page•1 - home | uw college of education...page•10. 0.1 10 100 1,000 10,000 5,000 500 50 5 0.5...
TRANSCRIPT
Page • 1
The Finder Bookfor the Standard Celeration Chart
Owen R. WhiteUniversity of Washington
Autumn 2003
Page • 2
Table of ContentsAn Easy Way to Chart........................................ 3Finding FrequenciesLining Up with the Chart ............................ 4The Record Floor ................................................ 5Correct Frequencies ............................................ 6Error Frequencies................................................ 7The Record Ceiling ............................................ 8Finding Frequencies: Extra Practice ................ 10
Finding Celeration Line Slopes ........................... 11Finding Celerations: Extra Practice........... 12
Finding Percentages ......................................... 13Finding Percentages: Extra Practice.......... 14
Finding the Number of Days Between Dates .. 15Finding the Roots of Numbers......................... 16
Finding the Logarithms of Numbers................ 17Anatomy of a Finder ........................................ 18
Big Sheet of Finders (make a transparencyof this sheet, cut it up, and get 10 finders) ....... 19
More Reading .................................................. 20
See also the “Chart Book” for a description of theStandard Celeration Chart used with the Finderdiscussed in this book.
IntroductionThis book is designed to provide an overview ofthe Finder — a device for finding frequencies,celerations, and more, on the standard celerationchart.
Getting the FinderMany variations of the Finder have beendeveloped. The one described in this book is aconvenient size, and contains sufficient detail tomake fine distinctions in measurements. It is alsoflexible, and can be carried easily in a wallet orpocket. A page of 20 finders (both left- andright-handed versions) is provided at the back ofthis book. Making a transparency of that page,and then cutting it up, will provide aninexpensive source of Finders. More durablefinders made of heavier plastic have also beenmade, and might differ in some details from thefinder described in this book.The original Finder (made of heavy plastic) canbe ordered from:
Behavior Research CompanyBox 3351, Kansas City, KS 66103
Page • 3
An Easy Way toChart
1
50%6070
808590
95969798
99
99.599.699.799.8
99.9
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
x1
2
3
4
5
6
78910
1.5
1.25
2.5
Frequencies
45
20
106
30
LogsL'Celeration
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
The multiply/divide scales of the StandardCeleration Chart enable the use of a very specialdevice — the Frequency Finder.Many of the computations we make that mightotherwise require a calculator can beperformed by the frequency Finderdirectly on the standard chart: finding& plotting frequencies, accuracyratios, celeration (the rate of a learner’sprogress), the number of days between two events,percents, and (for the mathematically minded)even logarithms and roots.
Two versions of the frequency Finder areprovided here — one for left-handed charters(marked with an “L”), and one for right-handedcharters (marked with an “R”). In the examplesshown on the following pages we’ll use the Finderdesigned for right-handed charters. Left-handedcharters actually have it easier, though. When wesay, “flip the Finder” to use the celeration Finder,left-handers can usually ignore that step. Theycan use the Finder without flipping it over.
The background for the Finders shown above hasbeen left blank so you can make usable copies foryourself. Just print or photocopy this page ontotransparency film and cut out the version (left- orright-handed) that will suit your needs.
NOTE: A page of Finders can also be found at theback of this book. Just make a transparency film of thatpage, cut it up, and you’ll have plenty of right- and left-handed Finders to practice the skills covered in thisbook.
Page • 4
Finding Frequencies: Lining UpThe Frequency-Finder Scale allows us to compute frequencies, floors, andceilings directly on the Standard Chart. To illustrate with some “easy” numbers,let’s assume that we conducted an assessment on the 4th Sunday covered by thischart and got the following results:
assessment time = 10 minutes,correct count = 200,error count = 50,total possible count = 1000.
To use the Finder to plot those results:Put the scale on the Finder labeled “frequencies” on the day-line whereyou wish to chart your results (in this case, the 4th Sunday line on thechart), and put the point on the Finder corresponding to the assessmenttime (in this case, “10") on top of the “1" line of the chart.NOTE: After you have the Finder lined-up with the proper day-line and theassessment time (on the Finder) lined-up with the 1-line of the chart, DO NOTMOVE THE FINDER AGAIN until all your plotting for that day is completed.
(continued next page)
1
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
4th Sunday Line (the day on which these data should be charted)
Page • 5
Page • 6
Page • 7
4
Page • 8
5
Page • 9
The Frequency-Finder Scale allows us tocompute frequencies, floors, and ceilingsdirectly on the Standard Chart. Let’s seewhat happened with our “easy” numberswhen we wanted to chart our data on the 4thSunday line:
assessment time = 10 minutes,total possible count = 1000,
correct count = 200,error count = 50.
Ceiling = (total possible count) ÷ (assessment time in minutes) = 1000 ÷ 10 = 100
Correct Frequency = (correct count) ÷ (assessment time in minutes) = 200 ÷ 10 = 20
Error Frequency = (error count) ÷ (assessment time in minutes) = 50 ÷ 10 = 5
Record Floor = 1 ÷ (assessment time in minutes) = 1 ÷ 10 = 0.1
The Frequency-Finder HasConverted Your Counts & Time
Into Frequencies!
Page • 10
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
Friday20 Minutes200 Correct6 Errors
500 Possible
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
Monday30 Seconds (0.5 Min.)
15 Correct2 Errors
50 Possible
Note: when the timing isless than a minute, you canuse fractional minutes on the“regular” number scale, orthe “seconds” scale in themiddle of the finder to locate
the record floor.
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
Wednesday120 Minutes35 CorrectNo ErrorsNo Ceiling
Note: since there are “nocount” errors, the error markis charted as a “?” just below
the record floor.
?
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
Friday60 Minutes50 Correct100 ErrorsNo Ceiling
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
Sunday270 Minutes35 Correct4 Errors
200 Possible
Find the assessment time (in minutes) on the finder’s Frequency scale, place it on the “1" line of the Chart next to the properday; make a dashed line next the the “1" on the Finder, a “dot” next to the correct count on the Finder, an “x” next to the errorcount on the Finder, and another dash next to the total possible count on the Finder (if there is a total possible).
Finding Frequencies: Extra Practice
Page • 11
x1'Celeration
2
3
4
5
6
78910
1.5
1.25
2.5
Logs
(1) Flip the Finderso the ‘Celeration
Scale is on the right-hand side; make surethe finder is straightup-and-down on the
chart.
(2) Line up the “1"on the left-hand (nowthe Frequency side)of the Finder touchesthe ‘celeration line forwhich you want tofind the slope
(3) Read theslope of the‘celeration line atthe point where itcrosses the right-hand side of theFinder (now the‘celeration side).Here, the slopeis about x1.80
X1.80
(4) Label theanswer as “x”(“times”), sincethe line is goingup. In thiscase, the line ismultiplying by afactor of x1.80per week.
If the ‘celeration line is going down,flipthe finder upside-down, read theanswer, and label the answer as “÷”(“divideby”). In this case, the slope isgoing down by a factor of approximately÷2.5
÷2.50
Finding Celeration Line Slopes
Page • 12
Finding Celeration Line Slopes: Extra PracticeFind and label each of the celeration lines shown below. A Finder has already been correctly placed on each set of slopes.
The answers are provided at the bottom of the chart.
x1'Celeration
2
3
4
5
6
78910
1.5
1.25
2.5
Logs
(F)
(E)
(D)
(C)
(B)(A)
Accelerating Celeration Line Slopes: (A) x1.25;(B) x1.50; (C) x2.0; (D) x2.60; (E) x3.70 (F) x7.00
(G)
(H)(I)
(J)
(K)(L)
Decelerating Celeration Line Slopes:(G) ÷1.00 (flat line, no slope); (H) ÷1.3;(I) ÷1.45; (J) ÷1.80; (K) ÷2.7 (L) ÷4.0
Page • 13
Finding PercentagesAlthough percentages are generally more confusing that ratio (“x”-times; “÷”-divide) statements, they are very
commonly used, and it sometimes helps to convert correct and error frequencies to percentage-accuracy statementsto communicate with some people. Fortunately, since the Standard Celeration Chart displays the relationships
among frequencies as ratios, conversion to percentages is very simple.
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
(1) Use the“% of Total”
scale found inthe middle ofthe Finder.
(2) Put the “50%” marker on the percent scale next to the lowestof the two frequencies on the day in question — in this case, theerror frequency.
(3) Read the percentage at the point on the scale where thehigher of the two frequencies falls — in this case, 90%. In thiscase, since the correct frequency was the higher of the two, theanswer is “90% correct.” If the error frequency had been higher,the answer would have been “90% error” (or, by subtracting 90%from 100%, we could just as well say “10% correct”).
Special Notes:• If the correct and error frequencies are both on or above
the record floor, the % correct (or error) can never reach100%, since both frequencies are non-zero.
• If one frequency is on or above the record floor, and theother is below the record floor, the percentage of the oneon or above the floor is automatically 100% (all of the total).
• If both of the frequencies are below the record floor,percentages are undefined, and should be noted as “?%”.
Page • 14
Finding Percentages: Extra PracticeFor each of the frequency pairs shown above, determine the percent of correct behavior. Check your answerswith the numbers provided along the top of the chart. Note that if the error frequency is higher than the correctfrequency, the number shown on the finder will be the percentage of error behavior, and you’ll have to subtractfrom 100% to find the percentage of error behavior. If the two frequencies fall on top on one another, thepercent correct is automatically 50%; if one (but not both) of the frequencies is below the record floor, thepercentage of the frequency above the record floor is automatically 100%; and if both frequencies are belowthe record floor, percentages are essentially undefined (i.e., “?%”). If you don’t have a clear plastic finder, mark
the distance between the correct and error frequencies on the edge of a piece of paper, then hold thosefrequencies next the the % scale on the finder shown on this page to determine the answer.
?
??
90%correct
≈95%correct
50%correct
≈66%correct
80%correct
≈98%correct
≈42%correct (note:the error
frequency ishighest)
100%correct
?% correct(note: bothcorrects &errors arebelow thefloor)
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
45
20
106
30
R
50%6070
808590
95969798
99
99.599.699.799.8
99.9
1
Page • 15
Finding the Number of DaysBetween Two Dates
Part of the Successive Calendar Days scale of the chart is reproduced down the middle of the Finder. Todetermine the number of successive calendar days that fall between two points on the chart, line up the “0" of
that scale with the first point, then read the answer where it falls on the right of that scale.
10 20 30 40 50CalendarDays
% of Total
0
(1) Use the “Calendar Days” scale in themiddle of the Finder
(2) Line up the “0"on the CalendarDays scale withthe first date ofinterest.
(3) Read the answer at the point where thesecond date of interest crosses the Calendar Daysscale. In this case, 39 calendar days have passedsince this program began.
Page • 16
Finding the Numerical Roots of NumbersSince the Standard Celeration Chart uses a ratio scale, a straight line drawn at an angle across the chart
changes values in multiples, much like “compound interest.” To find the “nth root” of a number, simply draw astraight line on the cart from the “zeroth day” at a “frequency=1" line to the frequency line representing thenumber for which you want the root, on the Sunday line that represents the “nth” root you wish to find; then
read the answer as the slope of the line you drew (see pages 11 & 12 for a review of how to find slopes). If theslope is down (that is, to a fractional number), the root will equal the inverse of the divideby slope
(for example, a slope of ÷2.0 = 1/2.0 = 0.50)
Line drawn from the 0th Sunday @ 1/minute to the 2nd Sunday@ 4/minute. The slope of that line is x2.0 (times 2),
so the 2nd root (square root) of 4 is 2 (that’s pretty simple).
Line drawn from the 0th Sunday @ 1/minute to the 12th Sunday@ 50/minute.
The slope of that line isapproximately x1.38 (times 1.38),
so the 12th root of 50 is roughly 1.38
Page • 17
Finding the Logarithms of NumbersUnderlying the ratio scale of the Standard Celeration Chart is an equal-interval logarithmic scale.Whyshould you care? Well, if you ever want to manipulate your frequencies to perform summary analyses(like calculating the mean or standard deviation of a set of data), or to run statistical tests (like t-tests),you must convert your frequencies to logarithms first, or the results of your analyses will not accuratelyreflect what you see on the chart. Most statistical programs and spreadsheets have “functions” that canconvert your frequencies into logarithms for you, but if you find yourself away from such aids, it helps to
be able to at least approximate logarithmic transformations with the Finder
What are logarithms? Common logarithms (or “logs” for short) ask the question: 10 raised to whatpower will equal the number you want? That power is the log of the number. For example, 102 = 100,and 103 = 1000, so the log of 100 = 2, and the log of 100 = 3. By convention, any number raised to the
“zeroth{ (0th) power equals one, so the log of 1 = 100 = 1. The logs of numbers less than 1 arenegative — the log of 0.1 = -1; the log of 0.001 = -3.
It might be easier to remember how to find the integer portion of a log by noting that it is equal to thenumber of decimal places you would move away from 1.0 in order to make the number in question. Forexample, to make 1 into 1000, we must move the decimal 3 places to the right (a “positive” 3 places,since the number gets bigger), so the log of 1000 is 3; and to make 1 into 0.01 we must move the
decimal 2 places to the left (a “negative” number, since we’re making the number smaller), so the log of0.01 is -2.
The integer portion of a log is called it’s characteristic. All multiples of 10 will have a simple log that isan integer multiple of 1. Those values are shown next to their equivalents on the scale on the left side
of this chart.
If a number falls somewhere between two multiples of 10, the log will have a fractional part called themantissa. The mantissa of a log can be found by placing the “log” scale on the finder between the twomultiples of 10 surrounding the number in question. Of course, using the finder to estimate logs is notas precise as using a calculator or statistical program, but in a pinch, it will serve most of your needs.
0.0
1.0
2.0
3.0
-1.0
-2.0
-3.0
The integer portion of the log is called the characteristic, and it reflects the number of “cycles” (multiples of 10)above (a positive number) or below (a negative number) the number “1" you must move to find the number inquestion.
Logs
The fractional portion of the log is called the mantissa, and is found by using the “log” scale of the finder. Placethat scale on the cycle of the chart containing the number in question, and add the decimal portion to the lower ofthe two characteristics. In this case, the log of 50 is approximately 1.7 (perhaps a smidgen higher).
The “log” scale of the Finder.
Page • 18
0.1
10
100
1,000
10,000
5,000
500
50
5
0.5
Frequencies
1
Anatomy of a Finder
First, note that the Frequency scale of theFinder is the same size as the Frequencyscale of the chart. By folding back a chart soonly the Frequency scale shows, and makinga solid arrow at the “1" line, you can duplicatethe Frequency scale very easily.
x1'Celeration
2
3
4
5
6
78910
1.5
1.25
2.5
If you ever find yourself without your trusty finder, you can at least approximate the two major parts of the finder (theFrequency scale and the Celeration Scale) using a copy of the chart. To make sure you never run out of Finders,though, the following page, copied onto a transparency and cut-up, will give you 10 Finders at a time!
The Finder is 4-weeks wide
The‘CelerationScale is 4-times biggerthan a singlecycle on thechart (thatallows muchmore detail tobe drawn onthe scale).
Projecting lines through the major frequencylines where they cross the 1st week
... out to the 4th week
... tells you where to mark the major parts ofthe Celeration scale.
“Interpolate” points between those majormarkings as best you can.
Of course, the Calendar Scale at the bottom ofthe chart can be used just as it is to find the timebetween any two days on the chart.
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
10 20 30 40 50CalendarDays
% of Total
0
Page • 20
More ReadingThe following books provide additional information about the Standard Behavior chart and charting techniques. Theyare only available through “nontraditional” sources, but are well worth the effort to obtain.
Pennypacker, H.S., Gutierrez, A., & Lindsley, O.R. (2002). Handbook of the Standard Celeration Chart.Gainesville, FL: Xerographics, Inc. (email [email protected]; phone 352 375-0797).This book is an updated version of the first Standard Charting Book. It presents all the basicconventions for charting in an easy-to-understand manner, including the use of a version of thefrequency/celeration finder that is somewhat different in construction from the one discussed in thethis FinderBook.
Graf, Steve, and Lindsley, Og. (2002) Standard Celeration Charting 2002. Poland, Ohio: GrafImplements (7779 Lee Run Rd, Portland OH 44514-2510)
This book provides far more information about the details of the chart and related information. It alsoprovides numerous “practice sheets” for developing skill and fluency in evaluating chartedinformation.
White, Owen R. (2003) The Chart Book. Available for downloading as a PDF file from a class web sitemaintained by Owen White: http://courses.washington.edu/edspe510/ Just click on the “Readings”tab. The downloading link will appear a bit down the list of those readings.
This book provides an overview of the Standard Celeration chart and the conventions for its use.