Representation of data
Diagrams
- Choose a diagram to suit the data:
- stem-and-leaf (keeps the values, shows shape),
- box-and-whisker (lowest, three quartiles, highest),
- histogram (grouped data — bar area = frequency),
- cumulative frequency (running totals → median and quartiles).
Practice
In a histogram of grouped data, the frequency is shown by each bar's:
In a histogram, the area of each bar represents the frequency.
Averages and spread
- Central tendency: mean $\bar{x} = \dfrac{\sum x}{n}$, median (middle), mode (most common).
- Spread: range, interquartile range, and standard deviation $\sigma = \sqrt{\dfrac{\sum x^2}{n} - \bar{x}^2}$.
- Variance = $\sigma^2$.
Practice
For 10 values with Σx = 50, what is the mean?
Mean = Σx / n = 50 / 10 = 5.
Practice
For 10 values, Σx = 50 and Σx² = 300. What is the standard deviation? (√(Σx²/n − x̄²))
σ = √(300/10 − 5²) = √(30 − 25) = √5 ≈ 2.24.
You've got it
Key idea
- diagrams: stem-and-leaf, box plot, histogram (area = frequency), cumulative frequency
- averages: mean, median, mode; spread: range, IQR, standard deviation
- $\sigma = \sqrt{\dfrac{\sum x^2}{n} - \bar{x}^2}$, and variance $= \sigma^2$