Empirical Rule

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The Empirical Rule (also known as the 68-95-99.7 Rule) is a statistical principle that applies to normally distributed data (bell-shaped curves). Empirical Rule tells us how data is spread around the mean in such (bell-shaped) distributions.

Empirical Rule states that:

  • 68% of data falls within 1 standard deviation ($\sigma$) of the mean ($\mu$). In other words, 68% of the data falls within ±1 standard deviation ($\sigma$) of the mean ($\mu$). Range: $\mu-1\sigma$ to $\mu+1\sigma$.
  • 95% of data falls within 2 standard deviations ($\sigma$) of the mean ($\mu$). In other words, 95% of the data falls within ±2 standard deviations ($2\sigma$) of the mean ($\mu$). Range: $\mu-2\sigma$ to $\mu+2\sigma$.
  • 99.7% of data falls within 3 standard deviations ($\sigma$) of the mean ($\mu$). In other words, 99.7% of the data falls within ±3 standard deviations ($3\sigma$) of the mean ($\mu$). Range: $\mu-3\sigma$ to $\mu+3\sigma$.

Visual Representation of Empirical Rule

The empirical rule can be visualized from the following graphical representation:

Visual Representation of Empirical Rule

Key Points

  • Empirical Rule only applies to normal (symmetric, bell-shaped) distributions.
  • It helps estimate probabilities and identify outliers.
  • About 0.3% of data lies beyond ±3σ (considered rare events).

Numerical Example of Empirical Rule

Suppose adult human heights are normally distributed with Mean ($\mu$) = 70 inches and standard deviation ($\sigma$) = 3 inches. Then:

  • 68% of heights are between 67–73 inches ($\mu \pm \sigma \Rightarrow 70 \pm 3$ ).
  • 95% are between 64–76 inches ($\mu \pm 2\sigma\Rightarrow 70 \pm 2\times 3$).
  • 99.7% are between 61–79 inches ($\mu \pm 3\sigma \Rightarrow 70 ± 3\times 3$).

This rule is a quick way to understand variability in normally distributed data without complex calculations. For non-normal distributions, other methods (like Chebyshev’s inequality) may be used.

Real-Life Applications & Examples

  • Quality Control in Manufacturing: Manufacturers measure product dimensions (e.g., bottle fill volume, screw lengths). If the process is normally distributed, the Empirical Rule helps detect defects: If soda bottles have a mean volume of 500ml with $\sigma$ = 10ml:
    • 68% of bottles will be between 490ml–510ml.
    • 95% will be between 480ml–520ml.
    • Bottles outside 470ml–530ml (3$\sigma$) are rare and may indicate a production issue.
  • Human Height Distribution: The Heights of people in a population often follow a normal distribution. If the average male height is 70 inches (5’10”) with $\sigma$ = 3 inches:
    • 68% of men are between 67–73 inches.
    • 95% are between 64–76 inches.
    • 99.7% are between 61–79 inches.
  • Test Scores (Standardized Exams): The exam scores (SAT, IQ tests) are often normally distributed. If SAT scores have $\mu$ = 1000 and $\sigma$ = 200:
    • 68% of students score between 800–1200.
    • 95% score between 600–1400.
    • Extremely low (<400) or high (>1600) scores are rare.
  • Financial Market Analysis (Stock Returns): The daily stock returns often follow a normal distribution. If a stock has an average daily return of 0.1% with σ = 2%: If a stock has an average daily return of 0.1% with σ = 2%:
    • 68% of days will see returns between -1.9% to +2.1%.
    • 95% will be between -3.9% to +4.1%.
    • Extreme crashes or surges beyond ±6% are very rare (0.3%).
  • Medical Data (Blood Pressure, Cholesterol Levels): Many health metrics are normally distributed. If the average systolic blood pressure is 120 mmHg with $\sigma$ = 10:
    • 68% of people have readings between 110–130 mmHg.
    • 95% fall within 100–140 mmHg.
    • Readings above 150 mmHg may indicate hypertension.
  • Weather Data (Temperature Variations): The daily temperatures in a region often follow a normal distribution. If the average July temperature is 85°F with σ = 5°F:
    • 68% of days will be between 80°F–90°F.
    • 95% will be between 75°F–95°F.
    • Extremely hot (>100°F) or cold (<70°F) days are rare.

Why the Empirical Rule Matters

  • It helps in predicting probabilities without complex calculations.
  • It is used in risk assessment (finance, insurance).
  • It guides quality control and process improvements.
  • It assists in setting thresholds (e.g., medical diagnostics, passing scores).

FAQs about Empirical Rule

  • What is the empirical rule?
  • For what kind of probability distribution, the empirical rule is used.
  • What is the area under the curve (or percentage) if data falls within 1, 2, and 3 standard deviations?
  • Represent the rule graphically.
  • Give real-life applications and examples of the rule.
  • Why the empirical rule matters, describe.

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Neural Networks MCQs 3

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Challenge yourself with these Neural Networks MCQs covering key concepts like activation functions (ReLU, Tanh), optimizers (Adam), loss functions, GANs, vanishing gradients, and more! Perfect for ML beginners and AI enthusiasts. Evaluate your understanding and boost your neural networks expertise today! Let us start with the Neural Networks MCQs now.

Online Neural Networks MCQs with Answers

Online Neural Network MCQs with Answers

1. What function is commonly used as the loss function in a regression model with Keras?

 
 
 
 

2. Select the characteristics of the ReLU activation function.

 
 
 
 
 

3. Select the optimizers that use momentum to accelerate gradient vectors in the relevant direction.

 
 
 
 
 

4. What is a key characteristic of Generative Adversarial Networks (GANs)?

 
 
 
 

5. Which of the following neural network types are designed to handle long-term dependencies in sequential data?

 
 
 
 
 

6. What does an optimizer do in the context of training a neural network?

 
 
 
 

7. Which activation function is most likely to suffer from the vanishing gradient problem?

 
 
 
 

8. Select all characteristics that apply to the Tanh activation function.

 
 
 
 
 

9. Which type of neural network is best suited for image recognition tasks?

 
 
 
 

10. What is the primary purpose of a loss function in training a neural network?

 
 
 
 

11. Which of the following are characteristics of an effective loss function in neural network training?

 
 
 
 
 

12. Which of the following steps are involved in creating a regression model using a multilayer perceptron neural network?

 
 
 
 
 

13. What are some common metrics used to evaluate a regression model in Keras?

 
 
 
 
 

14. Which activation function is defined by the equation $f(x) = \frac{1}{1+e^{−x}}$.

 
 
 
 

15. Which of the following statements accurately describe the Adam optimizer?

 
 
 
 
 

16. What is the main advantage of using RMSprop over standard SGD?

 
 
 
 

17. Select all the scenarios where Mean Squared Error (MSE) would be a more suitable loss function than Binary Cross Entropy.

 
 
 
 
 

18. What are the primary functions of an artificial neuron in a neural network?

 
 
 
 
 

19. Which neural network architecture is most suitable for tasks involving sequential data, such as text or speech?

 
 
 
 

20. In the context of neural networks, what is the primary role of an optimizer?

 
 
 
 

Online Neural Networks MCQs with Answers

  • What are the primary functions of an artificial neuron in a neural network?
  • What does an optimizer do in the context of training a neural network?
  • Which activation function is most likely to suffer from the vanishing gradient problem?
  • Select the characteristics of the ReLU activation function.
  • Which activation function is defined by the equation $f(x) = \frac{1}{1+e^{−x}}$.
  • What is the primary purpose of a loss function in training a neural network?
  • Select all the scenarios where Mean Squared Error (MSE) would be a more suitable loss function than Binary Cross Entropy.
  • Select all characteristics that apply to the Tanh activation function.
  • What is the main advantage of using RMSprop over standard SGD?
  • Which of the following statements accurately describe the Adam optimizer?
  • What is a key characteristic of Generative Adversarial Networks (GANs)?
  • Which neural network architecture is most suitable for tasks involving sequential data, such as text or speech?
  • What function is commonly used as the loss function in a regression model with Keras?
  • Select the optimizers that use momentum to accelerate gradient vectors in the relevant direction.
  • In the context of neural networks, what is the primary role of an optimizer?
  • Which of the following neural network types are designed to handle long-term dependencies in sequential data?
  • What are some common metrics used to evaluate a regression model in Keras?
  • Which type of neural network is best suited for image recognition tasks?
  • Which of the following steps are involved in creating a regression model using a multilayer perceptron neural network?
  • Which of the following are characteristics of an effective loss function in neural network training?

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Power Query MCQs 11

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Test your knowledge of Power Query with these multiple-choice questions! Challenge yourself with these Power Query MCQs and see how well you know data transformations, M language, and query editing. Perfect for Excel users, data analysts, and BI professionals looking to sharpen their data transformation skills. See how well you know Power Query and boost your data skills today! Let us start with the Power Query MCQs now.

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Please go to Power Query MCQs 11 to view the test

Online Power Query MCQs with Answers

  • Which sources does Power Query allow us to Get Data from?
  • What would happen if you tried to create a query from data in the current workbook that is not part of a table or a named range?
  • In Australia, the first two digits represent the area code for a phone number, such as 0223789456. Consider a field that contains phone numbers in this format. What would be the appropriate option under Split Column to extract the area code?
  • If you are utilising Power Query primarily as a ‘working space’ without viewing the data in your spreadsheet within Close & Load to, you should choose the option:
  • After creating a new Table via Power Query, what would happen when the original data is edited or changed?
  • When getting data from another workbook, it is essential to transform the data within that workbook first.
  • What are some differences between Power Query and standard Excel?
  • Imdad was working in Power Query and loaded the data into a new worksheet. He notices that he has made an error and needs to undo one of his steps. What should he do?
  • When getting data from a database, unlike getting data from a spreadsheet, you have to transform the data at the source first.
  • There are currently three columns in Power Query: Street Address, City, and State, with data such as “42 Wallaby Way” (Street Address), “Sydney” (City), and “NSW” (State). What could we do to create a new column that displayed the full address as a single string, such as “42 Wallaby Way, Sydney, NSW”?
  • If you created a table in Excel after getting data from a database, changing the data in the new table will update the original database if you click Refresh.
  • An American company has 50 offices, one in each state, which all use their own Excel spreadsheet for their human resources data, but the parent company wants to maintain a separate spreadsheet that gets the data from all these files. What would be an efficient solution to this problem?
  • When getting data from a folder, the preview panel only shows a preview of the first file. Suppose there is an Australian company with an office in each of the 8 states and territories, where the parent office is in the state of New South Wales. If the 8 files are named as below, which file will appear in the preview pane, and why? Western Australia South Australia Northern Territory Tasmania Victoria Australian Capital Territory New South Wales Queensland
  • Suppose that we have created a new table by getting data from a folder that contains data from each of 5 branches of a company, each with its file. What should we do if we open two new branches – that maintains the existing structure and also gives unique information for each branch in its file?
  • Which aspect of getting data from a folder is similar to the result of an Append Query?
  • When using an Append Query, the two tables must:
  • Suppose we created 2 queries, one for Sydney and one for Other Instructors. We did not load these into the worksheet; and we only created a connection. Due to this setup, when choosing to append these queries, the result cannot be loaded into the worksheet – we can only create a connection.
  • When getting data from a PDF that contains a table in each of the 5 pages, what will we see in the preview panel?
  • When getting data from a PDF that contains a table in each of 5 pages and selecting multiple items, this will create as many queries as the number of items you have selected.
  • When getting data from a PDF that contains multiple pages, what would be a query that would be usually run, right at the end?

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