stats

Statistics for Advanced Experimental Design

Key Formulas and Concepts

Mean

Sample Mean: ( x̄ = (Σ xᵢ) / n )Where:(n) = size of the sample(xᵢ) = individual sample values

  • Summarizes entire sample

  • Might provide an estimate of entire population’s true mean

  • Unreliable w/ high variance/outliers

Standard Deviation

  • Sample Standard Deviation: ( S = √(Σ (x - x̄)² / (n - 1)) )

    • Aka: closeness/variability of data to the mean

  • Larger standard deviation indicates more variability, which implies that other variables influence the dependent variable

  • Smaller standard deviation indicates clustering around the mean, suggesting direct effects from the independent variable

    • better justification that x causes y

  • ± 1s = 68% of data

  • ± 2s = 95% of data

  • ± 3s = 99% of data

Standard Error of the Mean

  • ( SEM = S / √n )

  • Indicates how well the sample mean matches the true population mean.

    • determine confidence in data collected in sample

    • estimates how much the sample mean is likely to vary in repeated experiments

Chi-Square Test

  • ( χ² = Σ ((O - E)² / E) )Where:(O) = observed results(E) = expected results

    • Sum of ((observed - expected) squared) / expected

    • Degrees of freedom (df) = # of categories - 1

      • compare calculated value to critical value (p value) in chi-square distribution table

Degrees of Freedom

Calculated as: For Chi-Square: ( df = k - 1 ) (k = number of categories)

Important Statistical Insights

Statistics are essential for:

  • Understanding sample data meaning

  • Drawing conclusions from data

  • Supporting scientific arguments

  • Estimating data reliability

  • Effectively communicating findings

Advanced Statistical Tests

T-Test

  • A statistical test comparing means between two groups.

    • determine if observed effect reflect true population characteristics rather than sampling error (is it significant or due to chance/random ?)

Types:

  • Independent T-Test - Compares separate groups (e.g., control vs. treatment).

  • Paired T-Test - Compares the same group before/after treatment.

Hypotheses:

  • Null Hypothesis: No significant difference exists.

  • Alternative Hypothesis: Significant difference exists based on control vs. treatment.

Significance level: p = 0.05 → p < 0.05 then there is a significant difference → reject null hypothesis

ANOVA (Analysis of Variance)

Used when comparing means of more than two groups.

  • Helps determine if differences are due to chance. (like T-test)

Hypotheses:

  • Null Hypothesis: No significant difference among groups.

  • Alternative Hypothesis: Significant difference exists.

p-value

  • p < 0.05 → reject null + significant difference exists

Chi-Square Test

Compares observed data to expected data.

  • determines if differences are due to chance

  • represented by χ²

  • only used w/ numerical data / counting frequencies

    • NOT percentages or proportions

Hypotheses:

  • Null Hypothesis: No significant difference between observed and expected values.

  • Alternative Hypothesis: Significant differences exist.

Interpreting Results:

  • If calculated ( χ² ) is less than critical value (p < 0.05): Accept null hypothesis.

  • If greater than critical value (p > 0.05): Reject null hypothesis.