Standard Error, t-test, and Confidence Interval Cheat Sheet

Standard Error (SE)

Formula: SE = \frac{SD}{\sqrt{N}}
Plain Meaning: How precise is the sample mean?
Bigger N → smaller SE.
Analogy: Guessing class average height: 5 students = shaky guess, 100 students = precise guess.
APA Note: Rarely reported directly, but powers CI & t-tests.

Formula: t = \frac{M1 - M2}{SE}
Plain Meaning: Signal-to-noise ratio. Difference between means divided by uncertainty.
Analogy: Radio signal vs static: big signal compared to noise = strong evidence.
APA Example: t(24) = 3.4, p = .002

Formula: CI = M \pm (t^* \times SE)
Plain Meaning: Range of values where the true mean (or difference) likely falls.
Analogy: Dartboard: mean = bullseye, CI = ring where the darts cluster.
APA Example: 95% CI [6.9, 10.7] → improvement is 7–11 points.

Dataset: Pre-test (M = 66.8, SD = 4.2), Post-test (M = 75.7, SD = 3.9), N = 6.
Step 1: Compute SE
- SE = \frac{SD}{\sqrt{N}} = \frac{4.0}{\sqrt{6}} \approx 1.63 \Rightarrow \text{The sample mean is precise within } \sim 1.6 \text{ points.}
Step 2: Compute t-test
- t = \frac{M{\text{post}} - M{\text{pre}}}{SE} = \frac{75.7 - 66.8}{1.63} \approx 5.46 \Rightarrow \text{Big signal compared to noise.}
Step 3: Compute CI
- CI = \text{Mean Difference} \pm (t^* \times SE) = 8.9 \pm (2.57 \times 1.63)
- = [4.7, 13.1] \Rightarrow \text{True improvement is between 5 and 13 points.}$$
Step 4: APA Reporting
- “Students scored significantly higher on the post-test (M = 75.7, SD = 3.9) than on the pre-test (M = 66.8, SD = 4.2), t(5) = 5.46, p < .01. The 95% CI [4.7, 13.1] suggests the true improvement is between 5 and 13 points.”