AP Precalculus: Single-Variable Statistics
Master variance, standard deviation, outliers, and confidence intervals
π Understanding Statistics
Single-variable statistics summarizes and analyzes data from one variable. Key concepts include measures of spread (variance, standard deviation), identifying unusual values (outliers), and making inferences about populations using confidence intervals. These tools help us understand data variability and make informed decisions.
1 Variance & Standard Deviation
Variance measures how spread out data values are from the mean. Standard deviation is the square root of variance, in the same units as the original data.
Data: 4, 8, 6, 5, 7 (sample)
Mean: \(\bar{x} = \frac{4+8+6+5+7}{5} = 6\)
Deviations squared: \((4-6)^2 + (8-6)^2 + (6-6)^2 + (5-6)^2 + (7-6)^2 = 4+4+0+1+1 = 10\)
Sample variance: \(s^2 = \frac{10}{4} = 2.5\)
Sample std dev: \(s = \sqrt{2.5} \approx 1.58\)
Dividing by n-1 (instead of n) corrects for bias when estimating population variance from a sample. This is called Bessel's correction.
2 Outlier Detection & Effects
An outlier is a data point that is unusually far from other values. The IQR method provides a standard way to identify outliers.
Effects of Outliers
On Mean
Outliers pull the mean toward extreme values. Mean is NOT resistant to outliers.
On Median
Median is resistant β not affected much by outliers.
On Standard Deviation
Outliers increase standard deviation. Removing outliers usually decreases it.
On IQR
IQR is resistant β based on quartiles, not extreme values.
Data: 2, 4, 5, 6, 7, 8, 25
Q1 = 4, Q3 = 8: IQR = 8 - 4 = 4
Fences: Lower = 4 - 6 = -2, Upper = 8 + 6 = 14
Conclusion: 25 > 14 β 25 is an outlier β
3 Sampling Bias & Types
Sampling bias occurs when the sample is not representative of the population, leading to inaccurate conclusions.
| Type of Bias | Description | Example |
|---|---|---|
| Selection Bias | Some groups more likely to be selected | Surveying only mall shoppers |
| Non-response Bias | People who respond differ from non-responders | Only motivated people return surveys |
| Response Bias | Respondents give inaccurate answers | Embarrassing questions, leading wording |
| Undercoverage | Some population members have no chance of selection | Phone survey excludes those without phones |
| Voluntary Response | Participants self-select into study | Online polls attract strong opinions |
Use random sampling to give every member of the population an equal chance of being selected. This is the foundation of valid statistical inference.
4 Confidence Intervals
A confidence interval provides a range of plausible values for a population parameter, based on sample data.
Common Z* Values
Given: \(\bar{x} = 75\), \(s = 10\), \(n = 36\), find 95% CI for mean (Ο unknown)
Standard error: \(\frac{10}{\sqrt{36}} = \frac{10}{6} \approx 1.67\)
Margin of error: \(1.96 \times 1.67 \approx 3.27\)
95% CI: \(75 \pm 3.27 = (71.73, 78.27)\)
5 Interpreting Confidence Intervals
A confidence interval interpretation must include: confidence level, parameter of interest, and the interval values in context.
What Affects Interval Width?
Increasing Confidence
Higher confidence β Wider interval (more sure = less precise)
Increasing Sample Size
Larger n β Narrower interval (more data = more precise)
More Variability
Larger Ο or s β Wider interval
DON'T say "There's a 95% probability the parameter is in this interval." The parameter is fixed β either it's in the interval or it's not. The 95% refers to the method's long-run success rate.
6 Experiment Design Principles
A well-designed experiment allows us to establish cause and effect. Key principles ensure valid, reliable results.
Observational Study
Observe without intervention. Cannot establish causation β only association.
Experiment
Researcher imposes treatments. CAN establish cause and effect when well-designed.
Use simulations to model random processes and analyze variability. Repeat many times to understand what outcomes are likely or unusual.
π Quick Reference
Sample Variance
\(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\)
Standard Deviation
\(s = \sqrt{s^2}\)
IQR
\(Q_3 - Q_1\)
Outlier Test
\(x < Q_1 - 1.5 \cdot IQR\) or \(x> Q_3 + 1.5 \cdot IQR\)
CI for Mean
\(\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}\)
CI for Proportion
\(\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)