Q1. Scenario: In a dataset of house prices: [100k, 120k, 130k, 140k, 1M]. Why might the median be a better measure of central tendency than the mean?
Mean = (100+120+130+140+1000)/5 = 1490/5 = 298k. Median = 130k. The outlier 1M skews the mean upwards. Median is robust to outliers. In machine learning, feature scaling (like median normalization) is used when outliers are present.
Q2. Scenario: You have two stocks: Stock A returns: [5%, 7%, 6%, 8%, 4%]; Stock B returns: [2%, 15%, -5%, 10%, 3%]. Compare their average returns and risk (variance). Which is riskier?
Mean A = (5+7+6+8+4)/5 = 6%; Variance A = avg((xi-6)^2)= (1+1+0+4+4)/5=2. Mean B = (2+15-5+10+3)/5=5%; Variance B = (( -3)^2+10^2+(-10)^2+5^2+(-2)^2)/5 = (9+100+100+25+4)/5=238/5=47.6. B has lower mean and much higher variance → riskier. Variance measures spread, used in regularization (ridge/LASSO).
Q3. Scenario: A dataset of exam scores: 70,75,80,85,90,95,100. Compute mean, median, and variance. If we add a value 0 (a mistake), how do these change?
Original mean = (70+75+80+85+90+95+100)/7=595/7=85; median=85; variance = avg((xi-85)^2) = (225+100+25+0+25+100+225)/7=700/7=100. Add 0: new mean = 595/8=74.375; median becomes (80+85)/2=82.5; variance increases drastically. Median is more robust to this outlier.
Q4. Scenario: In feature scaling for machine learning, why do we often subtract mean and divide by standard deviation (z-score normalization)?
It centers the feature at zero (mean =0) and scales to unit variance (var=1). This prevents features with larger scales from dominating distance-based algorithms (k-NN, SVM) and helps gradient descent converge faster by making the loss landscape more spherical. It also improves numerical stability.
Q5. Scenario: A machine learning model's prediction errors on test set: [2, -1, 0, 3, -2]. Compute the mean error (bias) and the variance of errors. What does each tell about model performance?
Mean error = (2-1+0+3-2)/5 = 0.4; variance = avg((xi-0.4)^2) = (2.56+1.96+0.16+6.76+5.76)/5 = 17.2/5=3.44. Mean error indicates systematic bias (positive means under-prediction), variance indicates inconsistency. Low bias + low variance = ideal trade-off.