Frequency Distribution

Analyses start by viewing data. So, how to ‘view’ the data?

Understanding frequency distribution A frequency distribution is a representation, usually in the form of a table or graph, that illustrates the number of occurrences (frequency) of each unique value or range of values within a given dataset. This visual or tabular summary of data distribution allows us to understand the spread, shape, and central tendency of the data, providing initial understanding of the underlying patterns and structures in the dataset.

Examples of frequency distribution

And the measures of central tendency and spread.

(a): Bar chart (for discrete variable) (b): Histogram (for continuous variable)

Frequency Distribution and Density Function¶

Understanding PDF and CDF

Probability Density Function (PDF):

• A mathematical function representing the relative likelihood of observing a continuous random variable in a specific range of values.

• PDFs have non-negative values, and their total area under the curve equals.

• Different PDF shapes correspond to various probability distributions (e.g., normal, exponential).

Cumulative Distribution Function (CDF):

• Calculates the cumulative probability of a random variable being less than or equal to a given value.

• The CDF value at a point x corresponds to the area under the PDF curve to the left of x.

• CDFs are useful for assessing probabilities of events within a specific range.

The PDF and CDF are complementary tools in probability and statistics, providing insights into the distribution and likely outcomes of continuous random variables. PDFs offer a detailed view of relative likelihoods, while CDFs show the accumulated probabilities over a range of values. Together, they enable a comprehensive understanding of data distributions and facilitate various statistical analyses.

The PDF here is estimated from an empirical data series using Kernel Density Estimation (aka KDE).

If Y-axis shows count, then it should be a histogram, and if it shows ‘density’, then is should be either PDF or CDF.

Interpreting Frequency Distributions¶

Measure of Central Tendency

• Gravitational center: mean, aka arithmetic mean

• Middle value: median

• Most frequent value: mode

Measure of Spread

• Range and inter-quartile range

• Standard deviation and variance

Measure of Shape

• Symmetry: Mirrored between left and right of central line

• Skewness: Has a long tail on right (positive skew) or left (negative skew)

• Modality: Number of peaks

• Kurtosis: Steepness of peak compared to a normal distribution

what are the other ‘mean’ besides ‘arithmetic mean’?

Symmetry

Asymmetry

Modality

Kurtosis

Common Types of Frequency Distributions¶

Normal Distribution¶

The normal distribution, also known as the Gaussian distribution or the “bell curve,” is often used to model continuous variables in various fields. It is particularly useful in statistics because of the Central Limit Theorem, which states that the distribution of sample means approaches a normal distribution as the sample size increases. The normal distribution has applications in many areas, such as finance, psychology, and engineering.

Many real-world phenomena, such as heights, IQ scores, and measurement errors, tend to follow a normal distribution. As a result, the normal distribution plays a central role in statistics and is used as a basis for many statistical tests and analyses.

The normal distribution is defined by two parameters---the mean ($\mu$) and standard deviation ($\sigma$).

Poisson Distribution¶

The Poisson distribution is used to model discrete count data, such as the number of occurrences of a particular event within a specific time interval or spatial area. It is often applied to scenarios where events happen at a certain average rate, and each event is independent of the others. It has applications in various fields, such as telecommunications, insurance, and biology.

The Poisson distribution is ideal for modeling count data involving rare events. In many real-world situations, count data often represent occurrences of rare events.

The Poisson distribution is characterized by a single parameter, lambda (λ), which represents the average rate of events over a given time interval or a specific area.

Binomial Distribution¶

The binomial distribution is also used for discrete data, specifically for modeling the number of successes in a fixed number of independent trials. Each trial has only two possible outcomes (e.g., success or failure, true or false), and the probability of success remains the same across all trials. It is frequently applied in areas such as quality control, marketing research, and opinion polls.

The binomial distribution provides a straightforward way to calculate the probability of observing a specific number of successful outcomes in a fixed number of trials. This makes it easy to understand and communicate the results of analyses using binomial distributions.

The binomial distribution involves two parameters: number of trials and the probability of success (or being one of the two options).

Visualizing Frequency Distributions¶

Histogram¶

PDF plot¶

CDF plot¶

Box plot¶

Boxen plot¶

Joint grid plot for 2-variables¶