Page The Finder Book for the Standard Celeration Chart Owen R. White University of Washington 22
Page 2 Introduction This book is designed to provide an overview of the Finder a device for finding frequencies, celerations, and more, on the standard celeration chart. Getting the Finder Many variations of the Finder have been developed. The one described in this book is a convenient size, and contains sufficient detail to make fine distinctions in measurements. It is also flexible, and can be carried easily in a wallet or pocket. A page of 2 finders (both left- and right-handed versions) is provided at the back of this book. Making a transparency of that page, and then cutting it up, will provide an inexpensive source of Finders. More durable finders made of heavier plastic have also been made, and might differ in some details from the finder described in this book. A new heavy duty acrylic Finder, called the "Easy Charter", can be ordered from: Behavior Research Company Box 33, Kansas City, KS 3 ZZZ EHKDYLRUUHVHDUFKFRPSDQ\ FRP Table of Contents An Easy Way to Chart... 3 Finding Lining Up with the Chart... 4 The Record Floor... Correct... Error... 7 The Record Ceiling... 8 Finding : Extra Practice... Finding Celeration Line Slopes... Finding Celerations: Extra Practice... 2 Finding Percentages... 3 Finding Percentages: Extra Practice... 4 Finding the Number of Calendar Days... Finding the Roots of Numbers... Finding the Logarithms of Numbers... 7 Anatomy of a Finder... 8 Big Sheet of Finders (make a transparency of this sheet, cut it up, and get finders)... 9 More Reading... 2 See also the Chart Book for a description of the Standard Celeration Chart used with the Finder discussed in this book.
, 99.9, 99.8 99.7 99. 99. An Easy Way to Chart The multiply/divide scales of the Standard Celeration Chart enable the use of a very special device the Frequency Finder. Many of the computations we make that might otherwise require a calculator can be performed by the frequency Finder directly on the standard chart: finding & plotting frequencies, accuracy ratios, celeration (the rate of a learner s progress), the number of event days, percents, and (for the mathematically minded) even logarithms and roots. Two versions of the frequency Finder are provided here one for left-handed charters (marked with an L ), and one for right-handed charters (marked with an R ). In the examples shown on the following pages we ll use the Finder designed for right-handed charters. Left-handed charters actually have it easier, though. When we say, flip the Finder to use the celeration Finder, left-handers can usually ignore that step. They can use the Finder without flipping it over. 9 8 99.9, 99.8, 99.7 99. 99. 7, 99 99 98 98 97 9 9 4 97 9 9 9 3 9 8 8 2. 8 7 2 8 7, % %..2. 4 3 2. x 'Celeration L Logs R 4 3 2.. The background for the Finders shown above has been left blank so you can make usable copies for yourself. Just print or photocopy this page onto transparency film and cut out the version (left- or right-handed) that will suit your needs. NOTE: A page of Finders can also be found at the back of this book. Just make a transparency film of that page, cut it up, and you ll have plenty of right- and lefthanded Finders to practice the skills covered in this book. Page
Page 4 4th Sunday Line (the day on which these data should be charted) 99.9, 99.8, 99.7 99. 99. 99 98, 97 9 9 9 Finding : The Frequency-Finder Scale allows us to compute frequencies, floors, and ceilings 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 this chart and got the following results: assessment time = minutes, correct count = 2, error count =, total possible count =. 8 8 7 To use the Finder to plot those results: % R 4 3 2 Lining Up.. Put the scale on the Finder labeled frequencies on the day-line where you wish to chart your results (in this case, the 4th Sunday line on the chart), and put the point on the Finder corresponding to the assessment time (in this case, ") on top of the " line of the chart. NOTE: After you have the Finder lined-up with the proper day-line and the assessment time (on the Finder) lined-up with the -line of the chart, DO NOT MOVE THE FINDER AGAIN until all your plotting for that day is completed. (continued next page)
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Page 9 The Frequency-Finder Has Converted Your Counts & Time Into! The Frequency-Finder Scale allows us to compute frequencies, floors, and ceilings directly on the Standard Chart. Let s see what happened with our easy numbers when we wanted to chart our data on the 4th Sunday line: assessment time = minutes, total possible count =, correct count = 2, error count =. Record Floor = (assessment time in minutes) = =. Error Frequency = (error count) (assessment time in minutes) = = Correct Frequency = (correct count) (assessment time in minutes) = 2 = 2 Ceiling = (total possible count) (assessment time in minutes) = =
99.9, 99.8 Friday 2 Minutes 2 Correct Errors Possible 99.7 99. 99. 99 98 99.8, 97 9 9 99.9,, 9 99.7 99. 99. Page, 8 Wednesday 2 Minutes 3 Correct No Errors No Ceiling Note: since there are no count errors, the error mark is charted as a? just below the record floor. 99 98 7 % 97 9 9 9 99 98 7 R, 99.7 99. 99. 8 4 3 2 99.8.. R 4 3 2 99 9 98, 99.9, 9 99 9 % R.. Finding : Extra Practice 7 4 3 2 8 8, 97 9 9 7 7 Monday 3 Seconds (. Min.) Correct 2 Errors Possible 98 8, 99.7 99. 99. 8 8 % 99.8, 97 9 9 8. Note: when the timing is less than a minute, you can use fractional minutes on the regular number scale, or the seconds scale in the middle of the finder to locate the record floor., 97 9 9. Sunday 27 Minutes 3 Correct 4 Errors 2 Possible 99.7 99. 99. 8 % 99.8 99.9, 99.9, 8, Friday Minutes Correct Errors No Ceiling %? R 4 3 2.. R 4 3 2.. Find the assessment time (in minutes) on the finder s Frequency scale, place it on the " line of the Chart next to the proper day; make a dashed line next the the " on the Finder, a dot next to the correct count on the Finder, an x next to the error count on the Finder, and another dash next to the total possible count on the Finder (if there is a total possible).
9 8 7 4 (2) Line up the " (4) Label the answer as x ( times ), since the line is going up. In this case, the line is multiplying by a factor of x.8 per week. If the celeration line is going down,flip the finder upside-down, read the answer, and label the answer as ( divide by ). In this case, the slope is going down by a factor of approximately 2. 3 on the left-hand (now the Frequency side) of the Finder touches the celeration line for which you want to find the slope 2. X.8 2..2 x 'Celeration Logs () Flip the Finder (3) Read the slope of the celeration line at the point where it crosses the righthand side of the Finder (now the celeration side). Here, the slope is about x.8 so the Celeration Scale is on the righthand side; make sure the finder is straight up-and-down on the chart. Finding Celeration Line Slopes 2. Page
Page 2 Finding Celeration Line Slopes: Extra Practice Find 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. 9 8 (F) 7 (E) 4 (G) (D) 3 2. 2. (C) (H) (I) (B) (A) (J).2 x (K) 'Celeration (L) Logs Accelerating Celeration Line Slopes: (A) x.2; (B) x.; (C) x2.; (D) x2.; (E) x3.7 (F) x7. Decelerating Celeration Line Slopes: (G). (flat line, no slope); (H).3; (I).4; (J).8; (K) 2.7 (L) 4.
Page 3 99.9, 99.8, 99.7 99. 99. 99 98, 97 9 9 () Use the % of Total scale found in the middle of the Finder. 9 8 (3) Read the percentage at the point on the scale where the higher of the two frequencies falls in this case, 9%. In this case, since the correct frequency was the higher of the two, the answer is 9% correct. If the error frequency had been higher, the answer would have been 9% error (or, by subtracting 9% from %, we could just as well say % correct ). 8 7 (2) Put the % marker on the percent scale next to the lowest % of the two frequencies on the day in question in this case, the error frequency. Special Notes: R 4 3 2.. If the correct and error frequencies are both on or above the record floor, the % correct (or error) can never reach %, since both frequencies are non-zero. If one frequency is on or above the record floor, and the other is below the record floor, the percentage of the one on or above the floor is automatically % (all of the total). If both of the frequencies are below the record floor, percentages are undefined, and should be noted as?%. Finding Percentages Although 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 statements to communicate with some people. Fortunately, since the Standard Celeration Chart displays the relationships among frequencies as ratios, conversion to percentages is very simple.
?% correct 42% correct (note: 9% correct 9% correct % correct % correct 8% correct 98% correct the error frequency is highest) % correct (note: both corrects & errors are below the floor) 99.9, 99.8, 99.7 99. 99. 99 98, 97 9 9 9 8 8 7 %??? R 4 3 2.. Finding Percentages: Extra Practice For each of the frequency pairs shown above, determine the percent of correct behavior. Check your answers with the numbers provided along the top of the chart. Note that if the error frequency is higher than the correct frequency, the number shown on the finder will be the percentage of error behavior, and you ll have to subtract from % to find the percentage of error behavior. If the two frequencies fall on top on one another, the percent correct is automatically %; if one (but not both) of the frequencies is below the record floor, the percentage of the frequency above the record floor is automatically %; and if both frequencies are below the 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 those frequencies next the the % scale on the finder shown on this page to determine the answer. Page 4
Page Finding the Number of Calendar Days of an Event Part of the Successive Calendar Days scale of the chart is reproduced down the middle of the Finder. To determine the number of successive calendar days that fall between two points on the chart, line up the " of that scale with the calendar day before the first point, then read the answer where it falls on the right of that scale. (3) Read the answer at the point where the second date of interest crosses the Calendar Days scale. In this case, 39 calendar days have passed since this program began. (2) Line up the first t i c k a f t e r t h e " on the Calendar Days scale with the first date of interest. % of Total 2 3 4 Calendar Days () Use the Calendar Days scale in the middle of the Finder Updated by Malcolm Neely 22
Page Line drawn from the th Sunday @ /minute to the 2nd Sunday @ 4/minute. The slope of that line is x2. (times 2), so the 2nd root (square root) of 4 is 2 (that s pretty simple). Line drawn from the th Sunday @ /minute to the 2th Sunday @ /minute. The slope of that line is approximately x.38 (times.38), so the 2th root of is roughly.38 Finding the Numerical Roots of Numbers Since 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 a straight line on the cart from the zeroth day at a frequency=" line to the frequency line representing the number 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 & 2 for a review of how to find slopes). If the slope is down (that is, to a fractional number), the root will equal the inverse of the divideby slope (for example, a slope of 2. = /2. =.)
Page 7 3. The integer portion of the log is called the characteristic, and it reflects the number of cycles (multiples of ) above (a positive number) or below (a negative number) the number " you must move to find the number in question. The fractional portion of the log is called the mantissa, and is found by using the log scale of the finder. Place that scale on the cycle of the chart containing the number in question, and add the decimal portion to the lower of the two characteristics. In this case, the log of is approximately.7 (perhaps a smidgen higher). Finding the Logarithms of Numbers 2... Logs Underlying the ratio scale of the Standard Celeration Chart is an equal-interval logarithmic scale. Why should 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 accurately reflect what you see on the chart. Most statistical programs and spreadsheets have functions that can convert 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: raised to what power will equal the number you want? That power is the log of the number. For example, 2 =, and 3 =, so the log of = 2, and the log of = 3. By convention, any number raised to the zeroth{ (th) power equals one, so the log of = =. The logs of numbers less than are negative the log of. = -; the log of. = -3. -. It might be easier to remember how to find the integer portion of a log by noting that it is equal to the number of decimal places you would move away from. in order to make the number in question. For example, to make into, we must move the decimal 3 places to the right (a positive 3 places, since the number gets bigger), so the log of is 3; and to make into. we must move the decimal 2 places to the left (a negative number, since we re making the number smaller), so the log of. is -2. -2. The integer portion of a log is called it s characteristic. All multiples of will have a simple log that is an integer multiple of. Those values are shown next to their equivalents on the scale on the left side of this chart. -3. If a number falls somewhere between two multiples of, the log will have a fractional part called the mantissa. The mantissa of a log can be found by placing the log scale on the finder between the two multiples of surrounding the number in question. Of course, using the finder to estimate logs is not as precise as using a calculator or statistical program, but in a pinch, it will serve most of your needs. The log scale of the Finder.
, The Finder is 4-weeks wide Page 8,, First, note that the Frequency scale of the Finder is the same size as the Frequency scale of the chart. By folding back a chart so only the Frequency scale shows, and making a solid arrow at the " line, you can duplicate the Frequency scale very easily. 9 8 7 4 Projecting lines through the major frequency lines where they cross the st week... out to the 4th week.... tells you where to mark the major parts of the Celeration scale. Interpolate points between those major markings as best you can. 3 2. 2..2. Of course, the Calendar Scale at the bottom of the chart can be used just as it is to find the time between any two days on the chart. x 'Celeration Anatomy of a Finder If you ever find yourself without your trusty finder, you can at least approximate the two major parts of the finder (the Frequency 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 Finders at a time! The Celeration Scale is 4times bigger than a single cycle on the chart (that allows much more detail to be drawn on the scale).
% of Total % of Total 2 3 4 Calendar Days 2 3 4 Calendar Days % of Total % of Total 2 3 4 Calendar Days 2 3 4 Calendar Days % of Total % of Total 2 3 4 Calendar Days 2 3 4 Calendar Days % of Total % of Total 2 3 4 Calendar Days 2 3 4 Calendar Days % of Total % of Total 2 3 4 Calendar Days 2 3 4 Calendar Days
More Reading Page 2 The following books provide additional information about the Standard Behavior chart and charting techniques. They are only available through nontraditional sources, but are well worth the effort to obtain.. Chart Pennypacker, H.S., Gutierrez, A., & Lindsley, O.R. (22). Handbook of the Standard Celeration Cambridge, MA: Cambridge Center for Behavioral Studies (www.behavior.org) This book is an updated version of the first Standard Charting Book. It presents all the basic conventions for charting in an easy-to-understand manner, including the use of a version of the frequency/celeration finder that is somewhat different in construction from the one discussed in this book, (The Finder Book). Graf, Steve, and Lindsley, Og. (22) Standard Celeration Charting 22. Poland, Ohio: Graf Implements ( Order from Behavior Development Solutions at www.behaviordevelpmentsolutions.com/products/ 22-standard-celeration-charting-22.aspx") This book provides far more information about the details of the chart and related information. It also provides numerous practice sheets for developing skill and fluency in evaluating charted information. Kubina, R.M. and Yurich, K.L. (22) The Precision Teaching Book. Lemont, PA: Greatness Achieved Publishing Company. (Order from www.theprecisionteachingbook.com) This book blends the more formal language of Behavioral Analysis with the more common language of Precision Teaching. It is a guide for aspiring teachers and those who help others. It is a guide to make discoveries of human behavior and change, and an excellent teaching reference. (It includes a template for making a set of Dr. White's rate finders.) White, Owen R, and Neely, Malcolm D. (22) The Chart Book. Available for downloading as a PDF file from the University of Washington COE website. (http://education.washington.edu/areas/edspe/white/precision/readings/chartbook.pdf) This book provides an overview of the Standard Celeration chart and the conventions for its use.