Measures of Centrality

Context: FIT1043_MOC, FIT2086_MOC · the “typical value” of a sample · the first of the descriptive statistics · numerically interpreting numeric data FIT2086 framing: a statistic is any function $s(\mathbf{y})$ of a sample; the mean $\bar{y}=\frac{1}{n}{\sum }_j{y}_j$, median $\mathrm{med}(\mathbf{y})$ and mode are the simplest — later reused as estimators of population parameters (see Statistical Modelling and Inference).

Quick Revision

• 🎯 Objective: summarise a sample’s centre ➔ mean, median, mode.
• 📦 Core Components: mean (all values) | median (middle value) | mode (most frequent).
• ⚡ Key Constraint: the mean is sensitive to every value (outliers drag it); the median is resistant — their gap reveals skew.

📝 How It Works

1. What is a Statistic?

• Descriptive statistics ➔ numerically interpret key features of a dataset; usually lose information but gain comprehension (contrast inferential statistics).
• Statistic ➔ for a sample $y=(y_1,\dots ,y_n)$, any function $s(y)$ of the data.

2. The Three Measures

• Mean ➔ arithmetic average $\bar{y}=\frac{1}{n}{\sum }_{i=1}^n{y}_i$; uses all values.
• Median ➔ $\mathrm{med}(y)$ = value with 50% of samples below it; sort and take the middle.
• Mode ➔ the most frequently occurring value.

3. Mean vs Median (robustness)

• Mean ➔ any change to any value changes it; one huge value can move it arbitrarily.
• Median ➔ uses at most two middle values ➔ resistant to changes away from the middle.
• Worked ➔ $y=(1,2,3,4,5)\Rightarrow \bar{y}=3,\mathrm{med}=3$; $y=(1,2,3,4,50)\Rightarrow \bar{y}=12,\mathrm{med}=3$.

⚖️ Core Decision Matrix

Distribution	Mean vs Median	Tail
Symmetric	mean $\approx$ median	balanced
Positively skewed	mean $>$ median	long right tail
Negatively skewed	mean $<$ median	long left tail

When It Flips: the sign of (mean − median) diagnoses skew — the mean chases the long tail while the median stays put; equal ⇒ symmetric.

🧠 Active Recall

Why does changing one value from 5 to 50 move the mean from 3 to 12 but leave the median at 3?

• Hint: All-values vs middle-value.

Answer

• Short answer: The mean sums all values, so a large outlier inflates it; the median depends only on the middle position, unchanged when a non-middle value grows.
• Why: Resistance ➔ median uses at most two central values, so it’s robust to outliers; the mean is not.

How do you read skew from the mean and median?

• Hint: Mean chases the tail.

Answer

• Short answer: mean > median ⇒ positively skewed (right tail); mean < median ⇒ negatively skewed (left tail); mean ≈ median ⇒ symmetric.
• Why: Tail pull ➔ extreme values in the tail drag the mean toward that tail.