Describing Data: Centre and Spread

Descriptive statistics summarise a data set. Measures of centre:

• Mean — add all values and divide by how many: $\bar{x} = \dfrac{\sum x}{n}$.
• Median — the middle value once the data is sorted.
• Mode — the most frequent value.

Measures of spread: the range (max $-$ min) and the standard deviation, which measures typical distance from the mean.

The variance averages the squared distances from the mean; the standard deviation $\sigma$ is its square root, back in the original units.

Worked example. For $2,4,4,4,5,5,7,9$ the mean is $\dfrac{40}{8}=5$. The squared deviations are $9,1,1,1,0,0,4,16$, summing to $32$. Population variance $=\dfrac{32}{8}=4$, so $\sigma=\sqrt{4}=2$.