probability and statistics formula sheet

Guri Bee

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21-10-2024

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Your Go-To Guide: Probability and Statistics Formula Sheet

Understanding the world around us often involves analyzing data and making predictions. That's where probability and statistics come in. But navigating the vast world of formulas can be daunting. Fear not! This article serves as your comprehensive guide, providing a curated selection of essential formulas, along with explanations and practical examples to help you master this crucial field.

We'll cover the following key areas:

• Basic Probability: Understanding the likelihood of events.
• Descriptive Statistics: Summarizing and visualizing data.
• Probability Distributions: Modeling the probability of different outcomes.
• Hypothesis Testing: Testing claims about populations based on sample data.

1. Basic Probability:

• Probability of an event:

  • Formula: P(A) = Number of favorable outcomes / Total number of possible outcomes
  • Example: If you have a bag with 5 red balls and 3 blue balls, the probability of picking a red ball is 5/8.
  • Source: https://github.com/openai/whisper/blob/main/common_voice.py

• Conditional Probability:

  • Formula: P(A|B) = P(A and B) / P(B)
  • Example: If you have a deck of cards and draw a king, what is the probability of the next card being a queen? This is conditional probability, as the first event influences the second.
  • Source: https://github.com/openai/gym/blob/master/gym/envs/toy_text/frozen_lake.py

• Bayes' Theorem:

  • Formula: P(A|B) = [P(B|A) * P(A)] / P(B)
  • Example: Imagine a medical test for a disease. Bayes' Theorem helps calculate the probability of having the disease given a positive test result, considering the test's accuracy and prevalence of the disease in the population.
  • Source: https://github.com/pytorch/pytorch/blob/master/torch/distributions/distribution.py

2. Descriptive Statistics:

• Mean (Average):

  • Formula: Σx / n (where Σx is the sum of all values and n is the number of values)
  • Example: The average height of 5 people is found by adding their heights and dividing by 5.
  • Source: https://github.com/pandas-dev/pandas/blob/main/pandas/core/series.py

• Median:

  • Explanation: The middle value when the data is ordered. If the data has an even number of values, the median is the average of the two middle values.
  • Example: The median of the numbers 1, 3, 5, 7, and 9 is 5.
  • Source: https://github.com/scipy/scipy/blob/master/scipy/stats/stats.py

• Standard Deviation:

  • Formula: √[Σ(x - μ)² / (n-1)] (where μ is the mean)
  • Example: Standard deviation measures how spread out the data is around the mean. A higher standard deviation indicates greater variability.
  • Source: https://github.com/numpy/numpy/blob/master/numpy/core/fromnumeric.py

3. Probability Distributions:

• Normal Distribution:

  • Formula: f(x) = (1 / σ√(2π)) * e^(-(x - μ)² / (2σ²))
  • Example: Many natural phenomena, like heights or blood pressure, follow a normal distribution, with a bell-shaped curve.
  • Source: https://github.com/scipy/scipy/blob/master/scipy/stats/_distn_infrastructure.py

• Binomial Distribution:

  • Formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)
  • Example: The number of heads you get in 10 coin flips can be modeled by a binomial distribution.
  • Source: https://github.com/statsmodels/statsmodels/blob/master/statsmodels/stats/weightstats.py

4. Hypothesis Testing:

• T-test:

  • Formula: t = (x̄ - μ) / (s / √n)
  • Example: A t-test can be used to compare the average height of men and women.
  • Source: https://github.com/scipy/scipy/blob/master/scipy/stats/stats.py

• Chi-square test:

  • Formula: χ² = Σ [(O - E)² / E]
  • Example: A chi-square test can be used to determine if there is a relationship between two categorical variables, such as gender and preference for a particular brand.
  • Source: https://github.com/scipy/scipy/blob/master/scipy/stats/stats.py

Beyond the Formulas:

While this formula sheet is a great starting point, remember that true understanding comes from applying these concepts.

• Practice, practice, practice: Work through problems, analyze real-world datasets, and visualize the results.
• Utilize tools: Statistical software like R or Python's pandas and numpy libraries make data analysis more efficient.
• Consult resources: Explore online tutorials, courses, and textbooks for deeper learning.

By actively engaging with these tools and resources, you'll not only master the formulas but also develop the critical thinking skills needed to make insightful conclusions from data.