A.6 Calculating and summarizing data

What the outline says

Task A.6 in content area A, Data Collection and Graphing (13 of the 75 scored questions): "Calculate and summarize data in different ways (e.g., rate, mean duration, percentage)."

Count and rate

Count is the number of times a behavior occurred. Rate is count divided by the time observed, such as responses per minute or per hour. Rate is the better summary when sessions differ in length, because a longer session gives more chances for the behavior to occur.

• 24 hand raises in 30 minutes: 24 / 30 = 0.8 per minute.
• 15 words read in 90 seconds: 90 seconds is 1.5 minutes, so 15 / 1.5 = 10 per minute.
• To convert per minute to per hour, multiply by 60: 0.5 per minute = 30 per hour.

Use only the time you actually observed. If a 60-minute session included a 15-minute break with no data collection, 18 responses gives 18 / 45 = 0.4 per minute. To combine sessions, add all the counts and divide by all the time: 4 elopements in 2 hours plus 9 in 3 hours is 13 / 5 = 2.6 per hour. Averaging the two session rates (2 and 3 per hour) would give 2.5, which gives the shorter session too much weight.

Rate fits behavior that can happen at any time, such as greetings at recess. Percentage needs a set number of trials or opportunities.

Duration

• Total duration: add the length of every episode. Episodes of 2:30, 1:45 and 3:15 add to 6 minutes plus 90 seconds, which is 7 minutes 30 seconds.
• Mean duration: total duration divided by the number of episodes. Tantrums of 2, 4 and 9 minutes total 15 minutes; 15 / 3 = 5 minutes per tantrum.
• Percentage of time: total duration divided by session length, times 100. On-task episodes of 6, 10 and 8 minutes in a 40-minute class total 24 minutes; 24 / 40 x 100 = 60%.

Convert to one unit before you add. Episodes of 45 seconds, 1 minute 15 seconds and 30 seconds are 45 + 75 + 30 = 150 seconds, and the mean is 150 / 3 = 50 seconds.

Latency and interresponse time

Mean latency is the sum of the latencies divided by the number of trials. Latencies of 3, 5, 10 and 6 seconds total 24, so the mean is 6 seconds. A lower mean latency means the client starts sooner after the instruction.

Interresponse time (IRT) is the time between one response and the next. Four responses produce three IRTs, not four. Responses at minutes 2, 5, 9 and 11 give IRTs of 3, 4 and 2 minutes; the mean is 9 / 3 = 3 minutes. Dividing by the number of responses is a common error.

Percentages

A percentage is the part divided by the whole, times 100. The whole depends on what was measured:

• Percentage correct: correct trials / total trials. 14 of 20 = 70%. With 5 errors in 20 trials, 15 are correct: 75%.
• Percentage independent: count only what the plan counts. On a sheet reading + + − P + − + + P +, there are 6 independent correct out of 10, so 60%; prompted trials are not counted as correct if the plan says so.
• Percentage of task-analysis steps: 6 of 8 steps independent = 75%.
• Percentage of opportunities: 12 independent requests out of 15 opportunities = 80%.
• Percentage of intervals: scored intervals / total intervals. Work out the number of intervals first if needed: a 10-minute observation in 15-second intervals has 600 / 15 = 40 intervals, so 30 scored intervals is 75%.

Percentage of intervals is not percentage of time. In partial-interval recording, an interval counts if the behavior occurs at any point in it, so 50% of intervals means the behavior occurred at some point in half of them. Partial-interval recording tends to overestimate how much of the time behavior occurs.

To combine sessions, pool the trials. 6 of 10 in the morning and 18 of 20 in the afternoon is 24 of 30, which is 80%. Averaging 60% and 90% gives 75%, which is wrong because the sessions had different numbers of trials. Converting to percentage is what lets sessions with different numbers of trials be compared.

Mean, median and range

• Mean: add the values and divide by how many there are. Daily counts of 3, 0, 5, 2 and 4 total 14; 14 / 5 = 2.8. A zero day is still a day, so divide by 5, not 4.
• Median: put the values in order and take the middle one. For 9, 3, 12, 5, 6, the ordered list is 3, 5, 6, 9, 12, so the median is 6. With an even number of values, take the mean of the middle two.
• Range: highest minus lowest. For 7, 2, 12, 4, 5, the range is 12 − 2 = 10.

Trials to criterion

Trials to criterion is the number of trials (or sessions) a client needed to meet the mastery criterion. Fewer trials means the target was learned with less practice. When a criterion requires consecutive sessions, check that the streak was not broken: with a criterion of 90% or higher for 3 consecutive sessions and scores of 70, 80, 90, 95, 85, 90, 100, 90, the 85% in session 5 resets the count, so the criterion is met in session 8.

How this connects to other tasks

These summaries come from the measurement procedures in A.1 to A.3, and they are what you enter on graphs (A.4) and read for trends (A.7). The supervisor chooses which summary to use for each target; your part is to calculate it correctly and report it accurately.

Common mistakes

• Dividing time by count instead of count by time when calculating rate.
• Using the whole session length for rate when part of the session was not observed.
• Reporting total duration when the question asks for mean duration, or the reverse.
• Dividing by the number of responses instead of the number of gaps when finding mean IRT.
• Averaging percentages or rates from sessions of different sizes instead of pooling the raw numbers.
• Finding the median without first putting the values in order.
• Reading percentage of intervals as percentage of time.

How it's tested

Calculation items give all the numbers you need. The wrong options are usually the results of common slips: the total instead of the mean, the median or range instead of the mean, the inverse of a rate, or the complement of a percentage. Write out the formula, convert to one unit, and check that your answer is a sensible size before choosing.