Measures of Central Tendency

The mean is the sum of all values divided by the number of values:

Example: Test scores: 72, 85, 90, 65, 88

$$\bar{x} = \frac{72 + 85 + 90 + 65 + 88}{5} = \frac{400}{5} = 80$$

From a frequency table use $\bar{x} = \dfrac{\sum f \times x}{\sum f}$:

$$\bar{x} = \frac{790}{10} = 79$$

Mean: x-bar = (sum of x) / n, add all values, divide by count. Sensitive to outliers. Used when data is roughly symmetric. For grouped data, use midpoints as representatives. Always show the sum and the division as two separate steps.

Pause, copy the mean formula x̄ = Σx/n (sum all values, divide by count), the outlier sensitivity rule (one extreme value shifts the mean significantly), and when to use it (symmetric data, no outliers) into your book.

We just saw that the mean x̄ = Σx/n averages all values, but one extreme value can drag it far from where most data sits. That raises a question: are there measures of centre that don't get pulled by outliers? This card answers it → the median (middle value when ordered) and mode (most frequent value) are both resistant to outliers; the median is preferred whenever data is skewed or contains extreme values.

Median the middle value when data is ordered from smallest to largest.

Odd number of values: the median is the exact middle value.

Example: 12, 15, 18, 22, 25 → Median = 18

Even number of values: the median is the average of the two middle values.

Example: 12, 15, 18, 22 → Median = $\dfrac{15 + 18}{2} = 16.5$

Mode the value that appears most frequently.

Example: 3, 5, 5, 7, 8, 5, 9 → Mode = 5

A data set can be:

• Unimodal: one mode (most common).
• Bimodal: two modes.
• Multimodal: more than two modes.
• No mode: all values appear equally often.

The mode is the only measure of centre that applies to categorical data.

Median: middle value when data is ordered. For n odd: position (n+1)/2. For n even: average of positions n/2 and n/2+1. Mode: most frequent value(s). Median is preferred over mean when outliers are present.

Pause, copy the median position formulas (odd n: position (n+1)/2; even n: average positions n/2 and n/2+1) and the mode definition (most frequent value; data can have multiple modes or none) into your book.

We just saw the median and mode as alternatives to the mean for skewed data. That raises a question: how exactly does a single outlier change the mean vs the median, and how does the relationship between mean and median tell you about the shape of the distribution? This card answers it → an outlier pulls the mean toward it but leaves the median unchanged; positive skew (tail right) means mean > median; negative skew (tail left) means mean < median.

An outlier is a value far from the rest of the data. Outliers pull the mean toward them but do not affect the median.

Example, Salaries: $40K, $45K, $50K, $55K, $200K

Mean = $\dfrac{40+45+50+55+200}{5} = \$78K$ (misleading). Median = $50K (representative).

Skewed distributions:

• Right-skewed (positive skew): Long tail to the right. Mean > Median. Example: income distribution, house prices.
• Left-skewed (negative skew): Long tail to the left. Mean < Median.
• Symmetric: Mean ≈ Median ≈ Mode. Bell-shaped curves.

An outlier pulls the mean toward it but leaves the median unchanged. Positive skew (tail right): mean > median. Negative skew (tail left): mean < median. For skewed or outlier-affected data, the median is the better measure of centre.

Pause, copy the outlier effect rule (outlier shifts mean toward it, median unchanged), the skew–mean–median relationship (positive skew: mean > median; negative skew: mean < median), and the guideline (use median for skewed data or when outliers are present) into your book.