Statistical distributions

“Statistics is the grammar of science”, Karl Pearson

In the world of data science, there is a secret language that benefits those who understand it. Do you want to know what makes a data expert efficient? It’s having a profound understanding of the data. Unfortunately, you can’t have a friendly conversation with the data, but don’t worry, we have the next best solution.

Here are the top ten statistical concepts that you must have in your arsenal.  Whether you’re a budding data scientist, a seasoned professional, or merely intrigued by the inner workings of data-driven decision-making, prepare for an enthralling exploration of the statistical principles that underpin the world of data science. 

 

 10 statistical concepts you should know

 

1. Descriptive statistics: 

Starting with the most fundamental and essential statistical concept, descriptive statistics. Descriptive statistics are the specific methods and measures that describe the data. It’s like the foundation of your building. It is a sturdy groundwork upon which further analysis can be constructed. Descriptive statistics can be broken down into measures of central tendency and measures of variability. 

• Measure of Central Tendency: 

Central Tendency is defined as “the number used to represent the center or middle of a set of data values”. It is a single value that is typically representative of the whole data. They help us understand where the “average” or “central” point lies amidst a collection of data points.

There are a few techniques to find the central tendency of the data, namely “Mean” (average), “Median” (middle value when data is sorted), and “Mode” (most frequently occurring values).  

• Measures of variability: 

Measures of variability describe the spread, dispersion, and deviation of the data. In essence, they tell us how much each value point deviates from the central tendency. A few measures of variability are “Range”, “Variance”, “Standard Deviation”, and “Quartile Range”. These provide valuable insights into the degree of variability or uniformity in the data.   

 

 

 2. Inferential statistics: 

Inferential statistics enable us to draw conclusions about the population from a sample of the population. Imagine having to decide whether a medicinal drug is good or bad for the general public. It is practically impossible to test it on every single member of the population.

This is where inferential statistics comes in handy. Inferential statistics employ techniques such as hypothesis testing and regression analysis (also discussed later) to determine the likelihood of observed patterns occurring by chance and to estimate population parameters.

This invaluable tool empowers data scientists and researchers to go beyond descriptive analysis and uncover deeper insights, allowing them to make data-driven decisions and formulate hypotheses about the broader context from which the data was sampled. 

 

3. Probability distributions: 

Probability distributions serve as foundational concepts in statistics and mathematics, providing a structured framework for characterizing the probabilities of various outcomes in random events. These distributions, including well-known ones like the normal, binomial, and

Poisson distributions offer structured representations for understanding how data is distributed across different values or occurrences.

Much like navigational charts guiding explorers through uncharted territory, probability distributions function as reliable guides through the landscape of uncertainty, enabling us to quantitatively assess the likelihood of specific events.

They constitute essential tools for statistical analysis, hypothesis testing, and predictive modeling, furnishing a systematic approach to evaluate, analyze, and make informed decisions in scenarios involving randomness and unpredictability. Comprehension of probability distributions is imperative for effectively modeling and interpreting real-world data and facilitating accurate predictions. 

 

 

Read More —-> 7 types of statistical distributions with practical examples 

 

4. Sampling methods: 

We now know inferential statistics help us make conclusions about the population from a sample of the population. How do we ensure that the sample is representative of the population? This is where sampling methods come to aid us.

Sampling methods are a set of methods that help us pick our sample set out of the population. Sampling methods are indispensable in surveys, experiments, and observational studies, ensuring that our conclusions are both efficient and statistically valid. There are many types of sampling methods. Some of the most common ones are defined below. 

• Simple Random Sampling: A method where each member of the population has an equal chance of being selected for the sample, typically through random processes. 
• Stratified Sampling: The population is divided into subgroups (strata), and a random sample is taken from each stratum in proportion to its size. 
• Systematic Sampling: Selecting every “kth” element from a population list, using a systematic approach to create the sample. 
• Cluster Sampling: The population is divided into clusters, and a random sample of clusters is selected, with all members in selected clusters included. 
• Convenience Sampling: Selection of individuals/items based on convenience or availability, often leading to non-representative samples. 
• Purposive (Judgmental) Sampling: Researchers deliberately select specific individuals/items based on their expertise or judgment, potentially introducing bias. 
• Quota Sampling: The population is divided into subgroups, and individuals are purposively selected from each subgroup to meet predetermined quotas. 
• Snowball Sampling: Used in hard-to-reach populations, where participants refer researchers to others, leading to an expanding sample. 

 

5. Regression analysis: 

Regression analysis is a statistical method that helps us quantify the relationship between a dependent variable and one or more independent variables. It’s like drawing a line through data points to understand and predict how changes in one variable relate to changes in another.

Regression models, such as linear regression or logistic regression, are used to uncover patterns and causal relationships in diverse fields like economics, healthcare, and social sciences. This technique empowers researchers to make predictions, analyze cause-and-effect connections, and gain insights into complex phenomena. 

 

 

6. Hypothesis testing: 

Hypothesis testing is a key statistical method used to assess claims or hypotheses about a population using sample data. It’s like a process of weighing evidence to determine if there’s enough proof to support a hypothesis.

Researchers formulate a null hypothesis and an alternative hypothesis, then use statistical tests to evaluate whether the data supports rejecting the null hypothesis in favor of the alternative.

This method is crucial for making informed decisions, drawing meaningful conclusions, and assessing the significance of observed effects in various fields of research and decision-making. 

 

7. Data visualizations: 

Data visualization is the art and science of representing complex data in a visual and comprehensible form. It’s like translating the language of numbers and statistics into a graphical story that anyone can understand at a glance.

Effective data visualization not only makes data more accessible but also allows us to spot trends, patterns, and outliers, making it an essential tool for data analysis and decision-making. Whether through charts, graphs, maps, or interactive dashboards, data visualization empowers us to convey insights, share information, and gain a deeper understanding of complex datasets. 

 

 

Check out some of the most important plots for Data Science here. 

 

8. ANOVA (Analysis of variance): 

Analysis of Variance (ANOVA) is a statistical technique used to compare the means of two or more groups to determine if there are significant differences among them. It’s like the referee in a sports tournament, checking if there’s enough evidence to conclude that the teams’ performances are different.

ANOVA calculates a test statistic and a p-value, which indicates whether the observed differences in means are statistically significant or likely occurred by chance.

This method is widely used in research and experimental studies, allowing researchers to assess the impact of different factors or treatments on a dependent variable and draw meaningful conclusions about group differences. ANOVA is a powerful tool for hypothesis testing and plays a vital role in various fields, from medicine and psychology to economics and engineering. 

 

9. Time Series analysis: 

Time series analysis is a specialized field of statistics and data science that focuses on studying data points collected, recorded, or measured over time. It’s like examining the historical trajectory of a variable to understand its patterns and trends.

Time series analysis involves techniques for data visualization, smoothing, forecasting, and modeling to uncover insights and make predictions about future values.

This discipline finds applications in various domains, from finance and economics to climate science and stock market predictions, helping analysts and researchers understand and harness the temporal patterns within their data. 

 

10. Bayesian statistics: 

Bayesian statistics is a branch of statistics that takes a unique approach to probability and inference. Unlike classical statistics, which use fixed parameters, Bayesian statistics treat probability as a measure of uncertainty, updating beliefs based on prior information and new evidence.

It’s like continually refining your knowledge as you gather more data. Bayesian methods are particularly useful when dealing with complex, uncertain, or small-sample data, and they have applications in fields like machine learning, Bayesian networks, and decision analysis 

 

Statistical distributions help us understand a problem better by assigning a range of possible values to the variables, making them very useful in data science and machine learning. Here are 6 types of distributions with intuitive examples that often occur in real-life data. 

In statistics, a distribution is simply a way to understand how a set of data points are spread over some given range of values.  

For example, distribution takes place when the merchant and the producer agree to sell the product during a specific time frame. This form of distribution is exhibited by the agreement reached between Apple and AT&T to distribute their products in the United States. 

 

 

Types of statistical distributions 

There are several statistical distributions, each representing different types of data and serving different purposes. Here we will cover several commonly used distributions. 

1. Normal Distribution 
2. t-Distribution 
3. Binomial Distribution 
4. Poisson Distribution 
5. Uniform Distribution 

 

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1. Normal Distribution 

A normal distribution also known as “Gaussian Distribution” shows the probability density for a population of continuous data (for example height in cm for all NBA players). Also, it indicates the likelihood that any NBA player will have a particular height. Let’s say fewer players are much taller or shorter than usual; most are close to average height.  

The spread of the values in our population is measured using a metric called standard deviation. The Empirical Rule tells us that: 

• 68.3% of the values will fall between1 standard deviation above and below the mean 
• 95.5% of the values will fall between2 standard deviations above and below the mean 
• 99.7% of the values will fall between3 standard deviations above and below the mean 

 

Let’s assume that we know that the mean height of all players in the NBA is 200cm and the standard deviation is 7cm. If Le Bron James is 206 cm tall, what proportion of NBA players is he taller than? We can figure this out! LeBron is 6cm taller than the mean (206cm – 200cm). Since the standard deviation is 7cm, he is 0.86 standard deviations (6cm / 7cm) above the mean. 

Our value of 0.86 standard deviations is called the z-score. This shows that James is taller than 80.5% of players in the NBA!  

This can be converted to a percentile using the probability density function (or a look-up table) giving us our answer. A probability density function (PDF) defines the random variable’s probability of coming within a distinct range of values. 

 

2. t-distribution 

A t-distribution is symmetrical around the mean, like a normal distribution, and its breadth is determined by the variance of the data. A t-distribution is made for circumstances where the sample size is limited, but a normal distribution works with a population. With a smaller sample size, the t-distribution takes on a broader range to account for the increased level of uncertainty. 

The number of degrees of freedom, which is determined by dividing the sample size by one, determines the curve of a t-distribution. The t-distribution tends to resemble a normal distribution as sample size and degrees of freedom increase because a bigger sample size increases our confidence in estimating the underlying population statistics. 

For example, suppose we deal with the total number of apples sold by a shopkeeper in a month. In that case, we will use the normal distribution. Whereas, if we are dealing with the total amount of apples sold in a day, i.e., a smaller sample, we can use the t distribution. 

 

3. Binomial distribution 

A Binomial Distribution can look a lot like a normal distribution’s shape. The main difference is that instead of plotting continuous data, it plots a distribution of two possible discrete outcomes, for example, the results from flipping a coin. Imagine flipping a coin 10 times, and from those 10 flips, noting down how many were “Heads”. It could be any number between 1 and 10. Now imagine repeating that task 1,000 times. 

If the coin, we are using is indeed fair (not biased to heads or tails) then the distribution of outcomes should start to look at the plot above. In the vast majority of cases, we get 4, 5, or 6 “heads” from each set of 10 flips, and the likelihood of getting more extreme results is much rarer! 

 

4. Bernoulli distribution 

The Bernoulli Distribution is a special case of Binomial Distribution. It considers only two possible outcomes, success, and failure, true or false. It’s a really simple distribution, but worth knowing! In the example below we’re looking at the probability of rolling a 6 with a standard die.

If we roll a die many, many times, we should end up with a probability of rolling a 6, 1 out of every 6 times (or 16.7%) and thus a probability of not rolling a 6, in other words rolling a 1,2,3,4 or 5, 5 times out of 6 (or 83.3%) of the time! 

 

5. Discrete uniform distribution: All outcomes are equally likely 

Uniform distribution is represented by the function U(a, b), where a and b represent the starting and ending values, respectively. Like a discrete uniform distribution, there is a continuous uniform distribution for continuous variables.  

In statistics, uniform distribution refers to a statistical distribution in which all outcomes are equally likely. Consider rolling a six-sided die. You have an equal probability of obtaining all six numbers on your next roll, i.e., obtaining precisely one of 1, 2, 3, 4, 5, or 6, equaling a probability of 1/6, hence an example of a discrete uniform distribution. 

As a result, the uniform distribution graph contains bars of equal height representing each outcome. In our example, the height is a probability of 1/6 (0.166667). 

The drawbacks of this distribution are that it often provides us with no relevant information. Using our example of a rolling die, we get the expected value of 3.5, which gives us no accurate intuition since there is no such thing as half a number on a dice. Since all values are equally likely, it gives us no real predictive power. 

It is a distribution in which all events are equally likely to occur. Below, we’re looking at the results from rolling a die many, many times. We’re looking at which number we got on each roll and tallying these up. If we roll the die enough times (and the die is fair) we should end up with a completely uniform probability where the chance of getting any outcome is exactly the same 

 

6. Poisson distribution 

A Poisson Distribution is a discrete distribution similar to the Binomial Distribution (in that we’re plotting the probability of whole numbered outcomes) Unlike the other distributions we have seen however, this one is not symmetrical – it is instead bounded between 0 and infinity.  

For example, a cricket chirps two times in 7 seconds on average. We can use the Poisson distribution to determine the likelihood of it chirping five times in 15 seconds. A Poisson process is represented with the notation Po(λ), where λ represents the expected number of events that can take place in a period.

The expected value and variance of a Poisson process is λ. X represents the discrete random variable. A Poisson Distribution can be modeled using the following formula. 

The Poisson distribution describes the number of events or outcomes that occur during some fixed interval. Most commonly this is a time interval like in our example below where we are plotting the distribution of sales per hour in a shop. 

 

Conclusion: 

Data is an essential component of the data exploration and model development process. We can adjust our Machine Learning models to best match the problem if we can identify the pattern in the data distribution, which reduces the time to get to an accurate outcome.  

Indeed, specific Machine Learning models are built to perform best when certain distribution assumptions are met. Knowing which distributions, we’re dealing with may thus assist us in determining which models to apply. 

Statistical distributions help us understand a problem better by assigning a range of possible values to the variables, making them very useful in data science and machine learning. Here are 7 types of distributions with intuitive examples that often occur in real-life data.

Whether you’re guessing if it’s going to rain tomorrow, betting on a sports team to win an away match, framing a policy for an insurance company, or simply trying your luck on blackjack at the casino, probability, and distributions come into action in all aspects of life to determine the likelihood of events.

Having a sound statistical background can be incredibly beneficial in the daily life of a data scientist. Probability is one of the main building blocks of data science and machine learning. While the concept of probability gives us mathematical calculations, statistical distributions help us visualize what’s happening underneath.

 

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Having a good grip on statistical distribution makes exploring a new dataset and finding patterns within a lot easier. It helps us choose the appropriate machine-learning model to fit our data and speed up the overall process.

In this blog, we will be going over diverse types of data, the common distributions for each of them, and compelling examples of where they are applied in real life.

Before we proceed further, if you want to learn more about probability distribution, watch this video below:

 

 

Common Types of Data

Explaining various distributions becomes more manageable if we are familiar with the type of data they use. We encounter two different outcomes in day-to-day experiments: finite and infinite outcomes.

 

 

When you roll a die or pick a card from a deck, you have a limited number of outcomes possible. This type of data is called Discrete Data, which can only take a specified number of values. For example, in rolling a die, the specified values are 1, 2, 3, 4, 5, and 6.

Similarly, we can see examples of infinite outcomes from discrete events in our daily environment. Recording time or measuring a person’s height has infinitely many values within a given interval. This type of data is called Continuous Data, which can have any value within a given range. That range can be finite or infinite.

For example, suppose you measure a watermelon’s weight. It can be any value from 10.2 kg, 10.24 kg, or 10.243 kg. Making it measurable but not countable; hence, it is continuous. On the other hand, suppose you count the number of boys in a class; since the value is countable, it is discreet.

Types of Statistical Distributions

Depending on the type of data we use, we have grouped distributions into two categories, discrete distributions for discrete data (finite outcomes) and continuous distributions for continuous data (infinite outcomes).

 

Dig deeper into the Discrete vs Continuous data distributions

 

Discrete Distributions

Discrete Uniform Distribution: All Outcomes are Equally Likely

In statistics, uniform distribution refers to a statistical distribution in which all outcomes are equally likely. Consider rolling a six-sided die. You have an equal probability of obtaining all six numbers on your next roll, i.e., obtaining precisely one of 1, 2, 3, 4, 5, or 6, equaling a probability of 1/6, hence an example of a discrete uniform distribution.

As a result, the uniform distribution graph contains bars of equal height representing each outcome. In our example, the height is a probability of 1/6 (0.166667).

 

 

Uniform distribution is represented by the function U(a, b), where a and b represent the starting and ending values, respectively. Similar to a discrete uniform distribution, there is a continuous uniform distribution for continuous variables.

The drawbacks of this distribution are that it often provides us with no relevant information. Using our example of a rolling die, we get the expected value of 3.5, which gives us no accurate intuition since there is no such thing as half a number on a dice. Since all values are equally likely, it gives us no real predictive power.

 

 

Bernoulli Distribution: Single-trial with Two Possible Outcomes

The Bernoulli distribution is one of the easiest distributions to understand. It can be used as a starting point to derive more complex distributions. Any event with a single trial and only two outcomes follows a Bernoulli distribution. Flipping a coin or choosing between True and False in a quiz are examples of a Bernoulli distribution.

They have a single trial and only two outcomes. Let’s assume you flip a coin once; this is a single trail. The only two outcomes are either heads or tails. This is an example of a Bernoulli distribution.

Usually, when following a Bernoulli distribution, we have the probability of one of the outcomes (p). From (p), we can deduce the probability of the other outcome by subtracting it from the total probability (1), represented as (1-p).

It is represented by bern(p), where p is the probability of success. The expected value of a Bernoulli trial ‘x’ is represented as, E(x) = p, and similarly, Bernoulli variance is, Var(x) = p(1-p).

 

 

The graph of a Bernoulli distribution is simple to read. It consists of only two bars, one rising to the associated probability p and the other growing to 1-p.

Binomial Distribution: A Sequence of Bernoulli Events

The Binomial Distribution can be thought of as the sum of outcomes of an event following a Bernoulli distribution. Therefore, Binomial Distribution is used in binary outcome events, and the probability of success and failure is the same in all successive trials. An example of a binomial event would be flipping a coin multiple times to count the number of heads and tails.

 

Read more about Binomial Distribution and its importance in ML

 

Binomial vs Bernoulli distribution

The difference between these distributions can be explained through an example. Consider you’re attempting a quiz that contains 10 True/False questions. Trying a single T/F question would be considered a Bernoulli trial, whereas attempting the entire quiz of 10 T/F questions would be categorized as a Binomial trial. The main characteristics of Binomial Distribution are:

• Given multiple trials, each of them is independent of the other. That is, the outcome of one trial doesn’t affect another one.
• Each trial can lead to just two possible results (e.g., winning or losing), with probabilities p and (1 – p).

 

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A binomial distribution is represented by B (n, p), where n is the number of trials and p is the probability of success in a single trial. A Bernoulli distribution can be shaped as a binomial trial as B (1, p) since it has only one trial. The expected value of a binomial trial “x” is the number of times a success occurs, represented as E(x) = np. Similarly, variance is represented as Var(x) = np(1-p).

Let’s consider the probability of success (p) and the number of trials (n). We can then calculate the likelihood of success (x) for these n trials using the formula below:

 

 

For example, suppose that a candy company produces both milk chocolate and dark chocolate candy bars. The total products contain half milk chocolate bars and half dark chocolate bars. Say you choose ten candy bars at random and choosing milk chocolate is defined as a success. The probability distribution of the number of successes during these ten trials with p = 0.5 is shown here in the binomial distribution graph:

 

 

Poisson Distribution: The Probability that an Event May or May not Occur

Poisson distribution deals with the frequency with which an event occurs within a specific interval. Instead of the probability of an event, Poisson distribution requires knowing how often it happens in a particular period or distance. For example, a cricket chirps two times in 7 seconds on average. We can use the Poisson distribution to determine the likelihood of it chirping five times in 15 seconds.

A Poisson process is represented with the notation Po(λ), where λ represents the expected number of events that can take place in a period. The expected value and variance of a Poisson process is λ. X represents the discrete random variable. A Poisson Distribution can be modeled using the following formula.

 

Understand more about the Poisson Process and its properties

 

The main characteristics which describe the Poisson Processes are:

• The events are independent of each other.
• An event can occur any number of times (within the defined period).
• Two events can’t take place simultaneously.

 

 

The graph of Poisson distribution plots the number of instances an event occurs in the standard interval of time and the probability of each one.

Continuous Distributions

Normal Distribution: Symmetric Distribution of Values Around the Mean

Normal distribution is the most used distribution in data science. In a normal distribution graph, data is symmetrically distributed with no skew. When plotted, the data follows a bell shape, with most values clustering around a central region and tapering off as they go further away from the center.

The normal distribution frequently appears in nature and life in various forms. For example, the scores of a quiz follow a normal distribution. Many of the students scored between 60 and 80 as illustrated in the graph below. Of course, students with scores that fall outside this range are deviating from the center.

 

 

Here, you can witness the “bell-shaped” curve around the central region, indicating that most data points exist there. The normal distribution is represented as N(µ, σ2) here, µ represents the mean, and σ2 represents the variance, one of which is mostly provided. The expected value of a normal distribution is equal to its mean. Some of the characteristics which can help us to recognize a normal distribution are:

• The curve is symmetric at the center. Therefore mean, mode, and median are equal to the same value, distributing all the values symmetrically around the mean.
• The area under the distribution curve equals 1 (all the probabilities must sum up to 1).

 

Explore 9 key probability distributions in data science

 

68-95-99.7 Rule

While plotting a graph for a normal distribution, 68% of all values lie within one standard deviation from the mean. In the example above, if the mean is 70 and the standard deviation is 10, 68% of the values will lie between 60 and 80. Similarly, 95% of the values lie within two standard deviations from the mean, and 99.7% lie within three standard deviations from the mean. This last interval captures almost all matters. If a data point is not included, it is most likely an outlier.

 

 

Student t-Test Distribution: Small Sample Size Approximation of a Normal Distribution

The student’s t-distribution, also known as the t distribution, is a type of statistical distribution similar to the normal distribution with its bell shape but has heavier tails. The t distribution is used instead of the normal distribution when you have small sample sizes.

 

 

For example, suppose we deal with the total number of apples sold by a shopkeeper in a month. In that case, we will use the normal distribution. Whereas, if we are dealing with the total amount of apples sold in a day, i.e., a smaller sample, we can use the t distribution.

 

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Another critical difference between the student’s t distribution and the Normal one is that apart from the mean and variance, we must also define the degrees of freedom for the distribution. In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. A Student’s t distribution is represented as t(k), where k represents the number of degrees of freedom. For k=2, i.e., 2 degrees of freedom, the expected value is the same as the mean.

 

Degrees of freedom are in the left column of the t-distribution table.

 

Overall, the student t distribution is frequently used when conducting statistical analysis and plays a significant role in performing hypothesis testing with limited data.

Exponential Distribution: Model Elapsed Time between Two Events

Exponential distribution is one of the widely used continuous distributions. It is used to model the time taken between different events.

For example, in physics, it is often used to measure radioactive decay; in engineering, to measure the time associated with receiving a defective part on an assembly line; and in finance, to measure the likelihood of the next default for a portfolio of financial assets. Another common application of Exponential distributions in survival analysis (e.g., expected life of a device/machine).

 

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The exponential distribution is commonly represented as Exp(λ), where λ is the distribution parameter, often called the rate parameter. We can find the value of λ by the formula = 1/μ, where μ is the mean. Here, the standard deviation is the same as the mean. Var (x) gives the variance = 1/λ2

 

 

An exponential graph is a curved line representing how the probability changes exponentially. Exponential distributions are commonly used in calculations of product reliability or the length of time a product lasts.

Conclusion

Data is an essential component of the data exploration and model development process. The first thing that springs to mind when working with continuous variables is looking at the data distribution. We can adjust our machine-learning models to best match the problem if we can identify the pattern in the data distribution, which reduces the time to get to an accurate outcome.

 

 

Indeed, specific Machine Learning models are built to perform best when certain distribution assumptions are met. Knowing which distributions, we’re dealing with may thus assist us in determining which models to apply.