Statistics
What this domain tests
Descriptive statistics: measures of location and dispersion for raw and grouped data, quartiles from cumulative frequency, box-and-whisker plots and skewness, and how linear coding y = ax + b changes summary statistics.
Core rules
- Mean x̄ = Σx/n for raw data, Σfx/Σf for grouped data; median and quartiles by interpolation within the cumulative-frequency class.
- Sample variance s² = Σ(x - x̄)²/(n - 1); population variance uses n. Standard deviation is its square root.
- Coding y = ax + b gives ȳ = ax̄ + b and s_y = |a|s_x; variance scales by a², and the median and quartiles transform exactly like the mean.
- Positive skew: mean > median, with the longer whisker on the right of the box plot.
- Values beyond Q1 - 1.5 IQR or Q3 + 1.5 IQR are commonly treated as outliers.
Worked example
A data set x has mean 20 and variance 16. Find the mean and variance of y = 3x - 5. Mean: ȳ = 3(20) - 5 = 55. Variance: Var(y) = 3² × 16 = 144, so s_y = 12. The -5 shifts every value without changing spread; only the multiplier alters variance, and it does so by its square.
Traps
- Scaling variance by 3 instead of 9 when y = 3x - 5.
- Using n rather than n - 1 for a sample standard deviation.
- Interpolating a quartile without a consistent cumulative-frequency basis.
- Claiming symmetry from a sketch rather than comparing mean and median.