UPCAT Mathematics — Statistics & ProbabilityStudy Notes

Statistics is the science of collecting, organizing, analyzing, and interpreting data. Understanding basic terminology is crucial for solving statistical problems.

Key Definitions:

• Data: A collection of quantitative or qualitative information used for reasoning or calculation
• Variable: Any characteristic, number, or quantity that can be measured or counted
• Population: The complete set of all individuals sharing a common characteristic
• Sample: A subset selected from a population to represent that population

Types of Variables:

1. Qualitative Variables: Can be categorized but not numerically measured (e.g., gender, eye color)

2. Quantitative Variables: Numerical values that can be ordered

   • Discrete: Countable values (e.g., number of students)
   • Continuous: Infinite possible values within a range (e.g., height, weight)

Levels of Measurement:

• Nominal: Categories with no order (gender, nationality)
• Ordinal: Ranked categories with no precise differences (grades A, B, C)
• Interval: Ranked with precise differences, no true zero (temperature in °C)
• Ratio: Ranked with precise differences and true zero (height, weight, age)

Measures of central tendency describe the center or typical value of a dataset. The three main measures are mean, median, and mode.

1. MEAN (Arithmetic Average)
The mean is the sum of all values divided by the number of values.

Formula: Mean = (Sum of all values) ÷ (Number of values)

Step-by-Step Example:
Find the mean of test scores: 85, 90, 78, 92, 88, 85, 95

Step 1: Add all values
85 + 90 + 78 + 92 + 88 + 85 + 95 = 613

Step 2: Count the number of values
Number of scores = 7

Step 3: Divide sum by count
Mean = 613 ÷ 7 = 87.57

2. MEDIAN
The median is the middle value when data is arranged in order.

For Odd Number of Values:
Step 1: Arrange data in ascending order
Step 2: Find the middle position: (n+1)÷2
Step 3: The value at this position is the median

For Even Number of Values:
Step 1: Arrange data in ascending order
Step 2: Find the two middle values
Step 3: Calculate their average

Example with Odd Values:
Data: 12, 15, 18, 20, 25, 30, 35
Median position = (7+1)÷2 = 4th position
Median = 20

Example with Even Values:
Data: 10, 15, 20, 25, 30, 35
Middle positions: 3rd and 4th values
Median = (20 + 25)÷2 = 22.5

3. MODE
The mode is the value that appears most frequently.

Types of Mode:

• Unimodal: One mode
• Bimodal: Two modes
• Multimodal: More than two modes
• No mode: All values appear equally

Example:
Data: 3, 5, 7, 5, 9, 5, 12
The value 5 appears 3 times (most frequent)
Mode = 5 (unimodal)

Measures of variability describe how spread out the data values are from the center.

1. RANGE
Range is the difference between the maximum and minimum values.

Formula: Range = Maximum value - Minimum value

Example:
Test scores: 65, 70, 75, 80, 85, 90, 95
Range = 95 - 65 = 30

2. VARIANCE AND STANDARD DEVIATION
Variance measures the average squared deviation from the mean.
Standard deviation is the square root of variance.

Steps to Calculate Variance and Standard Deviation:

Step 1: Find the mean
Step 2: Find each deviation from the mean
Step 3: Square each deviation
Step 4: Find the average of squared deviations (variance)
Step 5: Take the square root of variance (standard deviation)

Detailed Example:
Data: 10, 12, 14, 16, 18

Step 1: Mean = (10+12+14+16+18)÷5 = 70÷5 = 14

Step 2: Deviations from mean:
10-14 = -4
12-14 = -2
14-14 = 0
16-14 = 2
18-14 = 4

Step 3: Square the deviations:
(-4)² = 16
(-2)² = 4
(0)² = 0
(2)² = 4
(4)² = 16

Step 4: Variance = (16+4+0+4+16)÷5 = 40÷5 = 8

Step 5: Standard deviation = √8 = 2.83

Interpretation:

• Small variance/standard deviation = data points close to mean
• Large variance/standard deviation = data points spread out from mean

Counting principles help us determine the number of ways events can occur, which is fundamental for probability calculations.

1. FUNDAMENTAL COUNTING PRINCIPLE
If event E₁ can occur in m ways and event E₂ can occur in n ways, then both events together can occur in m × n ways.

Example:
A student has 4 shirts and 3 pants. How many different outfits are possible?
Solution: 4 × 3 = 12 different outfits

Extended Example:
A phone number format: 09XX-XXX-XXXX

• First two digits: fixed as 09
• Third digit: can be 1,2,3,4,5,6,7,8,9 (9 choices)
• Fourth digit: can be 0,1,2,3,4,5,6,7,8,9 (10 choices)
• Remaining 7 digits: each can be 0-9 (10 choices each)

Total possibilities = 1 × 1 × 9 × 10 × 10⁷ = 9 × 10⁸

2. FACTORIAL NOTATION
n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1

Examples:
3! = 3 × 2 × 1 = 6
4! = 4 × 3 × 2 × 1 = 24
5! = 5 × 4 × 3 × 2 × 1 = 120

Note: 0! = 1 (by definition)

3. PERMUTATIONS
A permutation is an arrangement where ORDER MATTERS.

Formula for n objects taken r at a time:
ₙPᵣ = n!/(n-r)!

Step-by-Step Example:
In how many ways can 5 students be arranged in a row for a photo?

Solution:
n = 5 (total students)
r = 5 (all students)
₅P₅ = 5!/(5-5)! = 5!/0! = 5!/1 = 120 ways

Another Example:
How many ways can we select and arrange 3 officers (President, VP, Secretary) from 10 candidates?

Solution:
n = 10 (total candidates)
r = 3 (positions to fill)
₁₀P₃ = 10!/(10-3)! = 10!/7! = 10×9×8 = 720 ways

4. PERMUTATIONS WITH REPETITION
When objects are identical, we divide by the factorial of repeated objects.

Formula: n!/(n₁! × n₂! × n₃! × ...)

Example:
How many distinct arrangements of the letters in BANANA?

Solution:
Total letters = 6
A appears 3 times
N appears 2 times
B appears 1 time

Arrangements = 6!/(3! × 2! × 1!) = 720/(6 × 2 × 1) = 720/12 = 60

Combinations count arrangements where order doesn't matter, and probability measures the likelihood of events.

1. COMBINATIONS
A combination is a selection where ORDER DOESN'T MATTER.

Formula: ₙCᵣ = n!/[(n-r)! × r!] = ₙPᵣ/r!

Step-by-Step Example:
A committee of 3 students is chosen from 8 students. How many combinations are possible?

Step 1: Identify n and r
n = 8 (total students)
r = 3 (students to choose)

Step 2: Apply formula
₈C₃ = 8!/[(8-3)! × 3!] = 8!/(5! × 3!)

Step 3: Calculate
₈C₃ = (8×7×6×5!)/(5! × 3×2×1) = (8×7×6)/(3×2×1) = 336/6 = 56

Key Difference: Permutation vs Combination

• Selecting President, VP, Secretary from 8 people: ₈P₃ = 336 (order matters)
• Selecting 3-person committee from 8 people: ₈C₃ = 56 (order doesn't matter)

2. BASIC PROBABILITY
Probability measures the likelihood of an event occurring.

Formula: P(E) = Number of favorable outcomes / Total number of possible outcomes

Basic Concepts:

• Experiment: A process that produces outcomes
• Sample Space (S): Set of all possible outcomes
• Event (E): Subset of sample space
• Outcome: Individual result of an experiment

Properties of Probability:

1. 0 ≤ P(E) ≤ 1
2. P(certain event) = 1
3. P(impossible event) = 0
4. Sum of all probabilities in sample space = 1

Step-by-Step Example:
A fair die is rolled. What is the probability of getting an even number?

Step 1: Identify sample space
S = {1, 2, 3, 4, 5, 6}
Total outcomes = 6

Step 2: Identify favorable outcomes
Even numbers: E = {2, 4, 6}
Favorable outcomes = 3

Step 3: Calculate probability
P(even) = 3/6 = 1/2 = 0.5 or 50%

3. PROBABILITY WITH COMBINATIONS

Example:
A bag contains 5 red balls and 3 blue balls. What is the probability of selecting 2 red balls when drawing 2 balls without replacement?

Step 1: Calculate total ways to select 2 balls from 8
₈C₂ = 8!/(6!×2!) = (8×7)/(2×1) = 28

Step 2: Calculate ways to select 2 red balls from 5
₅C₂ = 5!/(3!×2!) = (5×4)/(2×1) = 10

Step 3: Calculate probability
P(2 red balls) = 10/28 = 5/14 ≈ 0.357 or 35.7%

Advanced probability concepts include compound events, conditional probability, and independence.

1. COMPLEMENT OF AN EVENT
The complement of event E (written as Ē or E') consists of all outcomes not in E.

Formula: P(Ē) = 1 - P(E)

Example:
If P(passing exam) = 0.85, then P(failing exam) = 1 - 0.85 = 0.15

2. ADDITION RULE
For any two events A and B:
P(A or B) = P(A) + P(B) - P(A and B)

For Mutually Exclusive Events (cannot occur simultaneously):
P(A or B) = P(A) + P(B)

Step-by-Step Example:
A card is drawn from a standard deck. What is the probability of drawing a red card or a face card?

Step 1: Identify probabilities
P(red card) = 26/52 = 1/2
P(face card) = 12/52 = 3/13
P(red face card) = 6/52 = 3/26

Step 2: Apply addition rule
P(red or face) = P(red) + P(face) - P(red and face)
P(red or face) = 26/52 + 12/52 - 6/52 = 32/52 = 8/13

3. MULTIPLICATION RULE

For Independent Events:
P(A and B) = P(A) × P(B)

Example:
Two coins are tossed. What is the probability of getting two heads?

Solution:
P(first head) = 1/2
P(second head) = 1/2
P(two heads) = 1/2 × 1/2 = 1/4

For Dependent Events:
P(A and B) = P(A) × P(B|A)

where P(B|A) is the probability of B given that A occurred.

4. CONDITIONAL PROBABILITY
P(B|A) = P(A and B) / P(A)

Step-by-Step Example:
A bag contains 3 red and 2 blue marbles. Two marbles are drawn without replacement. What is the probability the second marble is red given the first was red?

Step 1: After first red marble is drawn
Remaining marbles: 2 red, 2 blue (total 4)

Step 2: Calculate conditional probability
P(second red | first red) = 2/4 = 1/2

Alternative method using formula:
P(both red) = (3/5) × (2/4) = 6/20 = 3/10
P(first red) = 3/5
P(second red | first red) = (3/10) ÷ (3/5) = (3/10) × (5/3) = 1/2

5. TREE DIAGRAMS
Tree diagrams help visualize sequential probability problems.

Example:
A bag contains 4 red and 6 blue balls. Two balls are drawn with replacement. Find P(at least one red).

Method 1: Direct calculation
P(at least one red) = P(RR) + P(RB) + P(BR)
= (4/10)(4/10) + (4/10)(6/10) + (6/10)(4/10)
= 16/100 + 24/100 + 24/100 = 64/100 = 16/25

Method 2: Using complement
P(at least one red) = 1 - P(no red) = 1 - P(BB)
= 1 - (6/10)(6/10) = 1 - 36/100 = 64/100 = 16/25

Sampling involves selecting a subset of a population to make inferences about the entire population. Proper sampling techniques are crucial for obtaining reliable results.

1. WHY SAMPLE?

• Cost-effective: Cheaper than surveying entire population
• Time-efficient: Faster data collection
• Practical: Some populations are infinite or inaccessible
• Accuracy: Well-designed samples can be very representative

2. SAMPLING METHODS

A. SIMPLE RANDOM SAMPLING (SRS)
Every member has equal chance of selection.

Step-by-Step Process:
Step 1: Create a complete list of population members
Step 2: Assign each member a unique number
Step 3: Use random number generator to select sample

Example:
Select 50 students from 1,000 high school students.

• Number students 0001-1000
• Use random number table to select 50 numbers
• Survey students with selected numbers

B. STRATIFIED SAMPLING
Divide population into homogeneous groups (strata), then randomly sample from each group.

Step-by-Step Example:
Survey student satisfaction in a university with 40% freshmen, 30% sophomores, 20% juniors, 10% seniors.

For sample size of 200:
Step 1: Calculate strata sizes

• Freshmen: 40% × 200 = 80 students
• Sophomores: 30% × 200 = 60 students
• Juniors: 20% × 200 = 40 students
• Seniors: 10% × 200 = 20 students

Step 2: Randomly sample from each stratum

C. SYSTEMATIC SAMPLING
Select every kth member from ordered population.

Formula: k = Population size ÷ Sample size

Example:
Select 100 students from 2,000 students.
k = 2,000 ÷ 100 = 20
Start randomly (say #7), then select every 20th: 7, 27, 47, 67, ...

D. CLUSTER SAMPLING
Divide population into clusters, randomly select clusters, survey all members in selected clusters.

Example:
Study Manila public schools.

• Clusters: individual schools
• Randomly select 10 schools
• Survey all students in selected schools

E. CONVENIENCE SAMPLING (Non-probability)
Sample easily accessible members.

Warning: May introduce significant bias!

3. SAMPLE SIZE CONSIDERATIONS

Factors Affecting Sample Size:

• Population size: Larger populations need larger samples
• Desired precision: Higher precision needs larger samples
• Population variability: More variable populations need larger samples
• Confidence level: Higher confidence needs larger samples

Rule of Thumb:

• Sample size of 30+ usually adequate for normal approximation
• For proportions, need at least 5 successes and 5 failures

4. SAMPLING BIAS

Types of Bias:

A. Selection Bias
Sample doesn't represent population.

Example: Surveying only smartphone users about internet usage excludes those without smartphones.

B. Non-response Bias
Selected participants don't respond.

Example: Mail survey gets 20% response rate; non-responders may differ from responders.

C. Response Bias
Participants give inaccurate answers.

Example: Asking about illegal behavior - people may lie.

D. Volunteer Bias
Volunteers differ from general population.

5. MARGIN OF ERROR
Indicates precision of sample estimate.

Interpretation:
If poll shows 55% ± 3% support, true population percentage is likely between 52% and 58%.

Factors Reducing Margin of Error:

• Larger sample size
• Less population variability
• Better sampling method