Early Ideas with Probability Distributions and Data: Study Notes for Introductory Statistics

Probability Distributions and Data

Introduction to Probability Distributions

Probability distributions describe how the values of a variable are distributed across possible outcomes. They are foundational in statistics for understanding both populations and samples. Different types of variables (continuous, discrete, categorical) have different types of probability distributions.

• Continuous Variables: Can take any value within a range (e.g., actual volume of coffee dispensed, waistline measurements).

• Discrete Variables: Take specific, separate values (often counts, e.g., number of bacteria colonies).

• Categorical Variables: Represent categories or groups (e.g., type of pet: cat, dog, other).

Examples of probability distributions include the distribution of coffee dispensed by a vending machine, waistline measurements, counts of bacteria, and simulated random numbers.

Describing Data with Tables and Graphs

Frequency Distribution Tables

To understand a sample, we often summarize data using a frequency distribution table. This table groups data into intervals (called bins, intervals, or classes) and counts how many values fall into each bin.

• Bin (Interval/Class): A range of values that includes its left endpoint but not its right endpoint.

• Frequency: The number of sample points in each bin.

• Bins should not overlap, have no gaps, and together cover all sample values.

Example: Coffee vending machine sample data (50 measurements of dispensed ounces) can be grouped into bins as follows:

Bin (Ounces)	Frequency
11.40 – 11.50	1
11.50 – 11.60	2
11.60 – 11.70	2
11.70 – 11.80	4
11.80 – 11.90	6
11.90 – 12.00	3
12.00 – 12.10	4
12.10 – 12.20	7
12.20 – 12.30	8
12.30 – 12.40	7
12.40 – 12.50	4
12.50 – 12.60	3
12.60 – 12.70	2
Total	50

Additional info: Table reconstructed from context; some frequencies inferred for completeness.

Histograms

A histogram is a graphical representation of a frequency distribution. Each bin is represented by a rectangle whose height corresponds to the frequency of values in that bin. As the sample size increases, the histogram more closely resembles the underlying probability distribution curve.

• Choose the number of bins based on sample size (e.g., 10–15 bins for n = 50).

• Bin width should be consistent and chosen so that number of bins × bin width slightly exceeds the data range.

• Example calculation for bin width: For data ranging from 42.3 to 98.8 (range = 56.5), with 15 bins, bin width ≈ 4.0.

Steps to Create a Histogram

1. Sort data in ascending order.

2. Decide on the number of bins and calculate bin width.

3. Build the frequency distribution table by counting values in each bin.

4. Draw the histogram, with bin frequencies as rectangle heights.

Describing Data Numerically

Variables and Types

Each attribute measured in a population is called a variable. Variables can be:

• Numerical (Quantitative): Continuous (e.g., distance from home) or discrete (e.g., number of credits).

• Categorical (Qualitative): Binary (e.g., commutes: yes/no) or with more categories (e.g., major, minor).

Example: In a dataset of college students, variables might include major (categorical), credits (discrete numerical), and distance from home (continuous numerical).

Probability

Probability Distributions for Different Variable Types

• Continuous Variables: Probability is defined as the area under the distribution curve over an interval. The total area under the curve is 1.

• Discrete Variables: Probability is assigned to each possible value. The sum of all probabilities is 1.

• Categorical Variables: Probability is assigned to each category, often displayed in a table or pie chart.

For continuous variables, probabilities for intervals (left tail, right tail, or between two values) are found by calculating the area under the curve for that interval.

Example: Discrete Probability Distribution

Value	Probability
0	0.05
1	0.15
2	0.25
3	0.25
4	0.20
5	0.10

Sum of probabilities: $0.05 + 0.15 + 0.25 + 0.25 + 0.20 + 0.10 = 1$

Example: Categorical Probability Distribution

Category	Probability
Cats	0.30
Dogs	0.67
Other	0.03

Graphical representations include pie charts and bar charts.

Describing Data Numerically: Mode

Mode of a Distribution

The mode of a probability distribution is the value with the highest probability (the peak of the distribution curve).

• For a sample, the mode is the most frequently occurring value.

• A sample can have no mode, one mode, or multiple modes.

Examples:

• Sample: 4, 8, 8, 11, 12, 13 → Mode: 8

• Sample: 4, 8, 11, 12, 13 → No mode

• Sample: 4, 8, 8, 11, 12, 13, 13 → Modes: 8 and 13

Notation for Samples

Sample Size and Data Notation

• n: Sample size (number of values in the sample).

• x_1, x_2, ..., x_n: The values in the sample, where the subscript indicates the order or index.

• Sigma Notation (Summation): $\sum_{i=1}^{n} x_i$ means to sum all sample values from $x_1$ to $x_n$.

Example: For the coffee data, $n = 50$, $x_1 = 11.40$, $x_{50} = 12.61$.

Summary Table: Types of Variables

Type	Definition	Examples
Continuous Numerical	Any value in a range	Height, weight, coffee ounces
Discrete Numerical	Specific, separate values (often counts)	Number of credits, bacteria count
Categorical (Binary)	Two categories	Yes/No, Success/Failure
Categorical (Multiple)	Three or more categories	Major, pet type

Key Formulas

• Range: $\text{Range} = \text{Maximum} - \text{Minimum}$

• Sum of Sample Values: $\sum_{i=1}^{n} x_i$

• Probability (Discrete): $\sum_{i} P(x_i) = 1$

• Probability (Continuous): $\int_{-\infty}^{\infty} f(x) dx = 1$

Additional Info

• Choosing the number of bins and bin width is not an exact science; the goal is to best reveal the shape of the underlying distribution.

• Larger sample sizes produce histograms that more closely match the true distribution curve, reducing random variability ('noise').

• For categorical variables, probabilities can be communicated as percentages (e.g., 82% free throws made).