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.

t-test (t)

• Formula: $t = \frac{M<em>1 - M</em>2}{SE}$
• 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: $t(24) = 3.4, p = .002$

Confidence Interval (CI)

• 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.

Worked Example: Connecting SE, t, and CI

• 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
  • $t = \frac{M<em>{\text{post}} - M</em>{\text{pre}}}{SE} = \frac{75.7 - 66.8}{1.63} \approx 5.46 \Rightarrow \text{Big signal compared to noise.}$
• 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.”