Statistics & Probability
FROM BASICS TO HYPOTHESIS TESTING
Your ultimate premium study companion 4 packed with key definitions, formula tables, worked examples, real-life
applications, mnemonics, and practice questions with full answers. Whether you're cramming for finals or building a
rock-solid foundation, this guide has everything you need.
Key Definitions x Formula Tables û Worked Examples
Master the vocabulary of statistics Every essential formula, organized Step-by-step solutions to build real
and probability and ready to use understanding
n Real-Life Applications ' 30 Practice Questions
Business, psychology, and medicine use cases Full answers and high-yield exam strategies
,Key Definitions, Formula Tables &
Mnemonics
SECTIONS 2
~ Section 1: Key Definitions
Population vs. Sample Parameter vs. Statistic
A population is the entire group you want to study. A A parameter (e.g., ¼, Ã) describes a population. A
sample is a subset drawn from that population. statistic (e.g., x, s) describes a sample. Parameters are
Statistics infer population parameters from sample usually unknown; statistics estimate them.
statistics.
Variable Types Probability
Quantitative: numerical (discrete or continuous). The likelihood of an event occurring, expressed as a
Qualitative/Categorical: labels or categories (nominal number between and . P(A) = favorable outcomes
or ordinal). Knowing the type determines which test to / total outcomes. P = means impossible; P = means
use. certain.
Random Variable Distribution
A variable whose value is determined by a random A function describing all possible values and their
experiment. Discrete random variables take countable probabilities. Key distributions: Normal, Binomial,
values; continuous random variables take any value in Poisson, t-distribution, Chi-square, F-distribution.
a range.
x Section 2: Formula Tables
¸ High-Yield Formulas 4 These are the most frequently tested formulas. Memorize them cold before your exam.
Concept Formula Notes
Mean (Population) ¼ = £x / N Sum of all values ÷ count
Mean (Sample) x = £x / n Use for sample data
Variance (Population) ò = £(x2¼)² / N Average squared deviation
Variance (Sample) s² = £(x2x)² / (n2 ) Bessel's correction: n2
Standard Deviation à = :ò or s = :s² Square root of variance
Z-Score z = (x 2 ¼) / Ã Standardizes any value
Binomial Probability P(X=k) = C(n,k) · p_ · ( 2p){_ n trials, k successes
Poisson Probability P(X=k) = (»_ · e{») / k! » = average rate
Confidence Interval x ± z*(Ã/:n) Use t* when à unknown
t-Test Statistic t = (x 2 ¼ ) / (s/:n) df = n 2
Chi-Square Statistic Dz = £[(O2E)²/E] O=observed, E=expected
Correlation Coefficient r = £[(x2x)(y2y)] / [(n2 )s³s] 2 frf
í Section 3: Mnemonics for Probability Rules
"ADD for OR" "MULTIPLY for AND"
P(A or B) = P(A) + P(B) 2 P(A and B). Mnemonic: "OR P(A and B) = P(A) × P(B|A). Mnemonic: "AND means
means ADD, but don't double-dip!" Subtract the MULTIPLY 4 chain the chances!" For independent
overlap to avoid counting it twice. events, P(B|A) = P(B).
"FLIP for NOT" "68-95-99.7 Rule"
P(A') = 2 P(A). Mnemonic: "NOT means FLIP 4 For a normal distribution: % within Ã, % within
subtract from !" The complement always makes the Ã, . % within Ã. Mnemonic: "One-Two-Three: ,
total probability equal to . , !"
"PVALUE beats ALPHA ³ FAIL" Bayes' Theorem
If p-value < ³ ³ Reject H . If p-value g ³ ³ Fail to P(A|B) = [P(B|A)·P(A)] / P(B). Mnemonic: "Flip the
Reject H . Mnemonic: "If p is LOW, H must GO!" condition 4 update your belief!" Used to revise
probabilities with new evidence.
Pro Tip: Write each mnemonic on a flashcard and review them daily. Mnemonics reduce cognitive load during high-
pressure exams 4 your brain retrieves the rule automatically.
D
r Key Distributions at a Glance
Distribution Type Mean Variance Use Case
Normal Continuous ¼ ò Heights, test
scores
Binomial Discrete np np( 2p) Pass/fail trials
Poisson Discrete » » Events per time
unit
Uniform Continuous (a+b)/ (b2a)²/ Equal likelihood
Exponential Continuous /» /»² Time between
events
t-Distribution Continuous df/(df2 ) Small samples
, Worked Examples, Real-Life Applications &
High-Yield Methods
SECTIONS 2
û Section 4: Worked Examples 4 Step by Step
' High-Yield Method: Always write out every step. Partial credit is awarded in most exams for correct
methodology even if the final answer is wrong.
Example 1: Mean, Variance & Standard Example 2: Z-Score & Normal Distribution
Deviation
IQ scores: ¿ = ,Ã= . Find P(X < ).
Dataset: { , , , , }
. Z-score: z = ( 2 )/ = .
. Mean: x = ( + + + + )/ = / = . . Look up z = . in the standard normal table
. Deviations: ( 2 . )²= . , ( 2 . )²= . , . P(Z < . )= .
( 2 . )²= . , ( 2 . )²= . , ( 2 . )²= .
. Interpretation: . % of people score below
. Variance: s² = ( . + . + . + . + . )/( 2 ) =
. / = .
Mnemonic: "Z is the distance in standard
. Std Dev: s = : . j .
í
deviations 4 always standardize first!"
Example 3: Binomial Distribution Example 4: Hypothesis Testing (One-
Sample t-Test)
A fair coin is flipped times. Find P(exactly heads).
A sample of students has x = ,s= . Test H : ¿ =
. n= ,k= ,p= .
at ³ = . .
. C( , )=
. t=( 2 )/( /: )= / = .
. P= × ( . )v × ( . )t = ×( / )j .
. df = ; critical t = . (two-tailed)
. | . |< . ³ Fail to Reject H
. Conclusion: No significant difference from ¿ =
State Choose Test Calculate Make
Hypotheses &³ Statistic Decision
Every hypothesis test follows this exact four-step framework. Mastering this sequence means you can tackle any test 4 t-
test, z-test, chi-square, or ANOVA 4 with confidence and consistency.
n Section 5: Real-Life Applications
p Business & Finance í Psychology & Social # Medicine & Public Health
Science
Quality Control: Use control charts Diagnostic Tests: Sensitivity and
(mean ± Ã) to detect defective Survey Analysis: Likert scale data specificity use conditional
products on assembly lines analyzed with chi-square tests for probability (Bayes' Theorem)
A/B Testing: Companies like group differences Epidemiology: Relative risk and
Amazon run thousands of Clinical Trials: t-tests compare odds ratios quantify disease
hypothesis tests daily to optimize treatment vs. control group associations
conversion rates outcomes Drug Approval: FDA requires p <
Risk Assessment: Banks use Effect Size: Cohen's d measures . in randomized controlled trials
normal distributions to model loan practical significance beyond p- Survival Analysis: Kaplan-Meier
default probabilities values curves model time-to-event data in
Forecasting: Regression analysis Reliability: Cronbach's alpha oncology
predicts future sales from historical measures internal consistency of
data psychological scales
§ Section 6: High-Yield Methods 4 Deep Dive
1 2 3
Mean 4 The Center of Variance 4 Spread Standard Deviation 4
Gravity Quantified Interpretable Spread
The mean is the balance point of a Variance measures average The square root of variance, in the
distribution. Sensitive to outliers squared distance from the mean. same units as the data. The most
4 one extreme value can pull it far Always non-negative. Larger intuitive measure of spread. In a
from the "typical" value. Always variance = more spread. Use normal distribution, ~ % of data
report alongside standard sample variance (n2 ) to get an falls within standard deviation of
deviation for full context. unbiased estimate of population the mean.
variance.
4 5
Normal Distribution 4 The Bell Curve Hypothesis Testing 4 Decision
Framework
Symmetric, bell-shaped, defined entirely by ¿ and Ã.
The Central Limit Theorem guarantees that sample A formal procedure to test claims about populations.
means approach normality as n increases 4 the Key concepts: Type I Error (false positive, ³), Type II
foundation of inferential statistics. Error (false negative, ´), Power ( 2´). Always state H
and H¡ before collecting data.
FROM BASICS TO HYPOTHESIS TESTING
Your ultimate premium study companion 4 packed with key definitions, formula tables, worked examples, real-life
applications, mnemonics, and practice questions with full answers. Whether you're cramming for finals or building a
rock-solid foundation, this guide has everything you need.
Key Definitions x Formula Tables û Worked Examples
Master the vocabulary of statistics Every essential formula, organized Step-by-step solutions to build real
and probability and ready to use understanding
n Real-Life Applications ' 30 Practice Questions
Business, psychology, and medicine use cases Full answers and high-yield exam strategies
,Key Definitions, Formula Tables &
Mnemonics
SECTIONS 2
~ Section 1: Key Definitions
Population vs. Sample Parameter vs. Statistic
A population is the entire group you want to study. A A parameter (e.g., ¼, Ã) describes a population. A
sample is a subset drawn from that population. statistic (e.g., x, s) describes a sample. Parameters are
Statistics infer population parameters from sample usually unknown; statistics estimate them.
statistics.
Variable Types Probability
Quantitative: numerical (discrete or continuous). The likelihood of an event occurring, expressed as a
Qualitative/Categorical: labels or categories (nominal number between and . P(A) = favorable outcomes
or ordinal). Knowing the type determines which test to / total outcomes. P = means impossible; P = means
use. certain.
Random Variable Distribution
A variable whose value is determined by a random A function describing all possible values and their
experiment. Discrete random variables take countable probabilities. Key distributions: Normal, Binomial,
values; continuous random variables take any value in Poisson, t-distribution, Chi-square, F-distribution.
a range.
x Section 2: Formula Tables
¸ High-Yield Formulas 4 These are the most frequently tested formulas. Memorize them cold before your exam.
Concept Formula Notes
Mean (Population) ¼ = £x / N Sum of all values ÷ count
Mean (Sample) x = £x / n Use for sample data
Variance (Population) ò = £(x2¼)² / N Average squared deviation
Variance (Sample) s² = £(x2x)² / (n2 ) Bessel's correction: n2
Standard Deviation à = :ò or s = :s² Square root of variance
Z-Score z = (x 2 ¼) / Ã Standardizes any value
Binomial Probability P(X=k) = C(n,k) · p_ · ( 2p){_ n trials, k successes
Poisson Probability P(X=k) = (»_ · e{») / k! » = average rate
Confidence Interval x ± z*(Ã/:n) Use t* when à unknown
t-Test Statistic t = (x 2 ¼ ) / (s/:n) df = n 2
Chi-Square Statistic Dz = £[(O2E)²/E] O=observed, E=expected
Correlation Coefficient r = £[(x2x)(y2y)] / [(n2 )s³s] 2 frf
í Section 3: Mnemonics for Probability Rules
"ADD for OR" "MULTIPLY for AND"
P(A or B) = P(A) + P(B) 2 P(A and B). Mnemonic: "OR P(A and B) = P(A) × P(B|A). Mnemonic: "AND means
means ADD, but don't double-dip!" Subtract the MULTIPLY 4 chain the chances!" For independent
overlap to avoid counting it twice. events, P(B|A) = P(B).
"FLIP for NOT" "68-95-99.7 Rule"
P(A') = 2 P(A). Mnemonic: "NOT means FLIP 4 For a normal distribution: % within Ã, % within
subtract from !" The complement always makes the Ã, . % within Ã. Mnemonic: "One-Two-Three: ,
total probability equal to . , !"
"PVALUE beats ALPHA ³ FAIL" Bayes' Theorem
If p-value < ³ ³ Reject H . If p-value g ³ ³ Fail to P(A|B) = [P(B|A)·P(A)] / P(B). Mnemonic: "Flip the
Reject H . Mnemonic: "If p is LOW, H must GO!" condition 4 update your belief!" Used to revise
probabilities with new evidence.
Pro Tip: Write each mnemonic on a flashcard and review them daily. Mnemonics reduce cognitive load during high-
pressure exams 4 your brain retrieves the rule automatically.
D
r Key Distributions at a Glance
Distribution Type Mean Variance Use Case
Normal Continuous ¼ ò Heights, test
scores
Binomial Discrete np np( 2p) Pass/fail trials
Poisson Discrete » » Events per time
unit
Uniform Continuous (a+b)/ (b2a)²/ Equal likelihood
Exponential Continuous /» /»² Time between
events
t-Distribution Continuous df/(df2 ) Small samples
, Worked Examples, Real-Life Applications &
High-Yield Methods
SECTIONS 2
û Section 4: Worked Examples 4 Step by Step
' High-Yield Method: Always write out every step. Partial credit is awarded in most exams for correct
methodology even if the final answer is wrong.
Example 1: Mean, Variance & Standard Example 2: Z-Score & Normal Distribution
Deviation
IQ scores: ¿ = ,Ã= . Find P(X < ).
Dataset: { , , , , }
. Z-score: z = ( 2 )/ = .
. Mean: x = ( + + + + )/ = / = . . Look up z = . in the standard normal table
. Deviations: ( 2 . )²= . , ( 2 . )²= . , . P(Z < . )= .
( 2 . )²= . , ( 2 . )²= . , ( 2 . )²= .
. Interpretation: . % of people score below
. Variance: s² = ( . + . + . + . + . )/( 2 ) =
. / = .
Mnemonic: "Z is the distance in standard
. Std Dev: s = : . j .
í
deviations 4 always standardize first!"
Example 3: Binomial Distribution Example 4: Hypothesis Testing (One-
Sample t-Test)
A fair coin is flipped times. Find P(exactly heads).
A sample of students has x = ,s= . Test H : ¿ =
. n= ,k= ,p= .
at ³ = . .
. C( , )=
. t=( 2 )/( /: )= / = .
. P= × ( . )v × ( . )t = ×( / )j .
. df = ; critical t = . (two-tailed)
. | . |< . ³ Fail to Reject H
. Conclusion: No significant difference from ¿ =
State Choose Test Calculate Make
Hypotheses &³ Statistic Decision
Every hypothesis test follows this exact four-step framework. Mastering this sequence means you can tackle any test 4 t-
test, z-test, chi-square, or ANOVA 4 with confidence and consistency.
n Section 5: Real-Life Applications
p Business & Finance í Psychology & Social # Medicine & Public Health
Science
Quality Control: Use control charts Diagnostic Tests: Sensitivity and
(mean ± Ã) to detect defective Survey Analysis: Likert scale data specificity use conditional
products on assembly lines analyzed with chi-square tests for probability (Bayes' Theorem)
A/B Testing: Companies like group differences Epidemiology: Relative risk and
Amazon run thousands of Clinical Trials: t-tests compare odds ratios quantify disease
hypothesis tests daily to optimize treatment vs. control group associations
conversion rates outcomes Drug Approval: FDA requires p <
Risk Assessment: Banks use Effect Size: Cohen's d measures . in randomized controlled trials
normal distributions to model loan practical significance beyond p- Survival Analysis: Kaplan-Meier
default probabilities values curves model time-to-event data in
Forecasting: Regression analysis Reliability: Cronbach's alpha oncology
predicts future sales from historical measures internal consistency of
data psychological scales
§ Section 6: High-Yield Methods 4 Deep Dive
1 2 3
Mean 4 The Center of Variance 4 Spread Standard Deviation 4
Gravity Quantified Interpretable Spread
The mean is the balance point of a Variance measures average The square root of variance, in the
distribution. Sensitive to outliers squared distance from the mean. same units as the data. The most
4 one extreme value can pull it far Always non-negative. Larger intuitive measure of spread. In a
from the "typical" value. Always variance = more spread. Use normal distribution, ~ % of data
report alongside standard sample variance (n2 ) to get an falls within standard deviation of
deviation for full context. unbiased estimate of population the mean.
variance.
4 5
Normal Distribution 4 The Bell Curve Hypothesis Testing 4 Decision
Framework
Symmetric, bell-shaped, defined entirely by ¿ and Ã.
The Central Limit Theorem guarantees that sample A formal procedure to test claims about populations.
means approach normality as n increases 4 the Key concepts: Type I Error (false positive, ³), Type II
foundation of inferential statistics. Error (false negative, ´), Power ( 2´). Always state H
and H¡ before collecting data.
