Describing Data: Central Tendency and Variability

Measuring Spread: Variability

Knowing the average is not enough. For example, if one group of students scores 75, 75, and 75 on a test, and another group scores 50, 75, and 100, both groups have a mean of 75. However, their behaviors are very different. To describe this difference, we use measures of variability—the extent to which scores in a batch differ from each other.

Range and Standard Deviation

The simplest measure of variability is the range, which is the difference between the highest and lowest scores in a batch. While easy to calculate, the range is highly sensitive to outliers.

A more sophisticated and common measure is the standard deviation. This statistic calculates how much, on average, a batch of scores varies around its mean.

• A low standard deviation indicates that most scores are clustered closely around the mean.
• A high standard deviation indicates that the scores are spread out over a wider range of values.

Quantifying Relationships: Effect Size

After describing a single group, researchers often want to know the strength of a relationship between two different variables. This is captured by the effect size, a numerical estimate of the strength of a relationship.

Correlational Research and Scatterplots

In correlational studies, researchers use scatterplots to visualize the relationship between two variables. By looking at the dots on a graph, you can see the pattern: the more spread out the dots are, the weaker the relationship.

The strength and direction of this relationship are summarized by the correlation coefficient, abbreviated as $r$. The value of $r$ ranges from $-1.0$ to $+1.0$:

• $r = 1.0$: A perfect positive relationship (as one variable increases, the other increases).
• $r = -1.0$: A perfect negative relationship (as one variable increases, the other decreases).
• $r = 0$: No relationship between the variables.

In psychology, we generally follow these conventions for effect size:

• $.10$ = Weak effect
• $.30$ = Medium effect
• $.50$ or greater = Strong effect

Experimental Research and Cohen's $d$

In experimental studies, where researchers manipulate an independent variable to see its effect on a dependent variable, effect size is often expressed as $d$. This value tells us how many standard deviation units apart the two group means are.

$$\text{Effect Size } (d) \approx \frac{\text{Difference between means}}{\text{Standard deviation}}$$

A larger $d$ indicates less overlap between the groups, meaning the experimental manipulation had a more distinct impact. Common benchmarks include:

• $d = .20$ (Small)
• $d = .50$ (Medium)
• $d > .80$ (Large)

Critical Thinking Note: Be wary of media reports that exaggerate small differences. News outlets often use misleading scales or truncated y-axes in graphs to make a small effect size look like a massive one.