Descriptive Statistics and Data Representation
Descriptive statistics questions test summarising raw data and interpreting diagrams: histograms with unequal class widths, cumulative frequency curves, box-and-whisker plots, and measures of location and spread including the effect of coding.
Summarising data
- Mean x̄ = Σx/n, or Σfx/Σf for grouped data using class midpoints.
- Median and quartiles come from ordered data, or by interpolation on the cumulative frequency curve at n/2, n/4 and 3n/4.
- For grouped data the modal class is the class with greatest frequency; on a histogram the mode is read from the tallest bar drawn in frequency density.
- Range, interquartile range Q₃ − Q₁, and standard deviation are the standard spread measures.
Standard deviation and coding
- σ = √(Σ(x − x̄)²/n) = √(Σx²/n − x̄²). Follow the question if it specifies the divisor n − 1 for a sample.
- Coding y = (x − a)/b gives x̄ = bȳ + a and σ_x = |b|σ_y; variance scales by b², and adding a leaves spread unchanged.
Diagrams
- Histogram: frequency density = frequency ÷ class width, so the area of each bar equals its frequency. Unequal widths must be converted before plotting.
- Cumulative frequency curve: plot against upper class boundaries.
- Box plot: marks median, quartiles and extremes, with outliers beyond Q₁ − 1.5IQR or Q₃ + 1.5IQR.
- Skewness: Q₃ − Q₂ compared with Q₂ − Q₁ indicates positive or negative skew.
Worked example
Values of x are coded as y = (x − 20)/5, and the coded data give ȳ = 3 with s_y = 1.2. Then x̄ = 5(3) + 20 = 35, and the standard deviation of x is 5 × 1.2 = 6. The variance of x would be 36, not 1.44.
Traps
- Plotting frequencies rather than frequency densities when class widths differ.
- Interpolating at n instead of n/2, or using lower rather than upper boundaries.
- Forgetting to undo the coding when reporting statistics of the original data.
- Confusing standard deviation with variance in the final answer.