving the cursor from point to point on the scatterplot using the horizontal arrow keys. The screen opposite shows that the co-ordinates of the highlighted point are x=30 and y=66. Peter Jones and Chris Barling, 2001

18 Descriptive statistics 1.18 F Summarising bivariate data F.1 Determining the equation of the least squares line Step 1. Set up the calculation (1) Return to the STATS menu by pressing STAT, then use the horizontal arrow key to move the cursor to CALC and the down arrow key to option 8: LinReg(a + bx). (2) Press ENTER. This has the effect of pasting the LinReg(a + bx) command onto the Home Screen. (3) To complete the command we need to add the list containing the x-variable, BIRTH (retrieve from the LIST menu), followed by a comma (,) and the name of the list containing the y-variable, LIFE (also retrieve from the LIST menu). LinReg(a+bX) BIRTH, L LIFE L Xlist Ylist Note: The calculator adds a small L before the variable name to indicate that you are working with a list. Note: For experienced users: adding a Y1 to the LinReg(a+bx) command, as shown below, automatically plots the least squares line on the scatterplot. LinReg(a+bX) BIRTH, L LIFE,Y1 L Y1 can be retrieved from the VARS menu as follows: VARS [Y-VARS] (1:Funct ion) ENTER (1:Y1) ENTER. Step 2. Calculate Press ENTER, and the following results are generated. The equation of the least squares line is y = x (coefficients given to 4 significant figures) or, in terms of the variables we are working with, life expectancy = x birthrate Peter Jones and Chris, 2001

19 1.19 Descriptive statistics F.2 Plotting the least squares regression line on a scatterplot Step 1. Enter equation of least squares line into the graph plotter Press Y= and, opposite Y1=, type in X. (This is the right hand side of the equation of the least squares line.) Step 2. Plot the line Press GRAPH and the least squares line will be plotted on the scatterplot. See opposite. F.3 Reading values from the least squares line To read values off the least squares line, press TRACE and use the down arrow key to move the cursor onto the line graph. The diagram shows the cursor at the point X= and Y= on the graph. The cursor can then be moved along the line using the horizontal arrow keys and the appropriate values read from the screen. With the TRACE cursor (x) placed on the least squares line, we can find the value of Y for a particular value of x, say X=35, by simply typing in the value, 35, and then pressing ENTER. From the screen we see that when X=35, Y=54.8. Note: This can only be done for values of X that lie within the graph. If you want to go outside the X range, go to the WINDOW menu and extend the X range. F.4 Determining the value of Pearson's product moment correlation coefficient r The TI-83 Plus does not automatically display the values of the correlation coefficient r when fitting a least squares line to data. To obtain this statistic we have to first adjust the settings of the calculator as follows: Step 0. Adjusting the settings of the calculator (a once-only task) Starting from the Home Screen ( 2nd [QUIT] ( MODE )) press 2nd [CATALOG]( 0 ) to enter the CATALOGUE menu use the down arrow to scroll to DiagnosticOn press ENTER press ENTER again. Note: This will only need to be done again if the memory of your calculator has been cleared. Peter Jones and Chris Barling, 2001

20 Descriptive statistics 1.20 Step 1. Re-enter the LinReg(a+bx) command as follows: LinReg(a+bX) BIRTH, L LIFE L Xlist Ylist (See Section F.1, Step 1) Step 2. Calculate Press ENTER, and the following results are generated. As before, the equation of the least squares line is y = x However, we now also get the values of two further statistics, Pearson's correlation coefficient r r= (correct to 4 significant figures) and the Coefficient of determination r 2 r 2 = (correct to 4 significant figures) Exercise 1. The table shows the life expectancies of males and females in the period 1900 to Life expectancy (years) Year males females a b Construct a scatterplot showing female life expectancies on the vertical axis and male life expectancies on the horizontal axis. Determine: (1) The equation of the least squares line for this data. [y= x] (2) Pearson's correlation coefficient r and the coefficient of determination. [0.9674, ] Plot the least squares line on the scatterplot. c Predict female life expectancy when the male life expectancy reaches 80. (You will need to first rescale the x-axis to include x=80.) [88] Peter Jones and Chris, 2001