Measures of Center

Mean (Arithmetic Average)

The sample mean $\bar{x}$ is the sum of all values divided by the number of values:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}$

Properties of the mean:

• Uses every data value
• Sensitive to outliers and skewness
• The "balance point" of the distribution

Median

The median is the middle value when data is arranged in order.

• If $n$ is odd: median = middle value
• If $n$ is even: median = average of the two middle values

$\text{Median position} = \frac{n + 1}{2}$

Properties of the median:

• Resistant to outliers
• Better measure of center for skewed distributions

When to Use Which

| Situation | Best Measure | |-----------|-------------| | Symmetric distribution | Mean or Median (approximately equal) | | Skewed distribution | Median (more representative) | | Outliers present | Median (resistant) | | Further calculations needed | Mean (used in standard deviation, regression) |

Effect of Skewness

• Skewed right: Mean > Median (mean pulled toward tail)
• Symmetric: Mean ≈ Median
• Skewed left: Mean < Median

Weighted Mean

$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Used when different values have different weights (e.g., GPA calculation).

Trimmed Mean

Remove a percentage of the highest and lowest values, then calculate the mean. This makes it more resistant to outliers.

AP Tip: The AP exam often tests whether students know that the mean is affected by outliers while the median is not. Be ready to explain which measure is more appropriate in context.