Data Analysis and Statistics
The Data Analysis and Statistics domain tests how well you summarize a data set, describe its spread, use the normal distribution, and judge association between variables. Questions often present a small list of numbers or a description of a distribution and ask which summary is appropriate or which conclusion is justified.
Center and spread
- The mean is sensitive to outliers; the median is resistant. In a right-skewed distribution the mean exceeds the median; in a left-skewed one the mean is smaller.
- IQR = Q3 − Q1, the range of the middle half of the data. A value is a potential outlier if it lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
- Standard deviation measures a typical distance from the mean and is never negative. Adding a constant to every value shifts the mean but leaves the standard deviation unchanged; multiplying every value by k multiplies the standard deviation by |k|.
The normal distribution
- Standardize with z = (x − μ)/σ; the area under the curve corresponds to probability.
- The empirical rule: about 68% of values lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
Association
- Correlation r lies between −1 and 1; the sign gives direction and the magnitude gives strength, but r measures only linear association.
- A residual is observed minus predicted; a good fit produces small residuals with no pattern.
- Correlation does not establish causation, and an observed relationship may be driven by a lurking variable.
Worked example
Data: 4, 6, 7, 7, 9, 11, 15. The median is 7. Q1 is the median of 4, 6, 7, so Q1 = 6; Q3 is the median of 9, 11, 15, so Q3 = 11. IQR = 5, giving fences at 6 − 7.5 = −1.5 and 11 + 7.5 = 18.5, so there are no outliers. The mean is 59/7 ≈ 8.43, larger than the median because the single value 15 pulls the mean to the right.
Traps
- Reporting the mean for skewed data when the median is the better summary.
- Confusing Q1 and Q3 with the minimum and maximum values.
- Assuming a strong correlation implies a steep slope; correlation and slope are separate quantities.
- Reading a histogram's height as a count when the vertical axis shows relative frequency.