ML Notes

Statistics for Machine Learning Interviews

Essential statistics concepts for ML interviews: hypothesis testing, distributions, sampling, Bayesian thinking, and probability theory.

Statistics is fundamental to understanding and implementing machine learning algorithms. Here's what you need to know for interviews.

Probability Fundamentals

Basic Concepts

Probability = likelihood of event occurring (0 to 1)

Joint Probability: P(A and B) = P(A) * P(B|A)

Marginal Probability: P(A) = Σ P(A and B) for all B

Conditional Probability: P(A|B) = P(A and B) / P(B)

Independent Events: P(A|B) = P(A) if events are independent

Bayes' Theorem

Example

P(A|B) = P(B|A) * P(A) / P(B) Where: P(A|B) = Posterior (what we want to find) P(B|A) = Likelihood (probability of data given hypothesis) P(A) = Prior (our belief before seeing data) P(B) = Evidence (total probability of data)

Real-world example: Spam detection

Example

P(spam|words) = P(words|spam) * P(spam) / P(words) Where: - P(spam|words) = probability email is spam given certain words - P(words|spam) = likelihood of seeing those words in spam - P(spam) = prior probability an email is spam (e.g., 15%) - P(words) = probability of seeing those words overall

Distributions

Normal Distribution

Key properties:

Bell-shaped curve
Mean = Median = Mode
68% of data within 1 std dev
95% of data within 2 std devs
99.7% of data within 3 std devs

Used for:

Continuous variables (heights, test scores)
Central Limit Theorem (sample means approximate normal)
Assumption in many ML algorithms

Binomial Distribution

Example

P(X = k) = C(n,k) * p^k * (1-p)^(n-k) Parameters: - n = number of trials - p = probability of success - k = number of successes Examples: coin flips, conversion rates, classification correctness

Poisson Distribution

Used for count data (emails per day, accidents per year), rare events, approximates binomial when n large and p small.

Uniform Distribution

All values equally likely. Used for random sampling, baseline comparisons, initial weight distributions.

Hypothesis Testing

Framework

State hypotheses

H0 (null): no effect or difference
H1 (alternative): there is an effect

Choose significance level (α) - Typical: 0.05 (5%), More strict: 0.01

Calculate test statistic - Depends on test type (t-test, z-test, chi-square, etc.)

Determine p-value - If p < α: reject H0; If p ≥ α: fail to reject H0

Conclusion - Based on p-value and interpret in context

Common Tests

T-Test: Compare means of two groups

python example

from scipy import stats

# Independent samples t-test
t_stat, p_value = stats.ttest_ind(group1, group2)

# Paired t-test (same subjects, before/after)
t_stat, p_value = stats.ttest_rel(before, after)

if p_value < 0.05:
    print("Significant difference between groups")
else:
    print("No significant difference")

Chi-Square Test: Compare categorical variables

python example

from scipy.stats import chi2_contingency

contingency_table = pd.crosstab(var1, var2)
chi2, p_value, dof, expected = chi2_contingency(contingency_table)

ANOVA: Compare means across multiple groups

python example

from scipy.stats import f_oneway

f_stat, p_value = f_oneway(group1, group2, group3)

Correlation Test: Check if variables are related

python example

from scipy.stats import pearsonr, spearmanr

# Pearson (parametric)
r, p_value = pearsonr(x, y)

# Spearman (non-parametric, rank-based)
rho, p_value = spearmanr(x, y)

Type I and Type II Errors

Example

H0 True H0 False Reject H0 Type I Error (α) Correct (Power) Fail to Reject Correct Type II Error (β)

Type I Error (False Positive): Conclude effect exists when it doesn't. Probability = α. Example: flagging legitimate email as spam

Type II Error (False Negative): Conclude no effect when one exists. Probability = β. Power = 1 - β

Tradeoff: Decreasing α increases β. Can't minimize both simultaneously. Choose based on problem context.

Sampling Methods

Random Sampling

Every sample equally likely, no bias on average, used by stratified sampling.

Stratified Sampling

Divide population into strata (groups)
Sample proportionally from each stratum
Maintains population distribution

Example: Sample from 60% male, 40% female population → Ensure sample is also 60% male, 40% female

Advantages: More representative, reduces variance, better for imbalanced populations

Importance Sampling

Sample more from important regions, weight samples by probability of selection, used in ML for balanced datasets.

Central Limit Theorem

Key insight: Sample means follow normal distribution

Process:

Take random sample of size n from population
Calculate sample mean
Repeat many times
Distribution of sample means ≈ normal

Implications:

Works even if population not normal
Enables statistical inference (confidence intervals, hypothesis tests)
Sample size n ≥ 30 usually sufficient

Why important for ML:

Justifies using normal distribution for statistics
Foundation for many statistical tests
Explains why averaging models works

Confidence Intervals

Definition

Confidence interval: range of values likely to contain parameter

95% CI = [lower_bound, upper_bound]

"We're 95% confident the true value is in this range"

NOT: "95% probability true value is in this range" (The true value is fixed)

Calculation

CI = point_estimate ± critical_value * standard_error

For mean with normal distribution:

CI = x̄ ± 1.96 * (σ / √n)

For proportion:

CI = p̂ ± 1.96 * √(p̂(1-p̂)/n)

Example

import scipy.stats as stats mean = np.mean(data) se = stats.sem(data) # standard error ci = stats.t.interval(0.95, len(data)-1, loc=mean, scale=se) print(f"95% CI: [{ci[0]:.2f}, {ci[1]:.2f}]")

Bayesian Thinking

Frequentist vs Bayesian

FREQUENTIST:

Probability = long-run frequency
Parameters fixed but unknown
Confidence intervals based on repeated experiments

BAYESIAN:

Probability = degree of belief
Parameters are random variables
Prior beliefs + data → posterior beliefs

Bayesian Inference

1. Start with prior belief	P(θ)
2. See data	P(data\|θ)
3. Update to posterior	P(θ\|data)
Example	Is coin fair?
Prior	P(fair) = 50%
Observe	10 heads out of 10 flips
Posterior	P(fair\|10 heads) = very low (0.1%)

Advantage: Incorporates prior knowledge

A/B Testing Statistics

Sample Size Calculation

Need to know:

Baseline conversion rate (p)
Minimum detectable effect (lift)
Confidence level (typically 95%)
Statistical power (typically 80%)

python example

from scipy.stats import norm

def sample_size(p, effect_size, alpha=0.05, power=0.80):
    z_alpha = norm.ppf(1 - alpha/2)
    z_beta = norm.ppf(power)
    n = ((z_alpha + z_beta)**2 * p * (1-p)) / (effect_size**2)
    return n

# Example: baseline 5%, detect 10% lift
n = sample_size(0.05, 0.005)  # ~36,000 samples

Statistical Test for A/B Test

Example

from scipy.stats import chi2_contingency # Create contingency table contingency = np.array([[a, b], [c, d]]) chi2, p_value, dof, expected = chi2_contingency(contingency) if p_value < 0.05: print("Statistically significant difference") treatment_rate = c/(c+d) control_rate = a/(a+b) print(f"Treatment: {treatment_rate:.1%}, Control: {control_rate:.1%}")

Common Interview Questions

Q: What's the difference between correlation and causation?

A: Correlation means two variables move together. Causation means one causes the other. Correlation ≠ Causation. Could be coincidence, third variable causing both, reverse causation, or non-linear relationship.

Example: Ice cream sales & drowning deaths are correlated → Both caused by summer weather, not causation

Q: Explain Simpson's Paradox

A: Trend reverses when data is aggregated/disaggregated

Example: Treatment A vs B success rates

Overall: A > B
But by subgroup: A < B in each subgroup
Caused by different group sizes

Q: What's the power of a statistical test?

A: Power = 1 - β = probability of detecting effect when it exists

Power = 80% typical target
Depends on: effect size, sample size, significance level
Larger effect → higher power
Larger sample → higher power
Higher significance level → higher power

Q: Why collect more data?

A: Reduces standard error and variance

Smaller confidence intervals
More powerful hypothesis tests
Better model generalization
Smoother estimates

Summary

Key statistics for ML:

Probability and Bayes' theorem
Common distributions (normal, binomial)
Hypothesis testing framework
Confidence intervals
A/B testing for experiments
Understanding errors and power

Practice calculating these by hand during interviews to show mastery.

Exam Focus

Revise definitions, diagrams, examples, and short-answer points for Statistics for Machine Learning Interviews.

Interview Use

Prepare one clear explanation, one practical example, and one common mistake for this Machine Learning topic.

Search Terms

machine-learning, machine learning, machine, learning, interview, preparation, statistics, for

Continue learning this concept

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ML Notes

Statistics for Machine Learning Interviews

Essential statistics concepts for ML interviews: hypothesis testing, distributions, sampling, Bayesian thinking, and probability theory.

Statistics is fundamental to understanding and implementing machine learning algorithms. Here's what you need to know for interviews.

Probability Fundamentals

Basic Concepts

Probability = likelihood of event occurring (0 to 1)

Joint Probability: P(A and B) = P(A) * P(B|A)

Marginal Probability: P(A) = Σ P(A and B) for all B

Conditional Probability: P(A|B) = P(A and B) / P(B)

Independent Events: P(A|B) = P(A) if events are independent

Bayes' Theorem

Example

Real-world example: Spam detection

Example

Distributions

Normal Distribution

Key properties:

Bell-shaped curve
Mean = Median = Mode
68% of data within 1 std dev
95% of data within 2 std devs
99.7% of data within 3 std devs

Used for:

Continuous variables (heights, test scores)
Central Limit Theorem (sample means approximate normal)
Assumption in many ML algorithms

Binomial Distribution

Example

P(X = k) = C(n,k) * p^k * (1-p)^(n-k) Parameters: - n = number of trials - p = probability of success - k = number of successes Examples: coin flips, conversion rates, classification correctness

Poisson Distribution

Used for count data (emails per day, accidents per year), rare events, approximates binomial when n large and p small.

Uniform Distribution

All values equally likely. Used for random sampling, baseline comparisons, initial weight distributions.

Hypothesis Testing

Framework

State hypotheses

H0 (null): no effect or difference
H1 (alternative): there is an effect

Choose significance level (α) - Typical: 0.05 (5%), More strict: 0.01

Calculate test statistic - Depends on test type (t-test, z-test, chi-square, etc.)

Determine p-value - If p < α: reject H0; If p ≥ α: fail to reject H0

Conclusion - Based on p-value and interpret in context

Common Tests

T-Test: Compare means of two groups

python example

from scipy import stats

# Independent samples t-test
t_stat, p_value = stats.ttest_ind(group1, group2)

# Paired t-test (same subjects, before/after)
t_stat, p_value = stats.ttest_rel(before, after)

if p_value < 0.05:
    print("Significant difference between groups")
else:
    print("No significant difference")

Chi-Square Test: Compare categorical variables

python example

from scipy.stats import chi2_contingency

contingency_table = pd.crosstab(var1, var2)
chi2, p_value, dof, expected = chi2_contingency(contingency_table)

ANOVA: Compare means across multiple groups

python example

from scipy.stats import f_oneway

f_stat, p_value = f_oneway(group1, group2, group3)

Correlation Test: Check if variables are related

python example

from scipy.stats import pearsonr, spearmanr

# Pearson (parametric)
r, p_value = pearsonr(x, y)

# Spearman (non-parametric, rank-based)
rho, p_value = spearmanr(x, y)

Type I and Type II Errors

Example

H0 True H0 False Reject H0 Type I Error (α) Correct (Power) Fail to Reject Correct Type II Error (β)

Type I Error (False Positive): Conclude effect exists when it doesn't. Probability = α. Example: flagging legitimate email as spam

Type II Error (False Negative): Conclude no effect when one exists. Probability = β. Power = 1 - β

Tradeoff: Decreasing α increases β. Can't minimize both simultaneously. Choose based on problem context.

Sampling Methods

Random Sampling

Every sample equally likely, no bias on average, used by stratified sampling.

Stratified Sampling

Divide population into strata (groups)
Sample proportionally from each stratum
Maintains population distribution

Example: Sample from 60% male, 40% female population → Ensure sample is also 60% male, 40% female

Advantages: More representative, reduces variance, better for imbalanced populations

Importance Sampling

Sample more from important regions, weight samples by probability of selection, used in ML for balanced datasets.

Central Limit Theorem

Key insight: Sample means follow normal distribution

Process:

Take random sample of size n from population
Calculate sample mean
Repeat many times
Distribution of sample means ≈ normal

Implications:

Works even if population not normal
Enables statistical inference (confidence intervals, hypothesis tests)
Sample size n ≥ 30 usually sufficient

Why important for ML:

Justifies using normal distribution for statistics
Foundation for many statistical tests
Explains why averaging models works

Confidence Intervals

Definition

Confidence interval: range of values likely to contain parameter

95% CI = [lower_bound, upper_bound]

"We're 95% confident the true value is in this range"

NOT: "95% probability true value is in this range" (The true value is fixed)

Calculation

CI = point_estimate ± critical_value * standard_error

For mean with normal distribution:

CI = x̄ ± 1.96 * (σ / √n)

For proportion:

CI = p̂ ± 1.96 * √(p̂(1-p̂)/n)

Example

import scipy.stats as stats mean = np.mean(data) se = stats.sem(data) # standard error ci = stats.t.interval(0.95, len(data)-1, loc=mean, scale=se) print(f"95% CI: [{ci[0]:.2f}, {ci[1]:.2f}]")

Bayesian Thinking

Frequentist vs Bayesian

FREQUENTIST:

Probability = long-run frequency
Parameters fixed but unknown
Confidence intervals based on repeated experiments

BAYESIAN:

Probability = degree of belief
Parameters are random variables
Prior beliefs + data → posterior beliefs

Bayesian Inference

1. Start with prior belief	P(θ)
2. See data	P(data\|θ)
3. Update to posterior	P(θ\|data)
Example	Is coin fair?
Prior	P(fair) = 50%
Observe	10 heads out of 10 flips
Posterior	P(fair\|10 heads) = very low (0.1%)

Advantage: Incorporates prior knowledge

A/B Testing Statistics

Sample Size Calculation

Need to know:

Baseline conversion rate (p)
Minimum detectable effect (lift)
Confidence level (typically 95%)
Statistical power (typically 80%)

python example

from scipy.stats import norm

def sample_size(p, effect_size, alpha=0.05, power=0.80):
    z_alpha = norm.ppf(1 - alpha/2)
    z_beta = norm.ppf(power)
    n = ((z_alpha + z_beta)**2 * p * (1-p)) / (effect_size**2)
    return n

# Example: baseline 5%, detect 10% lift
n = sample_size(0.05, 0.005)  # ~36,000 samples

Statistical Test for A/B Test

Example

Common Interview Questions

Q: What's the difference between correlation and causation?

Example: Ice cream sales & drowning deaths are correlated → Both caused by summer weather, not causation

Q: Explain Simpson's Paradox

A: Trend reverses when data is aggregated/disaggregated

Example: Treatment A vs B success rates

Overall: A > B
But by subgroup: A < B in each subgroup
Caused by different group sizes

Q: What's the power of a statistical test?

A: Power = 1 - β = probability of detecting effect when it exists

Power = 80% typical target
Depends on: effect size, sample size, significance level
Larger effect → higher power
Larger sample → higher power
Higher significance level → higher power

Q: Why collect more data?

A: Reduces standard error and variance

Smaller confidence intervals
More powerful hypothesis tests
Better model generalization
Smoother estimates

Summary

Key statistics for ML:

Probability and Bayes' theorem
Common distributions (normal, binomial)
Hypothesis testing framework
Confidence intervals
A/B testing for experiments
Understanding errors and power

Practice calculating these by hand during interviews to show mastery.

Exam Focus

Revise definitions, diagrams, examples, and short-answer points for Statistics for Machine Learning Interviews.

Interview Use

Prepare one clear explanation, one practical example, and one common mistake for this Machine Learning topic.

Search Terms

machine-learning, machine learning, machine, learning, interview, preparation, statistics, for

Continue learning this concept

PreviousEssential Machine Learning Interview Questions NextDatasets for ML Practice