AP Precalculus: Single-Variable Statistics Formulas & Principles
1. Variance & Standard Deviation
- Population variance: \( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2 \)
- Sample variance: \( s^2 = \frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2 \)
- Standard deviation: \( \sigma = \sqrt{\sigma^2} \), \( s = \sqrt{s^2} \)
2. Outlier Detection & Effects
- An outlier: \( x \) is an outlier if \( x < Q_1 - 1.5 \times IQR \) or \( x > Q_3 + 1.5 \times IQR \)
- \( IQR = Q_3 - Q_1 \)
- Removing outlier typically decreases standard deviation and changes mean
3. Sampling Bias & Experiment Design
- Biased sample: Not representative of the population (selection, measurement, response bias, etc.)
- Avoid bias: Use random sampling, proper experiment design (control, randomize, replicate, block)
4. Confidence Intervals
- Mean, known \( \sigma \): \( \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \)
- Mean, unknown \( \sigma \): \( \bar{x} \pm t^* \frac{s}{\sqrt{n}} \)
- Proportion: \( \hat{p} \pm z^* \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{n} } \)
- Where \( z^* \) or \( t^* \): z-score (normal) or t-score (small samples)
5. Interpreting Confidence Intervals
- "We are ___% confident the true parameter lies within the interval."
- Increasing confidence increases width; increasing \( n \) decreases width
6. Experiment Design & Simulations
- Well-designed experiments have control, randomization, replication
- Use simulations to analyze variability/replication in practical situations