Describing results

A result is first a description of what the scores did. Descriptive statistics summarize a set of scores without yet claiming that the summary would hold in a larger population. The mean of this sample, the spread of this sample, the association in this sample: those are facts about the data in hand. Inferential claims, the next chapter, ask a further question. Skipping the description and reporting only a test is how a paper hides the shape of the data, including the floor effects, the outliers, and the curvilinear pattern a straight-line summary will miss. Cozby and Bates put description ahead of inference for that reason (Cozby & Bates, 2020).

A frequency distribution counts how many cases fall at each value or in each bin. It is the picture that shows ceiling and floor, skew, and a split into two clumps that a single average would smear together. Categorical variables are described by counts and proportions. Quantitative variables need a center and a spread. Choosing the picture is part of the method. A bar for groups, a histogram for a quantitative distribution, a scatterplot for a pair of quantitative variables: each picture answers a different question, and the wrong picture can make a null look like a law or a law look like noise.

The numbers in this chapter are about the sample that was run. They become claims about a population only with the inferential tools of the next chapter and, in full, with Statistics for Psychology. A methods student who can compute a careful description and refuse to narrate it as a universal has already avoided the most common overclaim.

The mode is the most common value. It is the right center for a categorical variable and a useful note when a distribution has two peaks. The median is the middle score when the scores are ordered. It resists a few extreme values. The mean is the arithmetic average. It uses every score, which makes it sensitive to extremes and makes it the center that later inferential procedures usually assume. A mean of incomes in a room with one enormous outlier describes a room that does not resemble any person in it. The median would have said so. Reporting the mean alone, for a badly skewed measure, is a choice to let the tail write the sentence.

Variability is how far scores sit from each other. The range is the distance from the smallest to the largest, and one odd score is the whole range. The variance averages the squared distances from the mean. The standard deviation is the square root of the variance, returned to the units of the measure, and it is the spread a reader should be able to say in a sentence: the typical distance from the mean, in these data, was about this large. A mean without a spread is an incomplete description. Two conditions can share a mean and differ wildly in spread, and the treatment may have changed the spread rather than the center. That result is invisible if only means are printed.

Standardizing a score, expressing it as how many standard deviations it sits from the mean, lets different measures be compared. The mechanics of z-scores and the normal model are developed properly in Statistics for Psychology. Here the idea is enough: a difference of five points is large or small depending on how spread out the scores already were. Effect size, below, is that idea applied to a comparison rather than to one person.

A correlation describes the direction and the strength of a straight-line association between two quantitative variables. The usual index, the Pearson correlation, runs from −1 to +1. A positive value means high scores travel with high scores. A negative value means high scores travel with low scores. The absolute value is the strength of the linear association. Zero means no straight-line association, which is not the same as no relationship. A curvilinear relationship, up and then down, can produce a correlation near zero and a very real pattern. The scatterplot is mandatory for that reason. A coefficient without the plot is a summary that might be summarizing the wrong shape.

Restriction of range shrinks a correlation when the sample covers only part of the variable. A test that predicts performance across the full range of applicants will look weaker if it is studied only among people who were already admitted. The relationship in the world can be larger than the relationship in the truncated sample. Outliers do the opposite damage: one point can invent a correlation or destroy one. The plot shows the point. The coefficient hides it.

Correlation is not causation. The opening chapter already gave the reasons, and they are concrete here. The correlation does not establish which variable came first. It does not rule out a third variable that produces both. A strong, replicated correlation is a successful prediction and a reason to design a study that could support a cause. Narrating it as “X produces Y” because the coefficient was large skips the design the verb requires. A partial association that shrinks when a third variable is held constant is a clue about that third variable, not a completed causal proof. The third variable may be a mediator, a confound, or a collider the analysis should not have controlled. Those distinctions are easy to get wrong, and this course’s rule is modest: a correlation licenses prediction, and the causal story stays labeled as a story until a design addresses time order and rivals.

An effect size states the magnitude of a result in units that do not depend on the sample size. For a difference between two means, Cohen’s d divides the difference by a standard deviation. Cohen offered conventions for thinking about d in many behavioral areas: about 0.2 as small, 0.5 as medium, and 0.8 as large (Cohen, 1988). The conventions are a shared language, not a law of nature. A “small” effect on a rare, serious outcome can matter more than a “large” effect on a trivial rating. The paper should give the effect size and the context, not only the adjective.

Effect size and statistical significance answer different questions, and the next chapter separates them properly. A tiny effect in a huge sample can be “significant.” A large effect in a tiny sample can fail a test. Description’s job is the size. Readers who are given only a star or a p value have been refused the description. Graph the means with an indication of spread. Report the correlation with the scatter in mind. Report d, or another stated effect size, next to the means that produced it.

Everything in this chapter can be done, and should be done, before anyone asks whether a result is surprising under a null hypothesis. That question is inference. The full distribution theory, the standard errors, and the tests themselves are Statistics for Psychology, waiting one course further on the map. The next chapter of this course stays at the level a methods reader needs in order not to overclaim: what a test says, what a confidence interval says, and the sentences that are larger than either one.