PSYC 304: EQUATIONS

What does Pearson’s r measure?

The strength and direction of a linear relationship between two variables (X and Y).

Formula for Pearson’s r

r = Σ(x−x̄)(y−ȳ) / √[Σ(x−x̄)² * Σ(y−ȳ)²]

Steps to calculate Pearson’s r

1. Find means of X and Y. 2. Compute (x−x̄) and (y−ȳ). 3. Multiply (x−x̄)(y−ȳ). 4. Sum those products. 5. Find Σ(x−x̄)² and Σ(y−ȳ)². 6. Plug into the formula.

Generic formula for Margin of Error (MOE)

MOE = (critical value) × (standard error)

MOE for one sample (mean or proportion)

MOE = t* × (s / √n)

MOE for two independent samples (means)

MOE = t* × √[(s₁²/n₁) + (s₂²/n₂)]

MOE for two paired samples (mean of differences)

MOE = t* × (sd / √nd)

What does a Z-score represent?

How many standard deviations a value is from the mean.

Formula for a Z-score (population)

z = (x − μ) / σ

Formula for a Z-score (sample)

z = (x − x̄) / s (technically a t-score if used for inference)

How to find t* using ESCI’s Distributions Tool

1. Open ESCI “Distributions” tool. 2. Select t-distribution. 3. Enter degrees of freedom (df = n−1). 4. Choose confidence level (e.g., 95%). 5. Read t* from the graph/output.

Symbol x̄

Sample mean

Symbol μ

Population mean

Symbol s

Sample standard deviation

Symbol σ

Population standard deviation

Symbol n

Sample size

Symbol p

Sample proportion

Symbol p₀

Hypothesized population proportion

Symbol t*

t critical value

Symbol z*

z critical value

Symbol SE

Standard error

Symbol df

Degrees of freedom

Symbol r

Pearson’s correlation coefficient

Symbol MOE

Margin of error

Symbol CI

Confidence interval

Symbol H₀

Null hypothesis

Symbol Hₐ

Alternative hypothesis