Introduction of Statistical Data Distributions

A statistical data distribution is a function that shows the possible values of a variable and how frequently they occur. It provides a mathematical description of the data’s behavior which indicate where most data points are concentrated and how they are spread out. Distributions can be represented in various forms such as probability density functions for continuous data or probability mass functions for discrete data.

• Probability Function: A probability function is used to assign probabilities to different possible outcomes in a dataset.
• Probability Density Function (PDF): It is used for continuous variables. It doesn’t give the exact probability of one value but instead shows the likelihood of a value falling within a certain range.
• Cumulative Distribution Function (CDF): This represents the probability that a variable takes a value less than or equal to a specific point.

We can classify statistical data distributions into two categories:

Discrete Distributions:

• A discrete distribution is used when the data can take only certain separate values. These values are countable like 1, 2, 3 and so on. You can’t have values in between like 1.5 or 2.7.
• For example the number of books on a shelf or the number of students in a class is discrete. Some common types of discrete distributions include the Binomial distribution, Poisson distribution and more.

Continuous Distributions:

• A continuous distribution is used when the data can take any value within a range including fractions and decimals. This is used for things you can measure like height, weight, temperature or time.
• For example someone can be 165.3 cm tall or a race can take 12.75 seconds. Common types of continuous distributions include the Normal distribution , Exponential distribution and more.

Common Statistical Distributions

There are some common statistical distribution that we use:

1. Normal Distribution

Normal distribution is one of the most commonly used distributions having a symmetrical bell shaped curve. In this most data points are close to mean and as you move away from it number of data points decreases. For example if we look at people's heights most people will be around a certain average height with fewer people being very short or very tall. 

2. Binomial Distribution

Binomial distribution is used when you perform a action multiple times and each time there are only two possible outcomes success or failure. For example if you flip a coin 10 times and want to know how many times you'll get heads you can apply the binomial distribution to calculate that probability. It helps you find the chances of getting a certain number of successes like heads out of a fixed number of tries.

3. Poisson Distribution

Poisson distribution is used to model events that happen randomly over a fixed time or space. For example think about how many cars pass through a toll booth in one hour or how many emails you get in a day. These events don’t happen at regular intervals. It’s useful when events are independent and occur at a constant average rate.

4. Exponential Distribution

The exponential distribution is closely related to the Poisson distribution but instead of counting how many events occur within a fixed time it focus on the time between events. For example if you're waiting for a bus the exponential distribution can help you estimate how long you have to wait before the next bus arrives.

5. Uniform Distribution

The uniform distribution is quite straightforward. In this type of distribution every outcome has an equal chance of occurring. Imagine rolling a fair six-sided die with each number from 1 to 6 has an equal probability of coming up.

6. Student’s T-Distribution

The t-distribution is similar to the normal distribution but it has heavier tails. It has more variability and is especially useful when dealing with small sample sizes. When researchers don’t know the population's standard deviation and have limited data they often use this distribution to make estimates about means.

Properties of Distributions

1. Mean (μ)

Mean is the average of all data points in a distribution helps us find the central point around which the data clusters giving us an idea of what a typical value looks like.

2. Variance (σ²) and Standard Deviation (σ)

It measures of how spread out the data points are around the mean. Variance calculates the average of the squared differences from the mean which helps us understand how much individual data points differ from that average. The standard deviation is simply the square root of the variance provides a more intuitive sense of this spread.

3. Skewness

In skewness we measures how symmetrical or asymmetrical a distribution is. If a distribution has longer tail on the right side it is said to be positively skewed and if it has longer tail on the left side it is called negatively skewed

4. Kurtosis

It refers to the "tailedness" of a distribution and indicate how much data is in the tails compared to the center. Kurtosis are of three types:

• Leptokurtic: Distributions with heavy tails.
• Platykurtic: Distributions with light tails.
• Mesokurtic: Distributions with tails similar to the normal distribution.

5. Mode

Mode means value that appears most frequently in a dataset. It indicates where most of the data points cluster and represents the peak of the distribution.

With these properties and statistical distribution we can easily analyse our data and can be helpful for EDA, machine learning model making and many more.

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• Engineering Mathematics