- Understand key measures of central tendency (mean, median, mode).
- Learn about measures of dispersion (range, variance, standard deviation).
- Explore data visualization techniques.
Measures of Central Tendency
Central tendency measures describe the center of a dataset. The most common measures are:
| Measure | Formula | Definition |
|---|---|---|
| Mean (Arithmetic Average) | \( \bar{x} = \frac{\sum x_i}{n} \) | Sum of all values divided by the number of values. |
| Median | Middle value when data is ordered | If \( n \) is odd, the middle value; if even, the average of two middle values. |
| Mode | Most frequently occurring value | A dataset can have no mode, one mode, or multiple modes. |
Measures of Dispersion
Dispersion measures how spread out the data points are. Key measures include:
| Measure | Formula | Definition |
|---|---|---|
| Range | \( \text{Range} = \max(x) - \min(x) \) | Difference between the largest and smallest value. |
| Variance | \( \sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 \) | Measures the squared deviation of each value from the mean. |
| Standard Deviation | \( \sigma = \sqrt{\sigma^2} \) | Square root of the variance, indicating the typical deviation from the mean. |
Exercises
- Question 1: Find the mean, median, and mode of {5, 8, 9, 7, 6, 5, 5, 10}.
- Question 2: Compute the range, variance, and standard deviation for {4, 5, 6, 7, 8, 9, 10}.
- Question 3: Draw a histogram for the dataset {2, 3, 3, 4, 4, 4, 5, 5, 6}.
- Answer 1: Mean = 6.875, Median = 6.5, Mode = 5.
- Answer 2: Range = 6, Variance = 4.67, Standard Deviation ≈ 2.16.
- Answer 3: The histogram below represents the dataset from Question 3: