Compound Events and Sample Spaces

Data Analysis, Statistics, and Probability

By the end of this lesson, you will be able to calculate measures of center and spread, interpret data displays, and compute basic probabilities.

Measures of Center

Mean: the arithmetic average.

$$\bar{x} = \frac{\text{sum of all values}}{\text{number of values}}$$

Example: Find the mean of 14, 18, 22, 11, 15.

$\bar{x} = \dfrac{14+18+22+11+15}{5} = \dfrac{80}{5} = 16$

Median: the middle value when data is ordered from least to greatest.

• Odd number of values: the middle value
• Even number of values: the mean of the two middle values

Example: Find the median of 5, 11, 13, 18, 22, 26.

6 values → two middle values: 13 and 18

Median = $\dfrac{13+18}{2} = 15.5$

Mode: the value(s) that appear most often.

• Can have one mode, multiple modes (bimodal, multimodal), or no mode.

Example: 3, 5, 5, 7, 9, 9, 9, 11 → mode = 9 (appears 3 times)

Choosing the Best Measure

Outliers (data points far from the rest) affect the mean significantly but not the median.

Rule of thumb:

• Symmetric distribution: mean ≈ median; either works
• Skewed distribution or outliers: median is a better representative measure

Example: Salaries: $40k, $42k, $45k, $48k, $250k (CEO).

Mean = $\dfrac{40+42+45+48+250}{5} = \dfrac{425}{5} = $85k$

Median = $45k

The $85k mean is misleading because of the CEO's salary. The $45k median is more representative.

Measures of Spread

Range: the difference between the maximum and minimum values.

$\text{Range} = \text{Max} - \text{Min}$

Mean Absolute Deviation (MAD): the average distance of each data point from the mean.

1. Find the mean
2. Find the distance (absolute difference) from the mean for each data point
3. Average those distances

Interquartile Range (IQR):

• Divide ordered data into four equal parts (quartiles)
• $Q_1$ = median of the lower half; $Q_3$ = median of the upper half
• $IQR = Q_3 - Q_1$
• IQR is resistant to outliers (it focuses on the middle 50% of data)