Science Practice 3: Data Interpretation

Original hypothetical retrieval experiment: researchers ask whether practicing recall improves one-week vocabulary memory relative to rereading. Eighty adult volunteers aged 18–24 respond to an advertisement at one university. A computer randomly assigns 40 to retrieval practice and 40 to rereading. Everyone first studies the same 24 unfamiliar word–meaning pairs for four minutes. For the next 12 minutes, retrieval participants see word cues, try to produce meanings, and then receive the correct answer; rereading participants see the complete pairs repeatedly. Both groups work in the same quiet room with identical materials and time. One week later, everyone writes meanings for all 24 cues without notes, with one point for each correct meaning (0–24). A scorer who does not know condition uses a predefined answer key. All 80 complete the test. Mean recall is 18/24 after retrieval and 14/24 after rereading. These are invented study summaries, not actual participants or a published finding.

The display below compares two assigned practice conditions, not two naturally different types of people. The horizontal categories are retrieval and rereading. The vertical axis is mean correct meanings on the same 24-item cued-recall test one week later. The height 18 is a group mean, not a claim that every retrieval participant scored 18 or that the group improved by 18 points.

The psychological concept is retention supported by retrieval practice: participants attempt to recall meanings during learning instead of only seeing them again. The statistical concept shown is a difference in group means. The operational outcome is one-week cued recall, not general intelligence, all academic achievement or the learner’s confidence. Read the design and outcome together to decide which concept the display represents.

Subtracting 14 from 18 gives a four-meaning average difference in this hypothetical sample. The common 0–24 scale makes that comparison meaningful. The chart includes no error bars or individual scores, so it does not reveal the overlap or spread within groups. It also cannot independently establish a p-value or tell us the effect on other tasks. The later inferential example explicitly supplies an analysis summary rather than deriving significance from bar height.

A shape alone rarely identifies a psychological process. Context describes how observations were produced. Suppose a hypothetical observer watches four adult volunteers repeatedly hear a harmless brief tone, at the same volume, every minute. Mean startle-response counts in successive four-trial blocks are 3, 2, 1 and 1. A declining response to repeated stimulation is consistent with habituation, a decrease in responding as a stimulus becomes familiar. The pattern alone does not establish its cause.

To identify the variables, state repetition block as the ordered condition and observed startle responses per four trials as the outcome. To identify the concept, connect repeated harmless stimulation with a decrease in response. Fatigue or differences in coding could also affect the counts. The procedure and pattern support a teaching application of habituation, while additional controls would be needed to isolate the cause.

Qualitative data can also depict a psychological idea. If a structured interview asks the same adult learners how they approached a hard task, descriptions such as ‘I broke it into smaller steps’ concern problem-solving strategies, not a numerical score of ability. Explain what the account describes and how it fits the concept; do not invent a scale from a phrase or diagnose a trait from one quotation.

Separate original teaching dataset: eight adult volunteers study the same 24 word–meaning pairs for four minutes, then complete the same 24-point cued-recall test after 20 minutes. Their scores, sorted from lowest to highest, are 8, 10, 10, 12, 14, 14, 16 and 20 correct meanings. There is no assigned comparison condition here. These eight values are not the individual records behind the 80-person experiment’s bar chart or its inferential summary.

Mean: add all scores (8 + 10 + 10 + 12 + 14 + 14 + 16 + 20 = 104) and divide by eight, giving 13 correct meanings. Median: with an even number of ordered scores, average the fourth and fifth values, (12 + 14) / 2 = 13. Mode: 10 and 14 each occur twice, more often than other values, so the dataset has two modes. Range: subtract the minimum from the maximum, 20 − 8 = 12 correct meanings.

Interpret those results in this task: the mean and middle location are 13 correct meanings, the most frequent observed scores are 10 and 14, and the full observed span is 12 meanings. None is the score that every learner obtained. The mean and median happen to be equal; that does not prove a normal distribution, especially in this small discrete sample.

The frequency bars below show how many of these eight people obtained each score. They are different from the previous graph, whose heights were group means. The axis label and the full raw list prevent that confusion. Calculate from individual scores or a frequency-weighted list, not by averaging the six distinct score labels without accounting for repeated values.

Standard deviation describes dispersion around the mean in the same units as the measurements. Two groups can have the same mean while one has tightly clustered scores and another varies widely. Range uses only the extremes; standard deviation uses deviations across all scores. Neither tells us which intervention caused the spread.

Supplemental calculation, with an explicitly chosen convention: treat the eight recall values as the entire descriptive set, using a population standard deviation. Their deviations from the mean of 13 are −5, −3, −3, −1, 1, 1, 3 and 7; their squared deviations sum to 104. Divide by eight to obtain variance 13, then take the square root: standard deviation ≈ 3.61 correct meanings. If estimating a wider population with the sample convention, divide by n − 1 = 7 instead, giving a sample SD ≈ 3.85. This calculation support is not a claim that the CED requires this particular formula or denominator.

A supplied SD of about 3.61 means this set has a characteristic spread of several correct meanings around 13. It is not the average score, a percentage correct, or a guarantee every score lies within one SD. The 68% rule belongs to a normal-model approximation, not to every dataset with a standard deviation.

Percentile rank describes relative standing in a specified comparison group. For a score of 14 in these eight values, choose the explicit teaching convention ‘percentage strictly below’: four of eight scores are below 14, so the rank is 50%. A midrank convention gives (4 + half of 2 ties) / 8 × 100 = 62.5%; an at-or-below convention gives 6 / 8 × 100 = 75%. Ties make conventions differ. Use the convention a problem or report supplies; these local examples do not establish an official AP tie rule.

A score of 14/24 is approximately 58.3% correct, which answers a different question from percentile rank. Percent correct concerns items answered; percentile concerns position relative to other people’s scores. Changing the comparison group can change percentile rank without changing the number correct. To interpret a reported 80th percentile, identify the comparison population and the report’s convention rather than saying the person answered 80% correctly.

A normal distribution is a symmetric bell-shaped model. Its mean, median and mode coincide at the center. Approximately 68% of values lie within one SD of the mean, 95% within two and 99.7% within three. Symmetry divides the area equally on either side. These are model percentages; a small dataset need not reproduce them exactly.

In the plotted hypothetical normal model, the mean is 100 score units and SD is 15. One SD below and above are 85 and 115; about 68% of the model lies between them. Two SD bounds are 70 and 130, enclosing about 95%; three SD bounds are 55 and 145, enclosing about 99.7%. The central-to-one-SD area on either side is about 34%; the one-to-two-SD band on either side is about 13.5%; the two-to-three-SD band is about 2.35%; about 0.15% remains beyond three SD in each tail. Rounded percentages are approximate.

The curve’s height is relative density, not the percentage scoring exactly 100. Percentages refer to areas over intervals. This normal model does not establish a test norm or imply that actual memory scores must be normal.

Skewness identifies the direction of a long tail. For invented wait times of 1, 2, 2, 3 and 12 minutes before a learning session, the high value makes a right or positive tail; mean = 4, median = 2 and mode = 2. The mean is pulled toward the long high tail. For a separate five-person practice score set of 2, 11, 12, 12 and 13 correct items out of 15, the low score produces a left or negative tail; mean = 10, median = 12 and mode = 12. A negative skew does not mean all values are negative or performance is bad.

Bimodal means two peaks in a distribution. Imagine an equal-sized class tested on an easy introductory task: beginners often score near 4/15 and experienced learners near 12/15. Two clusters may produce two peaks, and one middle average could describe neither cluster well. Mixing groups is one possible explanation; the shape alone does not prove that exactly two natural kinds of people exist. The eight-score set has two observed modes, but its small size cannot establish a stable underlying bimodal population.

Regression toward the mean concerns what happens when unusually extreme observations are followed by additional measurements, especially when measurements contain temporary influences or error. A person selected because of an extreme first score often has a less extreme later score. It does not require every individual to move closer on every trial, and it does not mean the person’s underlying ability necessarily changed.

Complete invented example: a group of adults takes equivalent 24-point recall tasks on two ordinary days with no training between them. Researchers select people who scored unusually low on day one for a proposed improvement program. Some low first scores reflect a temporarily distracting room or ordinary score variability, alongside lasting differences in ability. On a second equivalent test under ordinary conditions, the temporary influences may be less extreme, so average scores among those selected may rise even without the program.

A before–after rise alone would therefore not isolate a program effect. An appropriately designed comparison group selected using the same extreme criterion and tested again helps assess what change occurs without the program. More repeated observations provide a better picture of typical performance and reduce the chance that one extreme day is mistaken for a stable level. Simply adding different people to any dataset is not guaranteed to cause this pattern.

Likewise, unusually high first scores may be less high later; regression is toward the typical level rather than always upward or downward. Do not interpret a return toward the mean as evidence that praise harmed a high performer or punishment improved a low performer unless the design supports that causal claim. The measurement pattern supplies a competing explanation.

Separate original six-person display: six adult volunteers report their mean hours of sleep per night over a week, then complete the same 24-point recall task 20 minutes after study. Pairs of (hours, correct meanings) are (4, 7), (5, 10), (6, 9), (7, 13), (8, 14), (9, 13). No sleep duration is assigned. Each point is one person’s pair, not a condition mean. These six teaching values are not the records for the separate 160-adult survey, whose supplied r is +0.45.

The plotted pattern is positive: higher sleep reports tend to accompany higher recall in these six pairs. The independently calculated Pearson correlation is approximately r = +0.89. The sign describes direction; the magnitude |r| describes linear association strength on a scale from 0 to 1. A value near −1 can be as strong as one near +1, but has the opposite direction. A negative correlation is not an automatically weak one.

The coefficient is a unitless summary of linear association, not a slope. r = +0.89 does not mean an extra hour causes 0.89 more meanings recalled or that 89% of people improve. A near-zero coefficient can conceal a nonlinear pattern, so inspect the scatterplot and context as well as the statistic. This six-person teaching plot supports description of its points, not broad population certainty.

In a non-experimental design, strength concerns an association among measured variables; in an experiment, a difference between assigned conditions concerns a treatment comparison. A strong observed association still leaves directionality and third variables open. Stress or work schedule could influence both sleep and recall. No p-value is given for this six-person example or the 160-person survey; the plotted trend by itself does not authorize a significance conclusion.

Restated original experiment: 80 university volunteers aged 18–24 are randomly assigned 40 each to retrieval or rereading of 24 pairs, with identical four-minute initial study and 12-minute practice. One-week recall means are 18 and 14 correct meanings. For this example only, the hypothetical analysis report additionally supplies pooled within-group SD ≈ 6.67 meanings, standardized mean difference Cohen’s d = 0.60, and a two-sided independent-groups test p = .009 with a preselected α = .05. These are supplied teaching summaries under an equal-variance test model, not inferred from the bars or calculated from the separate eight-score dataset.

The unstandardized magnitude is 18 − 14 = 4 additional correct meanings on average. The supplemental standardized calculation is 4 / 6.67 ≈ 0.60: a difference of about six-tenths of a pooled within-group SD. Cohen’s d is named so the quantity has a meaning. This particular formula is teaching support; the framework requires interpreting effect sizes but does not prescribe one universal effect-size formula.

The CED gives generic descriptors: small usually 0.2 or below, medium usually 0.3–0.7, large usually 0.8 or above. By those supplied descriptors, 0.60 falls in the medium band. The source does not name one universal statistic, fill the gaps 0.21–0.29 and 0.71–0.79, or establish a rounding rule. Do not silently interpolate the missing labels or apply the bands as a universal rule for r, d and every other statistic. Identify the statistic, use any convention supplied by a question, and discuss magnitude in context. This is the recorded source ambiguity, not an extra AP requirement.

Statistical significance asks about evidence against a specified null model under an analysis’s assumptions. Here p = .009 means results at least this incompatible with the no-mean-difference null model would have probability about 0.9% under that model and assumptions. Because .009 < .05, the supplied test meets its stated significance threshold. This does not mean a 99.1% chance the hypothesis is true, a 0.9% chance the data arose ‘by chance’ in every sense, or proof of no confounds.

An effect can be small but statistically significant in a large study; a larger estimated effect can fail a significance threshold when uncertainty is high. Practical usefulness depends on consequences, costs and the task. Four additional correct meanings may matter in a particular learning context, but this experiment does not show transfer to complex understanding or benefits for every adult. Magnitude, uncertainty and applicability are all relevant.

The CED’s shorthand significance description must be read alongside design: significance can be calculated for experimental and non-experimental methods, but it cannot transform an association into causation. Random assignment, matched procedures and a suitable manipulation support a causal interpretation within the vocabulary experiment; the p-value alone does not. A significant sleep–recall association would still leave directionality and third variables unresolved.

Complete separate teaching example: researchers invite 12 adult evening learners from one community class to the same structured interview about barriers to recall practice. Participants are volunteers; no condition is assigned. The interviewer asks when they practice, what interrupts it and how they resume. Two independent coders compare descriptions and resolve disagreements using a written theme definition. The source summary reports recurring accounts of work-shift interruptions and difficulty finding quiet time, alongside accounts of cue cards making short practice sessions easier.

The descriptions support an interpretation that scheduling and environment can constrain practice and that portable cues may help some interviewed learners resume. They do not show how common each theme is across all adult learners, establish a treatment effect, or supply a numerical effect size or p-value. Researchers can infer a plausible contextual explanation while preserving the limited sample and the difference between reported experience and measured outcomes.

A stronger interpretation states both the pattern and its boundary: ‘These volunteers described work timing and noise as barriers, suggesting that access to suitable study circumstances should be examined alongside the choice of practice method.’ A weak interpretation says ‘Everyone prefers cue cards, so they are statistically proven best.’ The latter invents universality and statistical evidence absent from the source. A follow-up study could test the explanation, but the present descriptions do not already contain that test.