Probability & Statistics 1 (S1)

Topic: Representation of Data

Objective: Diagrams, Measures of Central Tendency and Dispersion

This section covers how to effectively represent data visually, calculate measures of central tendency (like mean, median, and mode), and determine measures of dispersion (like range, variance, and standard deviation). These techniques are crucial for understanding and interpreting datasets.

1. Data Representation: Diagrams

Visual representations of data help in identifying patterns and trends more easily. Common diagrams include:

• Histograms: Used for continuous data. The height of each bar represents the frequency of data within a specific range.
• Bar Charts: Used for categorical data. The height of each bar represents the frequency or proportion of each category.
• Pie Charts: Used to show proportions of categorical data. Each slice represents a category, and the size of the slice is proportional to its proportion.
• Scatter Plots: Used to show the relationship between two continuous variables. Each point represents a pair of values.

2. Measures of Central Tendency

Measures of central tendency aim to identify a "typical" value in a dataset.

• Mean: The arithmetic average of all values. Calculated by summing all values and dividing by the number of values. $$ \text{Mean} = \frac{\sum x_i}{n} $$
• Median: The middle value in a sorted dataset. If there's an even number of values, the median is the average of the two middle values.
• Mode: The value that appears most frequently in a dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (multimodal).

3. Measures of Dispersion

Measures of dispersion indicate the spread or variability of data.

• Range: The difference between the highest and lowest values in a dataset. $$ \text{Range} = \text{Maximum} - \text{Minimum} $$
• Variance: The average of the squared differences from the mean. It measures how spread out the data is from the mean. $$ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} $$
• Standard Deviation: The square root of the variance. It provides a more interpretable measure of dispersion in the original units of the data. $$ \text{Standard Deviation} = \sqrt{\text{Variance}} $$
• Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1). It represents the spread of the middle 50% of the data.

4. Choosing the Appropriate Measure

The choice of which measure to use depends on the type of data and the characteristics of the dataset:

• Mean: Suitable for data that is approximately normally distributed and does not contain outliers.
• Median: Suitable for data with outliers or skewed distributions.
• Mode: Suitable for categorical data or data with multiple peaks.
• Range: Simple to calculate but sensitive to outliers.
• Variance & Standard Deviation: Useful for understanding the overall spread of the data.

5. Example Calculation

Consider the following dataset: 4, 7, 1, 9, 5

1. Mean: $$ \frac{4 + 7 + 1 + 9 + 5}{5} = \frac{26}{5} = 5.2 $$ $$ $$
2. Median: First, sort the data: 1, 4, 5, 7, 9. The middle value is 5. $$ $$
3. Mode: All values appear once, so there is no mode. $$ $$
4. Range: 9 - 1 = 8 $$ $$
5. Variance: First, calculate the squared differences from the mean: $$ (4 - 5.2)^2 = 1.44 $$ $$ (7 - 5.2)^2 = 2.89 $$ $$ (1 - 5.2)^2 = 16.81 $$ $$ (9 - 5.2)^2 = 18.49 $$ $$ (5 - 5.2)^2 = 0.04 $$ Sum of squared differences: 1.44 + 2.89 + 16.81 + 18.49 + 0.04 = 49.67 Variance: $$ \frac{49.67}{5} = 9.934 $$
6. Standard Deviation: $$ \sqrt{9.934} \approx 3.15 $$ $$ $$