Appendix A — Common Statistical Symbols
Symbols are listed in rough alphabetical order. Some statistics are represented by more than one symbol, so some statistics are given more than once here.
More symbols are given in the APA Manual.
| Symbol | Symbol Name/ Pronunciation |
Meaning/Use/Interpretation |
|---|---|---|
| α | Alpha | The probability of a false positive (a Type 1 error). Specifically, the probability of finding a given effect, value, etc. when the null hypothesis is true |
| b | b-weight | The unstandardized coefficient of a term in a model, such as a general linear regression. In other words, the change in the outcome variable in terms of the units of the outcome and the given predictor/term. |
| β | Beta | Either: 1. The probability of a false negative (a Type 2 error) or 2. The standardized coefficient of a term in a model, such as a general linear regression |
| 1 - β | One minus beta; power | The power of a test, statistic, etc. The probability of not making a false negative, i.e., the chance of not missing a real effect |
| d | Cohen’s d | A measure of effect size for the difference between two means. It is computed simply as the difference in the means divided by the standard deviation of the means: \(\text{Cohen's }d = \frac{\text{Mean}_1 - \text{Mean}_2}{SD}\); it is therefore a standardized measure that allows one to compare the size of effects across conditions, analyses, studies, etc. |
| df | Degrees of freedom | Roughly1, the amount (and source) of information in a set of data and related analyses. Each degree of freedom can be used—“spent”—to estimate some value in or about the data. For example, computing the mean of the data uses one degree of freedom; some statistics/analyses take up more than one degree of freedom to compute/conduct, and often using more degrees of freedom to conduct an analysis results in more powerful insights2. |
| η2 | Eta-squared | A measure of effect size for terms in a model—often a general linear regression. It is equivalent to f2 |
| F | F-score | A statistic representing a value on a distribution (of the same name), used commonly to test for differences between groups/levels in an ANOVA. Conceptually, it represents the ratio of the size of an effect (e.g., a difference in means) to the amount of “error” in our measurement of the effect: i.e., a signal-to-noise ratio |
| f2 | The effect size of a term in a model, usually a general (or generalized) linear model. | |
| H0 | Null hypothesis | The hypothesis to be refuted, it is usually set to be either that there is no difference between groups (e.g., there is no difference in outcoms between an experimental and a control group) or that there is no association between two variables (e.g., that two variables are not correlated). Abelson (1995) presents an accessible explanation of the role of the null hypothesis starting on page 8. |
| H1 or HA |
Alternative hypothesis | The hypothesis that there is, e.g., a difference between groups or an effect of an intervention/manipulation. It is often the proposed outcome of a study to be tested |
| MAD | Median absolute difference | A median distance of scores from the median. It is analogous to the standard deviation for a mean |
| μ | Mu; mean | The mean of a population |
| n or N | The number (count) of a sample, number of measurements, etc. Sometimes capital N is used to denote the whole sample (or even sometimes for a population) while lower-case n is used for a sub-sample / subgroup of a whole sample. | |
| p | p-value | A probability; very often the probability of a false positive (a Type 1 error). More specifically, it indicates the probability that the researchers would have found the results that they did even though there was no real difference. More loosely and generally, saying, e.g., “we found something significant (p < .05)” is taken to mean that the researchers believe there is less than a 5% chance that their results are “false positives.” |
| r | Pearson correlation | A correlation between two continuous (interval/ratio) variables. It is also a measure of effect size; Cohen (1988) suggests that a correlation of .1 is “small,” that .3 is “medium,” and .5 is “large.” |
| rpb | Point-biserial correlation | A correlation between a dichotomous variable and a continuous variable. The computation is equivalent to that for a Pearson correlation, so point-biserial correlations can be computed along with (mixed in with) Pearson correlations |
| R2 | R-squared | A measure of the total proportion of variance in the data that a given model can account for. It is always a measure for an entire model (i.e., all of the terms in a mode, including interaction terms and the intercept, if included). Like r2, it represents the variance accounted for. However, it is usually computed differently and can thus even take on negative values in extreme cases—usually for extremely-poorly fitting models. It is similar to information criteria, like AIC and BIC. |
| ρ | Spearman’s rho | A correlation of the ranks of two ordinal (or even continuous) variables. It measures how much the ranking (from largest to smallest) of the values of one variable match the rankings of the values on an other variable. It is more robust than Pearson’s r and can be used for non-linear variables |
| σ2 | Sigma-squared; variance | Variance of a population |
| S2 | Variance | Variance of a sample |
| Σ | Sum | The sum (adding up) of a series of values, often the sum of all of the values of a given variable. The mean, for example, is the sum of scores divided by the number of scores, for example, the mean of variable X is: \(\overline{X} = \frac{\sum{X}}{N}\) |
| σ | Sigma; standard deviation | Standard deviation of a population |
| SD | Standard deviation | Standard deviation of a sample |
| t | t-score | A statistic representing a value on a distribution (of the same name), used commonly to test for differences between two (and only two) groups/levels. Like an F-score, it represents the ratio of the size of an effect (e.g., a difference in means) to the amount of “error” in our measurement of the effect: i.e., a signal-to-noise ratio. In fact, it is simply the square root of a F-score (i.e., \(\sqrt{F}=t\) or, equivalently, \(t^2=F\)) |
| τ | Kendall’s tau | A correlation between two ordinal variables. Like Spearman’s ρ, it measures the ranks, but only whether the placement of participants, etc. are in the same ranks on both variables (for example, of two NPs both rank the severity of scholiosis of a set of patients in the same order) |
| X | X is often used to represent a variable, often a dependent (input or predictor) variable | |
| \(\overline{X}\) | Mean of X; X overbar |
The mean of variable X |
| x-axis | The horizontal axis on a chart | |
| var(X) | Variance | Variance of a variable (here—and often—a variable is represented by X) |
| χ2 | Chi-squared / Chi-square | A statistic representing a value on a distribution (of the same name) that resembles a normal distribution. It is very commonly used to test significance (e.g., if the frequency of events differs between two populations—like if the prevalence of COPD varies between genders) |
| \(\overline{x}\) | Mean (average) | A measure of “central tendency.” The mean is the sum of the scores divided by the number of scores |
| Y | Y is often used to represent an outcome variable (e.g., an independent variable) | |
| y-axis | The vertical axis on a chart | |
| z | z-score | A standardized score, nearly always meaning standardized so that the (sample) mean is set to zero and the (sample) standard deviation is set to 1; this allows for direct comparisons between variables that have very different raw scores (e.g., comparing A1C levels to BMIs |
More specifically, the number of values that can be computed in a given information space. If I have an equation like \(x + 2 = 3\), then I have only one degree of freedom: \(x\) can only have one possible value (here, a 1 for the mathematically disinclined). If I instead have an equation like \(x + y = 3\), there are two values to compute, and so I have two degree of freedom. Importantly, notice that once I know one value—once I “spend” one degree of freedom, I then can determine the other value: If \(x\) = 1, then we can compute that \(y\) = 2; if \(x\) = 0, then we can compute that \(y\) = 3. Therefore, “spending” degrees of freedom to estimate values (e.g., to compute the sample mean) uses up some of the total information available, but also lets us more accurately determine the value of other estimates, such as how far away a given person’s score is from that mean. We can use a data set’s degrees of freedom to make a certain number of insights, but the total number of insights is strictly limited by the size of our dataset. We could use those degrees of freedom to make different insights, but not an infinite number of insights.↩︎
Devoting more of the available information to making a decision makes that decision more insightful. However, since any given set of data only has so many degrees of freedom—only so much information useful for making decisions—we should economize how much information we use to make a given decision.↩︎