Statistics & Probability - Cheat sheet
Your last-minute revision companion for mastering Statistics & Probability concepts, formulas, and problem-solving techniques for UPCAT and other CETs.
Basic Statistics Concepts
Formulas
Mean = Σx / n
Meaning
Σx = sum of all values, n = number of values
Watch out
Don't confuse with median - mean uses ALL values and is affected by outliers
When to use
Finding the average or central tendency of a dataset
Range = Maximum - Minimum
Meaning
Difference between highest and lowest values
Watch out
Range is easily affected by extreme values (outliers)
When to use
Finding the spread or dispersion of data
Variance = Σ(x - x̄)² / n
Meaning
x = individual values, x̄ = mean, n = number of values
Watch out
Remember to square the deviations - don't forget this step
When to use
Measuring how spread out data points are from the mean
Standard Deviation = √Variance
Meaning
Square root of variance
Watch out
Always take the positive square root
When to use
Expressing spread in the same units as the original data
Important facts
- Mean is affected by outliers, median is not
- For odd n: median = middle value; for even n: median = average of two middle values
- A dataset can have no mode, one mode, or multiple modes
- Arrange data in ascending order to find median easily
- Standard deviation is always non-negative
Key definitions
Population
Example
All students in a school
Complete set of all individuals or items of interest
Sample
Example
50 randomly selected students from the school
Subset selected from a population for study
Variable
Example
Height, age, test scores
Characteristic that can take different values
Data
Example
Heights: 160cm, 165cm, 170cm
Actual values collected for variables
Median
Example
For {3,5,7,9,11}, median = 7
Middle value when data is arranged in order
Mode
Example
For {2,3,3,4,5}, mode = 3
Most frequently occurring value
Diagrams to know
- Bar graphs for categorical data
- Histograms for continuous data
- Pie charts for parts of a whole
- Box plots showing quartiles
Sampling Methods
Formulas
Sample Size Rule: n ≥ 30 for normal approximation
Meaning
n = sample size
Watch out
Smaller samples may not represent population well
When to use
Determining if sample size is adequate for statistical inference
Important facts
- Random sampling reduces bias
- Larger samples generally give better estimates
- Convenience sampling may introduce bias
- Stratified sampling ensures representation of all groups
- Systematic sampling can miss patterns if list has cycles
Key definitions
Simple Random Sampling
Example
Drawing names from a hat
Every member has equal chance of selection
Stratified Sampling
Example
Sample students from each grade level
Population divided into groups, sample from each proportionally
Systematic Sampling
Example
Every 10th student from enrollment list
Select every kth member from ordered list
Cluster Sampling
Example
Select random barangays, survey all households in those barangays
Divide into clusters, randomly select clusters, survey all in selected clusters
Convenience Sampling
Example
Survey friends or classmates
Select easily accessible members (non-probability)
Counting Principles
Formulas
Fundamental Counting Principle: k₁ × k₂ × k₃ × ... × kₙ
Meaning
k₁, k₂, etc. = number of ways for each event
Watch out
Events must be independent - one doesn't affect the others
When to use
Counting total ways for sequence of independent events
n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1
Meaning
n = positive integer, 0! = 1, 1! = 1
Watch out
Remember that 0! = 1, not 0
When to use
Counting arrangements of n distinct objects
ₙPᵣ = n!/(n-r)!
Meaning
n = total objects, r = objects to arrange
Watch out
Order matters in permutations - AB ≠ BA
When to use
Arrangements where order matters
ₙCᵣ = n!/[r!(n-r)!]
Meaning
n = total objects, r = objects to choose
Watch out
Order doesn't matter in combinations - AB = BA
When to use
Selections where order doesn't matter
Common values
Value
1
Symbol
0!
Quantity
0!
Value
1
Symbol
1!
Quantity
1!
Value
2
Symbol
2!
Quantity
2!
Value
6
Symbol
3!
Quantity
3!
Value
24
Symbol
4!
Quantity
4!
Value
120
Symbol
5!
Quantity
5!
Important facts
- For permutations: order matters, use ₙPᵣ
- For combinations: order doesn't matter, use ₙCᵣ
- ₙCᵣ = ₙCₙ₋ᵣ (choosing r is same as leaving n-r)
- ₙP₀ = 1 and ₙC₀ = 1
- ₙPₙ = n! (all arrangements of n objects)
Key definitions
Permutation
Example
Arranging 3 books: ABC, ACB, BAC are different
Arrangement where order matters
Combination
Example
Choosing 3 books: ABC = ACB = BAC
Selection where order doesn't matter
Factorial
Example
5! = 5 × 4 × 3 × 2 × 1 = 120
Product of all positive integers up to n
Probability
Formulas
P(E) = n(E)/n(S)
Meaning
n(E) = favorable outcomes, n(S) = total outcomes
Watch out
Ensure all outcomes are equally likely
When to use
Finding probability of any event
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Meaning
P(A ∪ B) = A or B, P(A ∩ B) = A and B
Watch out
Don't forget to subtract P(A ∩ B) to avoid double counting
When to use
Probability of either event A or B occurring
P(A') = 1 - P(A)
Meaning
P(A') = probability of complement of A
Watch out
Complement means everything except A
When to use
Finding probability of event NOT happening
P(A ∩ B) = P(A) × P(B) [if independent]
Meaning
For independent events only
Watch out
Only works when events are independent
When to use
Probability of both independent events occurring
P(B|A) = P(A ∩ B)/P(A)
Meaning
P(B|A) = probability of B given A occurred
Watch out
P(A) cannot be zero
When to use
Conditional probability when events are dependent
Important facts
- Probability is always between 0 and 1
- P(impossible) = 0, P(certain) = 1
- Sum of all probabilities in sample space = 1
- For mutually exclusive events: P(A ∩ B) = 0
- For independent events: P(B|A) = P(B)
Key definitions
Sample Space
Example
Rolling die: S = {1,2,3,4,5,6}
Set of all possible outcomes
Event
Example
Getting even number: E = {2,4,6}
Subset of sample space
Mutually Exclusive
Example
Getting heads and tails in one coin toss
Events cannot occur simultaneously
Independent Events
Example
Two coin tosses
One event doesn't affect the other
Conditional Probability
Example
P(rain | cloudy sky)
Probability of B given A has occurred
Diagrams to know
- Tree diagrams for sequential events
- Venn diagrams for event relationships
- Probability tables for conditional probability
Must remember
- P(E) = Favorable outcomes ÷ Total outcomes (basic probability)
- 0! = 1 (crucial for permutation/combination formulas)
- ₙPᵣ for arrangements (order matters), ₙCᵣ for selections (order doesn't matter)
- P(A') = 1 - P(A) (complement rule)
- Mean uses all values, median is middle value, mode is most frequent
- For mutually exclusive events: P(A ∩ B) = 0
- For independent events: P(A ∩ B) = P(A) × P(B)
- Sample space contains ALL possible outcomes
- Variance = average of squared deviations from mean
- Probability always between 0 and 1, inclusive
Last minute tips
- Always check if events are independent before multiplying probabilities
- For combination problems, ask yourself: does order matter? If no, use ₙCᵣ
- When finding median, always arrange data in order first
- In probability trees, multiply along branches, add across different paths
- Check your probability answers - they should never exceed 1 or be negative
Comparison tables
Mean vs Median vs Mode
| Measure | Definition | When to Use | Affected by Outliers |
|---|---|---|---|
| Mean | Sum ÷ Count | Normal distributions | Yes - very sensitive |
| Median | Middle value | Skewed distributions | No - resistant to outliers |
| Mode | Most frequent | Categorical data | No - shows typical value |
Permutations vs Combinations
| Type | Order Matters? | Formula | Example |
|---|---|---|---|
| Permutation | Yes | ₙPᵣ = n!/(n-r)! | Arranging letters ABC |
| Combination | No | ₙCᵣ = n!/[r!(n-r)!] | Choosing 3 from 5 students |
Sampling Methods Comparison
| Method | Process | Advantage | Disadvantage |
|---|---|---|---|
| Simple Random | Equal chance for all | Unbiased | May miss groups |
| Stratified | Sample from each stratum | Represents all groups | Complex to implement |
| Systematic | Every kth member | Easy to implement | May miss patterns |
| Cluster | Random clusters, all members | Cost effective | Less precise |
Diagrams
Statistics and Probability Overview
Complete overview of all major topics in Statistics and Probability chapter
Problem-Solving Decision Tree
Decision flowchart to help determine which formula or method to use for different problem types
Probability Problem Solving Steps
Step-by-step process for solving probability problems systematically