ZIMSEC A Level · Statistics · Form 5
Types of data
Qualitative and quantitative; discrete and continuous. Displays: stem-and-leaf diagrams, histograms (frequency density = frequency ÷ class width), box-and-whisker plots and cumulative frequency graphs.
Measures of location
Mean x̄ = Σx/n (or Σfx/Σf); median; mode. For grouped data use class mid-points.
Measures of spread
- Range; interquartile range (Q3 − Q1).
- Variance s² = Σx²/n − x̄² ; standard deviation s = √variance.
Worked example: Find the mean and standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.
Σx = 40, n = 8 → x̄ = 5
Σx² = 4 + 16 + 16 + 16 + 25 + 25 + 49 + 81 = 232
Variance = 232/8 − 25 = 4 → s = 2
Outliers and skewness
An outlier is often defined as a value more than 1.5 × IQR beyond the quartiles. Positive skew: mean > median, long right tail; negative skew: mean < median.
Coding
If y = (x − a)/b, then ȳ = (x̄ − a)/b and s_y = s_x/b.
Check yourself
- Find the median of 3, 8, 6, 2, 9, 5.
- The IQR of a data set is 12 and Q3 = 40. Above what value is a data point an outlier?
- A data set has mean 50 and standard deviation 5. Find the mean and SD of y = (x − 50)/5.
Show answers
- 5.5
- 40 + 1.5 × 12 = 58
- Mean 0; SD 1
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