MASTERING STATISTICS FOR DATA SCIENCE: From Fundamentals to Industry Applications

Overview

This article provides a comprehensive journey through statistics, starting from basic descriptive concepts to advanced analytical techniques. It explains each formula in simple language, clarifies the meaning of symbols and terms, and demonstrates how to apply methods using both SAS and Python. Real-world case studies from Clinical SAS and BFSI analytics illustrate practical applications. Learners will gain hands-on skills in data exploration, probability, hypothesis testing, regression, and predictive modeling, preparing them for industry challenges and interviews.

This version explains every statistical formula in layman’s terms. It clarifies each symbol used (like Σ for summation, σ for standard deviation, ρ for correlation coefficient, etc.) and includes examples in both SAS and Python. Domain-specific case studies from Clinical SAS and BFSI analytics are highlighted.

EXPLANATION OF SYMBOLS USED IN STATISTICS

• Σ (Sigma): Summation, meaning “add everything together”.
• σ (Small Sigma): Standard deviation, a measure of spread.
• σ²: Variance, the square of standard deviation.
• μ (Mu): Population mean (average).
• ρ (Rh**):** Crrelation coefficient, strength of relationship.
• P(E): Probability of event E occurring.
• O: Observed value.
• E: Expected value.
• Xᵢ: Individual data point.
• n: Number of observations.
• H₀: Null hypothesis.
• H₁: Alternative hypothesis.
• α (Alpha): Significance level, commonly 0.05.

CODE EXAMPLES

Python Code Snippets

import numpy as np
# Mean Calculation
np.mean([1,2,3])
        

from scipy import stats
# t-test example
data = [5,6,7]
stats.ttest_1samp(data,6)
        

SAS Code Snippets

proc means data=dataset; var bp; run;
        

proc ttest data=dataset h0=6; var score; run;
        

These examples will appear alongside respective topics below.

MODULE 3: DESCRIPTIVE STATISTICS

Mean (Average)

Formula: Xˉ=ΣXin\bar{X} = \frac{\Sigma X_i}{n} Explanation: Add all values (ΣXᵢ) and divide by the number of values (n). Example: Mean blood pressure of patients in a trial. Python: np.mean(data) SAS: proc means data=dataset; var bp; run; Case Study (Clinical SAS): CROs compute mean lab results to monitor drug impact.

Variance (σ²) & Standard Deviation (σ)

Formula: σ2=Σ(Xi−Xˉ)2n\sigma^2 = \frac{\Sigma (X_i - \bar{X})^2}{n}; σ=σ2\sigma = \sqrt{\sigma^2} Explanation: Measure how far data points are from the mean. Case Study (BFSI): Variance in stock returns measures risk.

MODULE 4: PROBABILITY FUNDAMENTALS

Probability

Formula: P(E) = Favourable / Total outcomes Explanation: Chance of an event happening. Case Study: Probability of an adverse drug reaction.

Python Code:

import random
trials = 10000
success = sum([1 for _ in range(trials) if random.randint(1,6) == 4])
print('Estimated Probability:', success/trials)
        

SAS Code:

data prob;
  trials = 10000;
  p = 1/6;
run;
proc print data=prob; run;
        

Conditional Probability

Formula: P(A|B) = P(A∩B) / P(B) Explanation: Probability of event A given that B has occurred. Case Study (BFSI): Probability of loan default given low credit score.

Python Code:

# Conditional probability simulation
import pandas as pd
loans = pd.DataFrame({'default':[1,0,1,0,1],'low_score':[1,1,0,0,1]})
p_a_and_b = len(loans[(loans['default']==1) & (loans['low_score']==1)]) / len(loans)
p_b = len(loans[loans['low_score']==1]) / len(loans)
print('P(default|low_score)=', p_a_and_b/p_b)
        

SAS Code:

data loans; input default low_score; datalines; 1 1 0 1 1 0 0 0 1 1 ; run;
proc freq data=loans; tables default*low_score / nopercent norow nocol; run;
        

Bayes’ Theorem

Formula: P(H|E) = [P(E|H) * P(H)] / P(E) Explanation: Updates the probability of a hypothesis H when new evidence E is observed. Case Study (Clinical SAS): Probability a patient has a disease after a positive test.

Python Code:

# Bayes example calculation
P_H = 0.01  # prior probability of disease
P_E_given_H = 0.9  # test sensitivity
P_E_given_not_H = 0.05  # false positive rate
P_E = P_E_given_H*P_H + P_E_given_not_H*(1-P_H)
P_H_given_E = (P_E_given_H*P_H) / P_E
print('P(Disease|Positive Test)=', P_H_given_E)
        

SAS Code:

data bayes; 
  P_H = 0.01; P_E_H = 0.9; P_E_notH = 0.05; 
  P_E = P_E_H*P_H + P_E_notH*(1-P_H); 
  P_H_E = (P_E_H*P_H)/P_E; 
run; 
proc print data=bayes; run;
        

MODULE 5: INFERENTIAL STATISTICS

Hypothesis Testing

Concept: Compares sample data to a claim.

• H₀: No effect/difference.
• H₁: There is an effect/difference. Formula for t-test: t = (X̄ - μ) / (s / √n)
• s: sample standard deviation. Case Study: Testing if a new medicine is more effective than the old one.

Python Code:

from scipy import stats
import numpy as np
data = np.array([68,71,69,72,70,73,67])
t_stat, p_val = stats.ttest_1samp(data,70)
print('t-statistic:', t_stat, 'p-value:', p_val)
        

SAS Code:

proc ttest data=dataset h0=70;
  var bp;
run;
        

Confidence Intervals

Formula: CI = X̄ ± Z * (σ / √n)

• Z: Z-score at desired confidence level. Case Study (BFSI): Calculating 95% CI for average credit card spend.

Python Code:

import numpy as np
import scipy.stats as st
data = np.array([100,110,120,130,140])
mean = np.mean(data)
ci = st.t.interval(alpha=0.95, df=len(data)-1, loc=mean, scale=st.sem(data))
print('95% CI:', ci)
        

SAS Code:

proc means data=dataset clm alpha=0.05; var spend; run;
        

p-values and Errors

Explanation: p-value < α indicates evidence against H₀. Type I error: false positive. Type II error: false negative.

MODULE 6: CORRELATION & REGRESSION

Correlation (ρ)

Formula: ρ = Cov(X,Y) / (σ_X σ_Y) Explanation: Measures strength of linear relationship. Case Study (BFSI): Relationship between income and loan repayment.

Python Code:

import pandas as pd
import numpy as np
data = pd.DataFrame({'income':[20,30,40],'repayment':[1,0,1]})
print(data.corr())
        

SAS Code:

proc corr data=dataset; var income repayment; run;
        

Linear Regression

Formula: Y = a + bX Explanation: Predicts Y using X. Case Study: Predicting treatment response from dosage.

Python Code:

from sklearn.linear_model import LinearRegression
import pandas as pd
data = pd.DataFrame({'dose':[1,2,3,4],'response':[2,4,6,8]})
X = data[['dose']]
y = data['response']
model = LinearRegression().fit(X,y)
print('Predicted:', model.predict([[5]]))
        

SAS Code:

proc reg data=dataset; model response = dose; run;
        

Multiple Regression

Formula: Y = a + b1X1 + b2X2 + ... Case Study: BFSI predicting loan recovery from multiple features.

Python Code:

X = data[['dose','age']]
model = LinearRegression().fit(X,y)
        

SAS Code:

proc reg data=dataset; model response = dose age; run;
        

MODULE 6: CORRELATION & REGRESSION

Correlation (ρ)

Formula: ρ=Cov(X,Y)σXσY\rho = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} Explanation: Measures strength of relationship (-1 to +1). Case Study (BFSI): Correlation between income and loan repayment.

Linear Regression

Formula: Y=a+bXY = a + bX Explanation: Predicts Y from X. Case Study: Predicting treatment response based on dosage.

MODULE 7: ANOVA & CHI-SQUARE (With SAS & Python Codes)

ANOVA

Formula: F = MS_between / MS_within Explanation: Tests if the means of three or more groups are significantly different. Case Study: Comparing mean recovery times for three drugs.

Python Code:

from scipy import stats
g1=[85,90,88]; g2=[78,82,80]; g3=[90,95,93]
F,p=stats.f_oneway(g1,g2,g3)
print('F-statistic:',F,'p-value:',p)
        

SAS Code:

proc anova data=dataset; class treatment; model recovery=treatment; run;
        

Formula: F=MSbetweenMSwithinF = \frac{MS_{between}}{MS_{within}}

• MS: Mean squares. Case Study: Comparing mean recovery times for three drugs.

Chi-Square

Formula: χ² = Σ (O - E)² / E Explanation: Compares observed vs expected frequencies to test independence of categorical variables. Case Study (Clinical SAS): Testing if side effects differ by age group.

Python Code:

import numpy as np
from scipy.stats import chi2_contingency
obs = np.array([[50,30],[20,40]])
chi2,p,dof,exp=chi2_contingency(obs)
print('Chi2:',chi2,'p-value:',p)
        

SAS Code:

proc freq data=dataset; tables age_group*side_effect / chisq; run;
        

Formula: χ2=Σ(O−E)2E\chi^2 = \Sigma \frac{(O - E)^2}{E} Explanation: Compares observed vs expected counts. Case Study (Clinical SAS): Testing if side effects differ by age group.

MODULE 8: ADVANCED CONCEPTS

• Logistic Regression: P=11+e−(a+bX)P = \frac{1}{1 + e^{-(a + bX)}} predicts binary outcomes.
• Time Series: Analyzes trends and seasonality in data. Case Study (BFSI): Forecasting credit card spending trends.

MODULE 9: STATISTICAL MODELING IN DATA SCIENCE

Concept: Combines all techniques to build predictive models. Case Study: BFSI firms model credit risk using regression and probability models.

MODULE 10: CAPSTONE PROJECT & INTERVIEW PREPARATION

• Work with clinical and BFSI datasets.
• Apply formulas, build models in SAS and Python.
• Prepare for interviews with theorem and formula knowledge.

PROJECT PROBLEM DEFINITIONS

1. Clinical Trial Safety Analysis: Analyze adverse event data to determine if drug dosage influences the occurrence of side effects using Chi-Square and Logistic Regression in SAS and Python.
2. Credit Risk Prediction (BFSI): Build a regression model to predict the probability of loan default based on customer demographics and transaction history.
3. Time Series Forecasting: Use historical claims data to forecast insurance claim volume over the next 12 months.
4. Drug Efficacy Comparison: Apply ANOVA to compare the effectiveness of multiple drug treatments in reducing blood pressure.

SAMPLE INTERVIEW QUESTIONS AND ANSWERS

Q1: Explain the Central Limit Theorem in simple terms. A: It states that when you take many samples from any population, the sample means will follow a normal distribution as the sample size grows, even if the population itself is not normal.

Q2: What is the difference between correlation and causation? A: Correlation measures a relationship between two variables, but it does not imply that one causes the other.

Q3: When would you use a Chi-Square test? A: When testing whether two categorical variables are related, such as age group and side-effect occurrence.

Q4: What is p-value? A: It is the probability of observing the data if the null hypothesis is true. A low p-value (<0.05) suggests strong evidence against the null.

Q5: How do you handle missing data in a dataset? A: Strategies include imputation (mean/median for numerical, mode for categorical), using advanced techniques like regression imputation, or excluding the missing cases.

Q6: Give an example where you applied statistics in a real project. A: For example, I applied logistic regression to predict patient survival probability in a clinical dataset, interpreting odds ratios for key variables.

ADDITIONAL SAS AND PYTHON CODE WITH CASE STUDY

Case Study: Clinical Trial Safety Analysis

• Objective: Assess if adverse events are related to drug dosage.

Python Code:

import pandas as pd
import statsmodels.api as sm
data = pd.read_csv('clinical_events.csv')
model = sm.Logit(data['AdverseEvent'], sm.add_constant(data['Dosage']))
result = model.fit()
print(result.summary())
        

SAS Code:

proc logistic data=clinical_events;
  model AdverseEvent(event='1') = Dosage;
run;
        

Case Study: BFSI Credit Risk Prediction

• Objective: Predict loan default probability.

Python Code:

from sklearn.linear_model import LogisticRegression
import pandas as pd
data = pd.read_csv('loan_data.csv')
X = data[['income','age','balance']]
y = data['default']
model = LogisticRegression().fit(X, y)
print(model.predict_proba([[30000,40,5000]]))
        

SAS Code:

proc logistic data=loan_data;
  model default(event='1') = income age balance;
run;
        

• All statistical theorems with symbol explanations.
• Applying formulas with SAS & Python.
• Domain problem-solving in Clinical SAS & BFSI.
• Confidently handling projects & interviews.