Section 01

The Big Idea — Two Operations, One Relationship

For most of calculus, derivatives and integrals feel like two completely separate tools. A derivative measures how fast something is changing at an instant — a slope. An integral measures the total accumulation of something over an interval — an area. They seem to answer opposite kinds of questions.

The Fundamental Theorem of Calculus (FTC) is the discovery that they are not separate at all. They are inverse operations — the same relationship that addition has with subtraction, or multiplication has with division. Differentiate an integral and you get back the original function. Integrate a derivative and you get back the original function (up to a constant). This single fact is why calculus works as one unified subject instead of two disconnected topics.

Section 02

Formal Statement — FTC Part 1 and Part 2

Part 1 tells you that integration and differentiation are inverse processes. Part 2 is the practical payoff: it hands you a way to compute the exact value of a definite integral without summing infinitely many rectangles — just find an antiderivative and plug in the two endpoints.

Section 03

Interactive: FTC Part 1 — Differentiating Accumulation

Pick a function f(t). Watch its accumulation function F(x) get built via integration, then differentiated right back to f(x) — confirming Part 1 in each case.

Section 04

Interactive: FTC Part 2 — Evaluating Definite Integrals

Four definite integrals, each solved the FTC way: find an antiderivative, then subtract F(a) from F(b). No rectangles, no limits of Riemann sums.

Section 05

The Odometer and Speedometer Model

Every car dashboard already contains a working demonstration of FTC. The speedometer shows your instantaneous speed — a rate. The odometer shows your total distance traveled — an accumulation of that rate over time.

Section 06

Interactive: Watching Area Accumulate

Let f(t) = t (a straight line through the origin). As x grows, the shaded area under the line from 0 to x grows too — that shaded area IS F(x). Click through increasing values of x and watch the accumulation function build up, one snapshot at a time.

Section 07

Why the Two Parts Are Really One Theorem

Part 1 and Part 2 can feel like two separate facts, but they’re two views of the exact same relationship, just pointed in opposite directions.

Section 08

Worked Example — Projectile Motion

A ball is thrown upward. Its velocity (accounting for gravity) is v(t) = −9.8t + 20 meters per second, where t is measured in seconds. Find the ball’s total displacement between t = 0 and t = 2 seconds.

Section 09

Common Mistakes to Avoid

Reversing F(a) and F(b)

The formula is F(b) − F(a), upper bound minus lower bound — never the other way around. Flipping it flips the sign of your answer.

Worrying about “+C”

For definite integrals, the constant of integration always cancels: (F(b)+C) − (F(a)+C) = F(b) − F(a). Any antiderivative works — you don’t need to find “the” one.

Ignoring discontinuities

If f has a vertical asymptote or jump inside [a,b], you can’t blindly apply FTC across it — the interval must be split or treated as an improper integral.

Confusing definite and indefinite

∫f(x)dx (indefinite) is a family of functions plus C. ∫ₐᵇf(x)dx (definite) is a single number. Only definite integrals get evaluated via FTC Part 2 directly.

Section 10

Real-World Applications

Physics — Motion

Displacement from velocity, velocity from acceleration, work done from a variable force — all direct FTC applications in mechanics.

Economics

Total cost is the integral of marginal cost. Consumer and producer surplus are computed as areas — definite integrals evaluated via FTC.

Medicine

Total drug absorbed into the bloodstream over time is the integral of the concentration-rate function — critical for dosage modeling.

Electrical Engineering

Total electric charge is the integral of current over time: Q = ∫ I(t) dt — a direct real-world use of FTC Part 2.

Probability

A cumulative distribution function (CDF) is the integral of a probability density function (PDF) — FTC Part 1 connects the two directly.

Biology

Total population growth over a period is the integral of the growth-rate function, used constantly in ecological and epidemiological modeling.

Section 11

Differentiation vs Integration — Comparison Table

Section 12

Code Examples — Python

Verifying FTC symbolically with SymPy

from sympy import *

x, t = symbols('x t')

# FTC Part 1: differentiate an accumulation function
f = t
F = integrate(f, (t, 0, x))       # F(x) = x**2/2
check = diff(F, x)                    # should equal f
print(F, "->", check)              # x**2/2 -> x

# FTC Part 2: evaluate a definite integral
result = integrate(x**2, (x, 0, 3))
print(result)                        # 9

result2 = integrate(sin(x), (x, 0, pi))
print(result2)                       # 2

Numerical integration with SciPy (matching the analytic answer)

from scipy.integrate import quad
import numpy as np

# ∫₀² e^x dx  — should be ~6.389
result, error = quad(lambda x: np.exp(x), 0, 2)
print(f"{result:.4f}")                 # 6.3891

# Projectile displacement: v(t) = -9.8t + 20, from t=0 to t=2
displacement, error = quad(lambda t: -9.8*t + 20, 0, 2)
print(f"{displacement:.1f} meters")     # 20.4 meters

Section 13

Knowledge Quiz

Click a question to expand it, then pick your answer.

Section 01

The Difference — A Lock Code vs. A Fruit Bowl

Imagine two everyday situations. First: you’re setting a 3-digit code on a padlock using digits 1, 2, and 3. The code 1-2-3 is completely different from 3-2-1 — they open different locks (or rather, only the exact sequence you set will open yours). Order matters. This is a permutation.

Second: you’re picking 3 fruits from a bowl to make a smoothie — an apple, a banana, and an orange. It doesn’t matter if you grabbed the apple first or the orange first — you end up with the exact same smoothie ingredients either way. Order doesn’t matter. This is a combination.

Section 02

Formal Definitions — The nPr and nCr Formulas

Both formulas start from the same n items, choosing r of them — they only differ in whether order counts.

Section 03

Interactive: Permutation or Combination?

Click through 5 real scenarios. Try to guess before reading the reasoning — this is the exact judgment call you’ll need to make on every word problem you encounter.

Section 04

The Factorial Foundation

Both formulas are built entirely from factorials, so it’s worth getting comfortable with them first. n! (read “n factorial”) means multiply every whole number from n down to 1.

Section 05

Interactive: Real-Life Permutations Gallery

Four real scenarios where order matters, each fully calculated.

  Permutations in the Real World

Olympic Podium PIN Code Seating Order Password

8 sprinters race in the 100m final. How many different ways can gold, silver, and bronze be awarded?

n = 8 sprinters, r = 3 medal positions
P(8,3) = 8! / (8−3)! = 8! / 5!
= 8 × 7 × 6
= 336 possible podium outcomes

336 ways — no repetition (a sprinter can’t win two medals), order matters (gold ≠ silver ≠ bronze)

A phone lock uses a 4-digit PIN, digits 0–9, and digits CAN repeat (like 1-1-2-2). How many possible PINs exist?

10 choices for each of 4 positions
10 × 10 × 10 × 10
= 10⁴
= 10,000 possible PINs

10,000 combinations — this is a “permutation with repetition”: order matters AND digits can repeat

5 friends want a group photo, standing in a row. How many different ways can they line up?

n = 5 friends, all 5 positions filled
5! = 5 × 4 × 3 × 2 × 1
= 120 possible lineups

120 ways — this is the special case where r = n (arranging ALL the items), so it’s just n!

A password requires exactly 4 DIFFERENT letters (no repeats), chosen from the 26-letter alphabet. How many such passwords exist?

n = 26 letters, r = 4 positions, no repeats
P(26,4) = 26! / (26−4)! = 26! / 22!
= 26 × 25 × 24 × 23
= 358,800 possible passwords

358,800 ways — order matters (ABCD ≠ DCBA) and no letter can repeat

Section 06

Interactive: Real-Life Combinations Gallery

Four real scenarios where order doesn’t matter — only which items get chosen.

  Combinations in the Real World

Lottery Committee Pizza Toppings Poker Hand

A lottery draws 6 numbers from a pool of 49. How many different winning combinations are possible?

n = 49 numbers, r = 6 drawn
C(49,6) = 49! / [6! × 43!]
= (49×48×47×46×45×44) / 720
= 13,983,816 possible combinations

~14 million combinations — {1,2,3,4,5,6} is the same winning ticket regardless of the order the balls were drawn in

A club with 10 members needs to select a 3-person committee, with no distinct roles. How many possible committees exist?

n = 10 members, r = 3 chosen
C(10,3) = 10! / [3! × 7!]
= (10×9×8) / (3×2×1)
= 720 / 6 = 120 possible committees

120 ways — {Alice, Bob, Carla} is one single committee, regardless of who was “picked first”

A pizza place offers 8 toppings. How many different 3-topping pizzas can you order?

n = 8 toppings, r = 3 chosen
C(8,3) = 8! / [3! × 5!]
= (8×7×6) / (3×2×1)
= 336 / 6 = 56 possible pizzas

56 unique pizzas — a mushroom-olive-pepper pizza is identical to a pepper-mushroom-olive pizza

A standard deck has 52 cards. How many different 5-card poker hands are possible?

n = 52 cards, r = 5 dealt
C(52,5) = 52! / [5! × 47!]
= (52×51×50×49×48) / 120
= 2,598,960 possible hands

~2.6 million hands — the order cards were dealt in doesn’t change what hand you’re holding

Section 07

The “Does Order Matter?” Test

A reliable 3-step process for any word problem you encounter:

Section 08

Worked Example — Building a Real Password System

Let’s combine everything into one realistic problem: a website requires passwords with exactly 2 different letters (no repeats) followed by 3 digits (digits CAN repeat). How many total passwords are possible?

Section 09

Permutations and Combinations with Repetition

The formulas above assume each item can only be chosen once. Real life often allows repeats — here’s how the math changes.

Worked Example — Ice Cream Scoops

An ice cream shop has 5 flavors. You order 3 scoops, and you’re allowed to repeat a flavor (like 2 scoops chocolate + 1 scoop vanilla). Order doesn’t matter — a cup with chocolate-chocolate-vanilla is the same order regardless of which scoop went in first.

Section 10

Pascal’s Triangle Connection

Every entry in Pascal’s Triangle IS a combination value. Row n, position k gives you exactly C(n,k) — no calculation required, just count and look it up.

Look at row n=6: the entries are C(6,0)=1, C(6,1)=6, C(6,2)=15, C(6,3)=20, and so on. Each number is the sum of the two numbers diagonally above it — which is exactly Pascal’s Rule: C(n,k) = C(n−1,k−1) + C(n−1,k).

Section 11

Real-World Domains

Cryptography

Key space size for a cipher is a permutation calculation — the number of possible keys directly determines how brute-forceable an encryption scheme is.

Genetics

Counting possible DNA sequences, or the number of ways alleles can combine in offspring, relies directly on permutation and combination formulas.

Tournament Brackets

The number of possible ways a single-elimination bracket can play out, or how many unique round-robin schedules exist, is pure combinatorics.

Scheduling

Assigning employees to shifts, or students to exam time slots, is a permutation/combination problem — especially when constraints (no repeats, fixed roles) apply.

Quality Control

Choosing a random sample of r items from a batch of n to inspect for defects is a classic combination — order of inspection doesn’t matter.

Game Design

Loot drop tables, card game deck compositions, and probability balancing in games all rely on combinatorics to calculate fair odds.

Section 12

Code Examples — Python

Using math.perm() and math.comb() (Python 3.8+)

import math

# Olympic podium — 3 medals from 8 finalists, order matters
print(math.perm(8, 3))     # 336

# Lottery — choose 6 numbers from 49, order doesn't matter
print(math.comb(49, 6))    # 13983816

# Password letters — 2 of 26, no repeats, order matters
print(math.perm(26, 2))    # 650

# Pizza toppings — choose 3 of 8, order doesn't matter
print(math.comb(8, 3))     # 56

# PIN code — permutation WITH repetition (not built into math module)
pin_combos = 10 ** 4
print(pin_combos)          # 10000

# Ice cream scoops — combination WITH repetition (stars and bars)
ice_cream = math.comb(5 + 3 - 1, 3)
print(ice_cream)          # 35

Generating the actual arrangements with itertools

from itertools import permutations, combinations

runners = ['A', 'B', 'C', 'D']

# All ordered podium outcomes (top 3 of 4 runners)
podiums = list(permutations(runners, 3))
print(len(podiums))      # 24  =  4P3
print(podiums[:3])
# [('A','B','C'), ('A','B','D'), ('A','C','B'), ...]

# All unordered committees (any 3 of 4 people)
committees = list(combinations(runners, 3))
print(len(committees))   # 4  =  4C3
print(committees)
# [('A','B','C'), ('A','B','D'), ('A','C','D'), ('B','C','D')]

Section 13

Knowledge Quiz

Click a question to expand it, then pick your answer.

Q4 — Why is choosing a 5-card poker hand a combination rather than a permutation?

Because there are only 52 cards in a deck Because face cards are different from number cards Because the order the cards were dealt in doesn’t change which hand you’re holding Because repetition is allowed in card games

A hand of {A, K, 7, 3, 2} is the same hand no matter which card was dealt first. Since order doesn’t affect the outcome, it’s counted with combinations: C(52,5) = 2,598,960.

Section 01

What is an Outlier?

An outlier is a data point that lies far away from the rest of the observations in a dataset. It is a value that is unusually high or low compared to the typical range of the data.

Consider the salaries of five employees: £28k, £31k, £29k, £33k, £420k. That last value — £420k — is an outlier. It could be the CEO’s salary, a data entry error, or a completely different type of employee included by mistake. Each explanation calls for a different response.

Section 02

Why Outliers Matter — The Impact on Statistics

Outliers can have a dramatic or minimal effect depending on which statistical measure you use. The mean is highly sensitive. The median is almost immune. Understanding this asymmetry is one of the most practical skills in applied statistics.

Salary example — the CEO effect

A team of 10 employees. Nine earn between £28,000 and £35,000. One is the CEO earning £500,000.

Section 03

Types of Outliers

Not all outliers are the same. Understanding why a value is an outlier determines what you should do with it.

Section 04

How to Detect Outliers — 4 Methods

IQR Method — Worked Example

Dataset: 12, 15, 14, 10, 18, 14, 13, 100, 16, 11

Section 05

Interactive Outlier Detector

Enter your own dataset (comma-separated numbers) and choose a detection method. The detector will flag outliers instantly and show you the full statistical breakdown.

Section 06

Visualising Outliers with Box Plots

A box plot (also called a box-and-whisker plot) is the standard visual tool for outlier detection. The box shows the IQR (Q1 to Q3). The line inside is the median. The whiskers extend to the Tukey fences. Individual dots beyond the whiskers are outliers.

Section 07

How to Handle Outliers — 5 Strategies

After detecting an outlier, you have five options. The right one depends on the type of outlier and your analytical goal.

Section 08

When You Should NEVER Remove an Outlier

The phrase “never ignore them” in the title of this article is deliberate. Outliers are sometimes the most important data points. Here is when removing them would be a serious mistake:

Section 09

Real-World Examples

Section 10

Python Code

IQR Method

import numpy as np

data = np.array([12, 15, 14, 10, 18, 14, 13, 100, 16, 11, 9, 17])

Q1 = np.percentile(data, 25)
Q3 = np.percentile(data, 75)
IQR = Q3 - Q1

lower_fence = Q1 - 1.5 * IQR
upper_fence = Q3 + 1.5 * IQR

outliers = data[(data < lower_fence) | (data > upper_fence)]
clean    = data[(data >= lower_fence) & (data <= upper_fence)]

print(f"Q1={Q1}, Q3={Q3}, IQR={IQR}")
print(f"Fences: [{lower_fence:.1f}, {upper_fence:.1f}]")
print(f"Outliers: {outliers}")   # → [100]
print(f"Clean data: {clean}")

Z-Score Method

from scipy import stats
import numpy as np

data = np.array([12, 15, 14, 10, 18, 14, 13, 100, 16, 11])

z_scores = np.abs(stats.zscore(data))
threshold = 3.0

outliers = data[z_scores > threshold]
print(f"Z-scores: {z_scores.round(2)}")
print(f"Outliers (|z| > {threshold}): {outliers}")  # → [100]

Modified Z-Score (most robust)

import numpy as np

def modified_zscore(data):
    median = np.median(data)
    mad = np.median(np.abs(data - median))   # Median Absolute Deviation
    return 0.6745 * (data - median) / mad

data = np.array([12, 15, 14, 10, 18, 14, 13, 100, 16, 11])
m_scores = np.abs(modified_zscore(data))
outliers = data[m_scores > 3.5]
print(f"Outliers: {outliers}")  # → [100]

Winsorising (capping outliers)

from scipy.stats import mstats
import numpy as np

data = np.array([12, 15, 14, 10, 18, 14, 13, 100, 16, 11])

# Winsorise at 5th and 95th percentiles
winsorised = mstats.winsorize(data, limits=[0.05, 0.05])
print(f"Original:   {data}")
print(f"Winsorised: {np.array(winsorised)}")
# 100 is replaced with the 95th percentile value

# Log transformation for right-skewed data
log_data = np.log1p(data)   # log(1+x) handles zeros safely
print(f"Log:        {log_data.round(2)}")
# 100 → 4.62, rest compressed into smaller range

Box plot visualisation with matplotlib

import matplotlib.pyplot as plt
import numpy as np

data = [12, 15, 14, 10, 18, 14, 13, 100, 16, 11, 9, 17]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))

# Box plot
bp = ax1.boxplot(data, vert=False, patch_artist=True,
                 flierprops=dict(marker='o', color='red', markersize=8))
bp['boxes'][0].set_facecolor('#99f6e4')
ax1.set_title('Box Plot — red dots are outliers')
ax1.set_xlabel('Value')

# Dot plot
ax2.scatter(data, [1]*len(data), alpha=0.7, s=80,
           c=['#e11d48' if x > 21 else '#0d9488' for x in data])
ax2.set_title('Dot Plot — red = outlier')
ax2.set_yticks([])

plt.tight_layout()
plt.show()

Section 11

Knowledge Quiz

Six questions to test your understanding of outliers in statistics.

Section 01

What is a Normal Distribution?

A normal distribution (also called a Gaussian distribution or bell curve) is a way that data arranges itself around a central value — with most values clustered near the middle, and fewer and fewer values appearing as you move toward the extremes.

Plot it on a graph and you get the unmistakable bell shape: perfectly symmetric, peaked in the centre, tapering off equally on both sides. The curve never actually touches the x-axis — it just gets infinitely close at the extremes.

Section 02

Why Does Nature Produce Bell Curves?

This is the deep question. Why should human heights, leaf sizes, or measurement errors all form a bell? The answer comes from one of the most powerful theorems in all of mathematics: the Central Limit Theorem.

The Central Limit Theorem says: whenever a quantity is the result of many small, independent, random influences added together — the result will be normally distributed, no matter what distribution each individual influence has.

The logic in three steps

Section 03

Interactive: Coin Flips Become a Bell Curve

The clearest way to see the Central Limit Theorem is with coin flips. Flip a coin 10 times and count heads. Do it thousands of times. The histogram of “number of heads” will form a perfect bell curve.

Click Flip & Toss to simulate thousands of experiments. Watch the histogram grow into a bell shape before your eyes.

Section 04

Key Properties of the Normal Distribution

The normal distribution has a precise mathematical formula, but you don’t need to memorise it — you need to understand what its two parameters mean.

Section 05

Interactive Bell Curve Builder

Drag the sliders to change the mean (μ) and standard deviation (σ). Watch how the bell curve shifts position and changes shape in real time.

Section 06

The 68-95-99.7 Empirical Rule

One of the most useful facts about the normal distribution is that fixed percentages of data always fall within 1, 2, and 3 standard deviations of the mean — regardless of what μ and σ actually are.

Worked Example — Adult Male Heights

Adult male heights in the UK are approximately normally distributed with μ = 175 cm and σ = 7 cm. Using the empirical rule:

Section 07

Z-Scores — Standardising Any Normal Distribution

A z-score tells you how many standard deviations a value is away from the mean. It lets you compare values from completely different normal distributions on the same scale.

What z-scores mean

Section 08

Interactive Z-Score Calculator

Enter any value, mean, and standard deviation to instantly calculate the z-score and see what percentile it corresponds to.

Section 09

Real-World Examples

The normal distribution appears across almost every domain of science, engineering, and everyday life.

Section 10

When Data is NOT Normally Distributed

Not everything is a bell curve. Recognising when data isn’t normal is just as important as knowing when it is.

Section 11

Python Code

Plotting a bell curve

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

mu, sigma = 175, 7   # UK male heights: mean=175cm, std=7cm

x = np.linspace(mu - 4*sigma, mu + 4*sigma, 300)
y = norm.pdf(x, mu, sigma)

plt.figure(figsize=(10, 5))
plt.plot(x, y, color='#6366f1', linewidth=2.5)

# Shade the 1σ, 2σ, 3σ regions
for n, color, label in [(1,'#10b981','68%'),(2,'#f59e0b','95%'),(3,'#ef4444','99.7%')]:
    mask = (x >= mu - n*sigma) & (x <= mu + n*sigma)
    plt.fill_between(x, y, where=mask, alpha=0.2, color=color, label=f'{label} within {n}σ')

plt.title('Normal Distribution of UK Male Heights')
plt.xlabel('Height (cm)')
plt.ylabel('Probability Density')
plt.legend()
plt.tight_layout()
plt.show()

Z-scores and probabilities

from scipy.stats import norm

mu, sigma = 175, 7
x = 185

# Z-score
z = (x - mu) / sigma
print(f"Z-score: {z:.2f}")           # → 1.43

# Probability of being BELOW this height
p_below = norm.cdf(x, mu, sigma)
print(f"P(height < 185): {p_below:.1%}")  # → 92.4%

# Probability of being ABOVE this height
p_above = 1 - p_below
print(f"P(height > 185): {p_above:.1%}")  # → 7.6%

# Probability of being BETWEEN two values
p_between = norm.cdf(182, mu, sigma) - norm.cdf(168, mu, sigma)
print(f"P(168 < height < 182): {p_between:.1%}")  # → 68.3%

# What height is at the 90th percentile?
p90 = norm.ppf(0.90, mu, sigma)
print(f"90th percentile height: {p90:.1f} cm")  # → 184.0 cm

Testing if data is normally distributed

from scipy.stats import shapiro, normaltest
import numpy as np

data = np.random.normal(loc=175, scale=7, size=100)

# Shapiro-Wilk test (best for n < 5000)
stat, p = shapiro(data)
print(f"Shapiro-Wilk: stat={stat:.3f}, p={p:.3f}")
# If p > 0.05 → fail to reject normality (data is likely normal)

# D'Agostino K² test
stat2, p2 = normaltest(data)
print(f"Normaltest:   stat={stat2:.3f}, p={p2:.3f}")

# Quick visual check
import scipy.stats as stats
import matplotlib.pyplot as plt
stats.probplot(data, dist="norm", plot=plt)
plt.title("Q-Q Plot — points on the line = normal")
plt.show()

Section 12

Knowledge Quiz

Six questions to test your understanding of the normal distribution.