- Q1: Explain the central limit theorem and give examples of when you can use it in a real-world problem?
- Q2: Briefly explain the A/B testing and its application? What are some common pitfalls encountered in A/B testing?
- Q3: Describe briefly the hypothesis testing and p-value in layman’s term? And give a practical application for them ?
- Q4: Given a left-skewed distribution that has a median of 60, what conclusions can we draw about the mean and the mode of the data?
- Q5: What is the meaning of selection bias and how to avoid it?
- Q6: Explain the long-tailed distribution and provide three examples of relevant phenomena that have long tails. Why are they important in classification and regression problems?
- Q7: What is the meaning of KPI in statistics
- Q8: Say you flip a coin 10 times and observe only one head. What would be the null hypothesis and p-value for testing whether the coin is fair or not?
- Q9: You are testing hundreds of hypotheses, each with a t-test. What considerations would you take into account when doing this?
- Q10: What general conditions must be satisfied for the central limit theorem to hold?
- Q11: What is skewness discuss two methods to measure it?
- Q12: You sample from a uniform distribution [0, d] n times. What is your best estimate of d?
- Q13: Discuss the Chi-square, ANOVA, and t-test
- Q14: Say you have two subsets of a dataset for which you know their means and standard deviations. How do you calculate the blended mean and standard deviation of the total dataset? Can you extend it to K subsets?
- Q15: What is the relationship between the significance level and the confidence level in Statistics?
- Q16: What is the Law of Large Numbers in statistics and how it can be used in data science ?
- Q17: What is the difference between a confidence interval and a prediction interval, and how do you calculate them?
Q1: Explain the central limit theorem and give examples of when you can use it in a real-world problem?
Answers:
The center limit theorem states that if any random variable, regardless of the distribution, is sampled a large enough times, the sample mean will be approximately normally distributed. This allows for studying the properties of any statistical distribution as long as there is a large enough sample size.
Important remark from Adrian Olszewski:
Examples of real-world usage of CLT:
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The CLT can be used at any company with a large amount of data. Consider companies like Uber/Lyft wants to test whether adding a new feature will increase the booked rides or not using hypothesis testing. So if we have a large number of individual ride X, which in this case is a Bernoulli random variable (since the rider will book a ride or not), we can estimate the statistical properties of the total number of bookings. Understanding and estimating these statistical properties play a significant role in applying hypothesis testing to your data and knowing whether adding a new feature will increase the number of booked riders or not.
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Manufacturing plants often use the central limit theorem to estimate how many products produced by the plant are defective.
Q2: Briefly explain the A/B testing and its application? What are some common pitfalls encountered in A/B testing?
A/B testing helps us to determine whether a change in something will cause a change in performance significantly or not. So in other words you aim to statistically estimate the impact of a given change within your digital product (for example). You measure success and counter metrics on at least 1 treatment vs 1 control group (there can be more than 1 XP group for multivariate tests).
Applications:
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Consider the example of a general store that sells bread packets but not butter, for a year. If we want to check whether its sale depends on the butter or not, then suppose the store also sells butter and sales for next year are observed. Now we can determine whether selling butter can significantly increase/decrease or doesn't affect the sale of bread.
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While developing the landing page of a website you create 2 different versions of the page. You define a criteria for success eg. conversion rate. Then define your hypothesis Null hypothesis(H): No difference between the performance of the 2 versions. Alternative hypothesis(H'): version A will perform better than B.
NOTE: You will have to split your traffic randomly(to avoid sample bias) into 2 versions. The split doesn't have to be symmetric, you just need to set the minimum sample size for each version to avoid undersample bias.
Now if version A gives better results than version B, we will still have to statistically prove that results derived from our sample represent the entire population. Now one of the very common tests used to do so is 2 sample t-test where we use values of significance level (alpha) and p-value to see which hypothesis is right. If p-value<alpha, H is rejected.
Common pitfalls:
- Wrong success metrics inadequate to the business problem
- Lack of counter metric, as you might add friction to the product regardless along with the positive impact
- Sample mismatch: heterogeneous control and treatment, unequal variances
- Underpowered test: too small sample or XP running too short 5. Not accounting for network effects (introduce bias within measurement)
Q3: Describe briefly the hypothesis testing and p-value in layman’s term? And give a practical application for them ?
In Layman's terms:
- Hypothesis test is where you have a current state (null hypothesis) and an alternative state (alternative hypothesis). You assess the results of both of the states and see some differences. You want to decide whether the difference is due to the alternative approach or not.
You use the p-value to decide this, where the p-value is the likelihood of getting the same results the alternative approach achieved if you keep using the existing approach. It's the probability to find the result in the gaussian distribution of the results you may get from the existing approach.
The rule of thumb is to reject the null hypothesis if the p-value < 0.05, which means that the probability to get these results from the existing approach is <95%. But this % changes according to task and domain.
To explain the hypothesis testing in Layman's term with an example, suppose we have two drugs A and B, and we want to determine whether these two drugs are the same or different. This idea of trying to determine whether the drugs are the same or different is called hypothesis testing. The null hypothesis is that the drugs are the same, and the p-value helps us decide whether we should reject the null hypothesis or not.
p-values are numbers between 0 and 1, and in this particular case, it helps us to quantify how confident we should be to conclude that drug A is different from drug B. The closer the p-value is to 0, the more confident we are that the drugs A and B are different.
Q4: Given a left-skewed distribution that has a median of 60, what conclusions can we draw about the mean and the mode of the data?
Answer: Left skewed distribution means the tail of the distribution is to the left and the tip is to the right. So the mean which tends to be near outliers (very large or small values) will be shifted towards the left or in other words, towards the tail.
While the mode (which represents the most repeated value) will be near the tip and the median is the middle element independent of the distribution skewness, therefore it will be smaller than the mode and more than the mean.
Mean < 60 Mode > 60
Answer:
Sampling bias is the phenomenon that occurs when a research study design fails to collect a representative sample of a target population. This typically occurs because the selection criteria for respondents failed to capture a wide enough sampling frame to represent all viewpoints.
The cause of sampling bias almost always owes to one of two conditions.
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Poor methodology: In most cases, non-representative samples pop up when researchers set improper parameters for survey research. The most accurate and repeatable sampling method is simple random sampling where a large number of respondents are chosen at random. When researchers stray from random sampling (also called probability sampling), they risk injecting their own selection bias into recruiting respondents.
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Poor execution: Sometimes data researchers craft scientifically sound sampling methods, but their work is undermined when field workers cut corners. By reverting to convenience sampling (where the only people studied are those who are easy to reach) or giving up on reaching non-responders, a field worker can jeopardize the careful methodology set up by data scientists.
The best way to avoid sampling bias is to stick to probability-based sampling methods. These include simple random sampling, systematic sampling, cluster sampling, and stratified sampling. In these methodologies, respondents are only chosen through processes of random selection—even if they are sometimes sorted into demographic groups along the way.
Q6: Explain the long-tailed distribution and provide three examples of relevant phenomena that have long tails. Why are they important in classification and regression problems?
Answer: A long-tailed distribution is a type of heavy-tailed distribution that has a tail (or tails) that drop off gradually and asymptotically.
Three examples of relevant phenomena that have long tails:
- Frequencies of languages spoken
- Population of cities
- Pageviews of articles
All of these follow something close to 80-20 rule: 80% of outcomes (or outputs) result from 20% of all causes (or inputs) for any given event. This 20% forms the long tail in the distribution.
It’s important to be mindful of long-tailed distributions in classification and regression problems because the least frequently occurring values make up the majority of the population. This can ultimately change the way that you deal with outliers, and it also conflicts with some machine learning techniques with the assumption that the data is normally distributed.
Answer:
KPI stands for key performance indicator, a quantifiable measure of performance over time for a specific objective. KPIs provide targets for teams to shoot for, milestones to gauge progress, and insights that help people across the organization make better decisions. From finance and HR to marketing and sales, key performance indicators help every area of the business move forward at the strategic level.
KPIs are an important way to ensure your teams are supporting the overall goals of the organization. Here are some of the biggest reasons why you need key performance indicators.
- Keep your teams aligned: Whether measuring project success or employee performance, KPIs keep teams moving in the same direction.
- Provide a health check: Key performance indicators give you a realistic look at the health of your organization, from risk factors to financial indicators.
- Make adjustments: KPIs help you clearly see your successes and failures so you can do more of what’s working, and less of what’s not.
- Hold your teams accountable: Make sure everyone provides value with key performance indicators that help employees track their progress and help managers move things along.
Types of KPIs Key performance indicators come in many flavors. While some are used to measure monthly progress against a goal, others have a longer-term focus. The one thing all KPIs have in common is that they’re tied to strategic goals. Here’s an overview of some of the most common types of KPIs.
- Strategic: These big-picture key performance indicators monitor organizational goals. Executives typically look to one or two strategic KPIs to find out how the organization is doing at any given time. Examples include return on investment, revenue and market share.
- Operational: These KPIs typically measure performance in a shorter time frame, and are focused on organizational processes and efficiencies. Some examples include sales by region, average monthly transportation costs and cost per acquisition (CPA).
- Functional Unit: Many key performance indicators are tied to specific functions, such finance or IT. While IT might track time to resolution or average uptime, finance KPIs track gross profit margin or return on assets. These functional KPIs can also be classified as strategic or operational.
- Leading vs Lagging: Regardless of the type of key performance indicator you define, you should know the difference between leading indicators and lagging indicators. While leading KPIs can help predict outcomes, lagging KPIs track what has already happened. Organizations use a mix of both to ensure they’re tracking what’s most important.
Q8: Say you flip a coin 10 times and observe only one head. What would be the null hypothesis and p-value for testing whether the coin is fair or not?
Answer:
The null hypothesis is that the coin is fair, and the alternative hypothesis is that the coin is biased. The p-value is the probability of observing the results obtained given that the null hypothesis is true, in this case, the coin is fair.
In total for 10 flips of a coin, there are 2^10 = 1024 possible outcomes and in only 10 of them are there 9 tails and one head.
Hence, the exact probability of the given result is the p-value, which is 10/1024 = 0.0098. Therefore, with a significance level set, for example, at 0.05, we can reject the null hypothesis.
Q9: You are testing hundreds of hypotheses, each with a t-test. What considerations would you take into account when doing this?
Answer: The main consideration when we have a large number of tests is that probability of getting a significant test due to chance alone increases. This will increase the type 1 error (rejecting the null hypothesis when it's actually true).
Therefore we need to consider the Bonferroni Effect which happens when we make many tests. Ex. If our significance level is 0.05 but we made a 100 test it means that the probability of getting a value inside the rejection rejoin is 0.0005, not 0.05 so here we need to use another significance level which's called alpha star = significance level /K Where K is the number of the tests.
Answer:
In order to apply the central limit theorem, there are four conditions that must be met:
1.** Randomization:** The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample.
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Independence: The sample values must be independent of each other.
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The 10% Condition: When the sample is drawn without replacement, the sample size should be no larger than 10% of the population.
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Large Sample Condition: The sample size needs to be sufficiently large.
Answer:
Skewness refers to a distortion or asymmetry that deviates from the symmetrical bell curve, or normal distribution, in a set of data. If the curve is shifted to the left or to the right, it is said to be skewed.Skewness can be quantified as a representation of the extent to which a given distribution varies from a normal distribution. There are two main types of skewness negative skew which refers to a longer or fatter tail on the left side of the distribution, while positive skew refers to a longer or fatter tail on the right. These two skews refer to the direction or weight of the distribution.
The mean of positively skewed data will be greater than the median. In a negatively skewed distribution, the exact opposite is the case: the mean of negatively skewed data will be less than the median. If the data graphs symmetrically, the distribution has zero skewness, regardless of how long or fat the tails are.
There are several ways to measure skewness. Pearson’s first and second coefficients of skewness are two common methods. Pearson’s first coefficient of skewness, or Pearson mode skewness, subtracts the mode from the mean and divides the difference by the standard deviation. Pearson’s second coefficient of skewness, or Pearson median skewness, subtracts the median from the mean, multiplies the difference by three, and divides the product by the standard deviation.
Answer:
Intuitively it is the maximum of the sample points. Here's the mathematical proof is in the figure below:
Answer:
Chi-square test A statistical method is used to find the difference or correlation between the observed and expected categorical variables in the dataset.
Example: A food delivery company wants to find the relationship between gender, location, and food choices of people.
It is used to determine whether the difference between 2 categorical variables is:
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Due to chance or
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Due to relationship
Analysis of Variance (ANOVA) is a statistical formula used to compare variances across the means (or average) of different groups. A range of scenarios uses it to determine if there is any difference between the means of different groups.
t_test is a statistical method for the comparison of the mean of the two groups of the normally distributed sample(s).
It comes in various types such as:
- One sample t-test:
Used to compare the mean of a sample and the population.
- Two sample t-tests:
Used to compare the mean of two independent samples and whether their population is statistically different.
- Paired t-test:
Used to compare means of different samples from the same group.
Q14: Say you have two subsets of a dataset for which you know their means and standard deviations. How do you calculate the blended mean and standard deviation of the total dataset? Can you extend it to K subsets?
Answer:
Q15: What is the relationship between the significance level and the confidence level in Statistics?###
Answer: Confidence level = 1 - significance level.
It's closely related to hypothesis testing and confidence intervals.
⏺ Significance Level according to the hypothesis testing literature means the probability of Type-I error one is willing to tolerate.
⏺ Confidence Level according to the confidence interval literature means the probability in terms of the true parameter value lying inside the confidence interval. They are usually written in percentages.
Answer: The law of large numbers states that as the number of trials in a random experiment increases, the average of the results obtained from the experiment approaches the expected value. In statistics, it's used to describe the relationship between sample size and the accuracy of statistical estimates.
In data science, the law of large numbers is used to understand the behavior of random variables over many trials. It's often applied in areas such as predictive modeling, risk assessment, and quality control to ensure that data-driven decisions are based on a robust and accurate representation of the underlying patterns in the data.
The law of large numbers helps to guarantee that the average of the results from a large number of independent and identically distributed trials will converge to the expected value, providing a foundation for statistical inference and hypothesis testing.
Q17: What is the difference between a confidence interval and a prediction interval, and how do you calculate them?
Answer:
A confidence interval is a range of values that is likely to contain the true value of a population parameter with a certain level of confidence. It is used to estimate the precision or accuracy of a sample statistic, such as a mean or a proportion, based on a sample from a larger population.
For example, if we want to estimate the average height of all adults in a certain region, we can take a random sample of individuals from that region and calculate the sample mean height. Then we can construct a confidence interval for the true population mean height, based on the sample mean and the sample size, with a certain level of confidence, such as 95%. This means that if we repeat the sampling process many times, 95% of the resulting intervals will contain the true population mean height.
The formula for a confidence interval is: confidence interval = sample statistic +/- margin of error
The margin of error depends on the sample size, the standard deviation of the population (or the sample, if the population standard deviation is unknown), and the desired level of confidence. For example, if the sample size is larger or the standard deviation is smaller, the margin of error will be smaller, resulting in a narrower confidence interval.
A prediction interval is a range of values that is likely to contain a future observation or outcome with a certain level of confidence. It is used to estimate the uncertainty or variability of a future value based on a statistical model and the observed data.
For example, if we have a regression model that predicts the sales of a product based on its price and advertising budget, we can use a prediction interval to estimate the range of possible sales for a new product with a certain price and advertising budget, with a certain level of confidence, such as 95%. This means that if we repeat the prediction process many times, 95% of the resulting intervals will contain the true sales value.
The formula for a prediction interval is: prediction interval = point estimate +/- margin of error
The point estimate is the predicted value of the outcome variable based on the model and the input variables. The margin of error depends on the residual standard deviation of the model, which measures the variability of the observed data around the predicted values, and the desired level of confidence. For example, if the residual standard deviation is larger or the level of confidence is higher, the margin of error will be larger, resulting in a wider prediction interval.
