Descriptive Statistics

Descriptive Statistics

Descriptive statistics are methods for summarizing and describing data characteristics.

Measures of Central Tendency

Mean (Average)

Sum of all values divided by number of observations.

\[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\]

Example: Data: 2, 4, 6, 8, 10 Mean = (2+4+6+8+10)/5 = 30/5 = 6

Median (Middle Value)

The value in the middle after data is sorted.

How to find: 1. Sort data from smallest to largest 2. If n is odd: median = value at position (n+1)/2 3. If n is even: median = average of values at positions n/2 and (n/2)+1

Mode

The value that appears most frequently in the data.

Measures of Dispersion

Range

\[Range = X_{max} - X_{min}\]

Variance

\[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\]

Standard Deviation

\[s = \sqrt{s^2}\]

Standard deviation shows how far data is spread from the mean.

Interpretation

• Small standard deviation → data tends to be close to the mean
• Large standard deviation → data is widely spread from the mean

Example Case

Test scores data: 75, 80, 85, 90, 95

• Mean = 85
• Median = 85
• Range = 95 - 75 = 20
• No mode (all values appear once)