🔢 Week 2 – NumPy & Pandas · Day 09
Day 9: NumPy Maths, Broadcasting & Random
⏱ 2 hours📶 Beginner → Intermediate🎯 100% self-study
Day 9 / 60
✅ Today you will learn
- Vectorised maths on whole arrays
- Broadcasting: maths between different shapes
- Aggregations: sum, mean, std, percentile, cumsum
- np.where, clip, unique and sorting
- Random numbers and a Monte Carlo simulation
🏢 In the real world
An e-commerce company wants to know: “If we give 10% off in every city but 20% off in metro cities, what will the new prices be for 50,000 products?” Broadcasting answers it in one line, and a random simulation estimates the risk.
Vectorised maths
Python
import numpy as np
prices = np.array([499, 1299, 2499, 89, 15999])
qty = np.array([3, 1, 2, 10, 1])
print("Revenue per item:", prices * qty)
print("With 18% GST :", (prices * 1.18).round(2))
print("Log prices :", np.log10(prices).round(2))
print("Total revenue :", (prices * qty).sum())Revenue per item: [ 1497 1299 4998 890 15999]
With 18% GST : [ 588.82 1532.82 2948.82 105.02 18878.82]
Log prices : [2.7 3.11 3.4 1.95 4.2 ]
Total revenue : 24683
Broadcasting
Broadcasting lets NumPy combine arrays of different shapes by stretching the smaller one – no copies, no loops. Rule: compare shapes from the right; sizes must be equal or 1.
| A shape | B shape | Result | Works? |
|---|---|---|---|
| (5,) | () scalar | (5,) | ✅ |
| (4, 3) | (3,) | (4, 3) | ✅ row repeated |
| (4, 3) | (4, 1) | (4, 3) | ✅ column repeated |
| (4, 3) | (4,) | — | ❌ 3 ≠ 4 |
Python
import numpy as np
# 3 cities × 4 products base prices
base = np.array([[100, 250, 400, 1200],
[110, 260, 420, 1250],
[ 95, 240, 390, 1180]])
city_discount = np.array([[0.20], [0.20], [0.10]]) # shape (3, 1): metro, metro, small city
product_gst = np.array([0.05, 0.12, 0.18, 0.18]) # shape (4,)
final = base * (1 - city_discount) * (1 + product_gst)
print(final.round(1))[[ 84. 224. 377.6 1132.8]
[ 92.4 233. 396.5 1180. ]
[ 89.8 241.9 414.2 1253.2]]
Python
import numpy as np
scores = np.array([[78, 82, 69], [92, 88, 95], [65, 70, 58], [88, 79, 84]])
col_mean = scores.mean(axis=0)
print("Centred (score - subject mean):")
print((scores - col_mean).round(1))
z = (scores - scores.mean(axis=0)) / scores.std(axis=0)
print("Z-scores:")
print(z.round(2))Centred (score - subject mean):
[[ -2.8 2.2 -7.5]
[ 11.2 8.2 18.5]
[-15.8 -9.8 -18.5]
[ 7.2 -0.8 7.5]]
Z-scores:
[[-0.26 0.35 -0.53]
[ 1.08 1.27 1.31]
[-1.51 -1.5 -1.31]
[ 0.7 -0.12 0.53]]
Aggregations
Python
import numpy as np
daily_sales = np.array([42, 55, 38, 61, 70, 95, 88, 47, 52, 49, 66, 91, 99, 58]) # in thousands ₹
print("sum", daily_sales.sum(), "| mean", daily_sales.mean().round(2), "| median", np.median(daily_sales))
print("std", daily_sales.std().round(2), "| min", daily_sales.min(), "| max", daily_sales.max())
print("25th/75th percentile:", np.percentile(daily_sales, [25, 75]))
print("Running total:", daily_sales.cumsum()[:7])
print("Day-on-day change:", np.diff(daily_sales)[:6])sum 911 | mean 65.07 | median 59.5
std 19.75 | min 38 | max 99
25th/75th percentile: [49.75 83.5 ]
Running total: [ 42 97 135 196 266 361 449]
Day-on-day change: [ 13 -17 23 9 25 -7]
np.where, clip, unique, sort
Python
import numpy as np
marks = np.array([35, 82, 47, 91, 28, 66, 73])
print(np.where(marks >= 40, "Pass", "Fail"))
print(np.select([marks >= 75, marks >= 40], ["Distinction", "Pass"], default="Fail"))
print(np.clip(marks + 15, 0, 100)) # grace marks, max 100
cities = np.array(["Pune", "Delhi", "Pune", "Agra", "Delhi", "Pune"])
vals, counts = np.unique(cities, return_counts=True)
print({str(v): int(c) for v, c in zip(vals, counts)})
print(np.sort(marks)[::-1], np.argsort(marks)[::-1][:3])['Fail' 'Pass' 'Pass' 'Pass' 'Fail' 'Pass' 'Pass']
['Fail' 'Distinction' 'Pass' 'Distinction' 'Fail' 'Pass' 'Pass']
[ 50 97 62 100 43 81 88]
{'Agra': 1, 'Delhi': 2, 'Pune': 3}
[91 82 73 66 47 35 28] [3 1 6]
Random numbers
Python
import numpy as np
rng = np.random.default_rng(42) # modern, reproducible generator
print(rng.integers(1, 7, size=10)) # 10 dice rolls
print(rng.random(3).round(3)) # 0–1 uniform
heights = rng.normal(loc=165, scale=8, size=1000) # mean 165 cm, sd 8
print("mean", heights.mean().round(1), "| sd", heights.std().round(1))
print(rng.choice(["UPI", "Card", "Cash"], size=8, p=[0.6, 0.25, 0.15]))[1 5 4 3 3 6 1 5 2 1]
[0.976 0.761 0.786]
mean 164.8 | sd 7.9
['Card' 'UPI' 'UPI' 'UPI' 'Card' 'Cash' 'UPI' 'Card']
Python
import numpy as np
import matplotlib.pyplot as plt
rng = np.random.default_rng(42)
heights = rng.normal(165, 8, 5000)
plt.hist(heights, bins=40, color="#0ea5e9", edgecolor="white")
plt.axvline(heights.mean(), color="red", linestyle="--", label=f"mean {heights.mean():.1f} cm")
plt.title("5,000 simulated student heights")
plt.xlabel("Height (cm)"); plt.ylabel("Students"); plt.legend()
plt.show()
Real example: Monte Carlo risk simulation
A small café sells between 80 and 160 cups a day (average 120). Each cup earns ₹35 profit and fixed costs (rent, staff, electricity) are ₹4,100 a day. What is the chance of making a loss in a month? We simulate 10,000 months.
Python
import numpy as np
rng = np.random.default_rng(7)
months, days = 10_000, 30
cups = rng.normal(120, 22, size=(months, days)).clip(80, 160) # broadcasting-friendly 2-D array
profit = (cups * 35 - 4100).sum(axis=1) # one value per month
print("Average monthly profit: ₹", round(profit.mean()))
print("Worst 5% month : ₹", round(np.percentile(profit, 5)))
print("Chance of a loss :", f"{(profit < 0).mean():.2%}")
print("Chance profit > ₹10,000:", f"{(profit > 10000).mean():.1%}")Average monthly profit: ₹ 3008
Worst 5% month : ₹ -3555
Chance of a loss : 22.37%
Chance profit > ₹10,000: 4.0%
💡 Why simulate?
When maths formulas get hard, generating thousands of random “what-if” worlds gives a reliable answer. Banks, insurers and supply-chain teams use Monte Carlo every day.
🏅 Job skills unlocked today
✔ Vectorise business calculations✔ Apply broadcasting across rows and columns✔ Summarise with percentiles and cumsum✔ Run reproducible random simulations
Practice tasks
🏋️ Practice – do it yourself
Task 1: Convert temperatures
[30, 35, 41, 28] °C to °F (F = C × 9/5 + 32) in one line.Show solution
Python
import numpy as np c = np.array([30, 35, 41, 28]) print(c * 9 / 5 + 32)
[ 86. 95. 105.8 82.4]
Task 2: A 3×3 array has sales of 3 shops (rows) for 3 months. Express each value as a % of its shop’s total.
Show solution
Python
import numpy as np s = np.array([[120, 150, 130], [80, 95, 125], [200, 180, 220]]) print((s / s.sum(axis=1, keepdims=True) * 100).round(1))
[[30. 37.5 32.5]
[26.7 31.7 41.7]
[33.3 30. 36.7]]
Task 3: Simulate 100,000 rolls of two dice and estimate the probability that the total is 7.
Show solution
Python
import numpy as np rng = np.random.default_rng(1) rolls = rng.integers(1, 7, size=(100000, 2)).sum(axis=1) print(round((rolls == 7).mean(), 4), "(exact = 1/6 = 0.1667)")
0.1662 (exact = 1/6 = 0.1667)
Quick quiz
🧠 Quick Quiz – test yourself
1. Shapes (4, 3) and (3,) broadcast to…
The (3,) row is repeated for all 4 rows.
2. np.where(a > 0, a, 0) does…
Where the condition is False, the third argument is used.
3. Why call default_rng(42)?
A fixed seed makes random results repeatable.
4. np.percentile(x, 50) equals…
The 50th percentile is the median.
5. cumsum of [2, 3, 5] is…
Running totals: 2, 2+3, 2+3+5.
Interview corner
💼 Interview Corner
Explain broadcasting with an example.
Subtracting a (3,) array of column means from a (100, 3) matrix: NumPy virtually repeats the mean row 100 times and subtracts element-wise, without loops or extra memory. Shapes are compared from the right and must be equal or 1.
What is a Monte Carlo simulation?
Estimating an uncertain quantity by running the process many times with random inputs and summarising the outcomes – for example the probability of a monthly loss or the range of project costs.
Summary
- Maths on arrays works element-wise – no loops.
- Broadcasting combines shapes like (4,3) with (3,) or (4,1).
- sum, mean, std, percentile, cumsum summarise data fast.
- default_rng(seed) gives reproducible simulations.
➡️ Next
Day 10: pandas – Series and DataFrames, the tables every data scientist uses all day.
Frequently Asked Questions
np.random.seed or default_rng?
New code should use rng = np.random.default_rng(seed). The old np.random.seed still works but the Generator API is faster and safer.
What is keepdims=True?
It keeps the reduced axis with size 1, so the result broadcasts back against the original array – handy for percentages of row totals.
