Statistics Basics Cheat Sheet — Mean, Std, p-value, CI, z-score | Dataplexa

Statistics Basics

mean · median · std · variance · z-score · p-value · confidence intervals · hypothesis testing

Sheet 1 of 6 Stats & Math Beginner Printable

Descriptive Statistics

must know first

Mean (Average)

x̄ = Σ xᵢ n

Sum of all values ÷ count of values. Sensitive to outliers.

Median

Middle value when sorted.
Even n: avg of two middle values

Not affected by outliers — use for skewed data.

Mode

Most frequently occurring value.
Can be multi-modal or none.

Only measure that works for categorical data.

Variance

s² = Σ(xᵢ − x̄)² n − 1

Average squared deviation from the mean. n−1 = sample variance (Bessel's correction).

Standard Deviation

s = √s² = √ Σ(xᵢ−x̄)² n−1

Square root of variance. Same units as data. Most common spread measure.

Range & IQR

Range = max − min
IQR = Q₃ − Q₁

IQR = Interquartile Range. Middle 50% of data. Outlier: < Q1−1.5·IQR or > Q3+1.5·IQR.

Quick Example — Dataset

Data: [4, 7, 13, 2, 1, 7]

Mean    = (4+7+13+2+1+7)/6 = 5.67
Sorted  = [1,2,4,7,7,13]
Median  = (4+7)/2    = 5.5
Mode    = 7          (appears twice)

Variance  = 14.67  (sample)
Std Dev   = 3.83
Range     = 13 - 1   = 12
Q1        = 2
Q3        = 7
IQR       = 7 - 2   = 5

Population vs Sample: Use n for population variance (σ²), n−1 for sample variance (s²). Most real-world data is a sample — use n−1 to get an unbiased estimate.

Z-Score & Standardization

standard normal

Z-Score Formula

z = x − μ σ

x = value · μ = mean · σ = std deviation

Interpretation

z =  0     → exactly at the mean
z =  1     → 1 std dev above mean
z = -2     → 2 std devs below mean
z =  1.96  → top 2.5% (one tail)
z = ±1.96 → 95% of data lies here
z = ±2.58 → 99% of data lies here

% of data	Within ±σ
68.27 %	±1σ from mean
95.45 %	±2σ from mean
99.73 %	±3σ from mean (Empirical Rule)

Standardization use case: Z-scores let you compare values from different distributions. Required before many ML algorithms (k-NN, SVM, neural nets).

Key Distributions

normal · binomial · Poisson

Normal Distribution

X ~ N(μ, σ²)
Mean = μ, Std Dev = σ, Symmetric bell shape

Central Limit Theorem: sample means are approximately normal for large n (n ≥ 30).

Binomial Distribution

P(X=k) = C(n,k) · p^k · (1−p)^n−k
Mean = np, Var = np(1−p)

n trials, p = prob of success each trial. Count of successes in n trials.

Poisson Distribution

P(X=k) = λ^k · e^−λ k! Mean = Var = λ

λ = expected events per interval. Use for rare events (call centre, website hits).

Hypothesis Testing

H₀ · H₁ · p-value · Type I & II errors

Null & Alternative Hypothesis

H₀: no effect / no difference
H₁: effect exists / difference

We test whether data gives enough evidence to reject H₀. We never "prove" H₀.

p-value

p-value ≤ α → reject H₀
p-value > α → fail to reject H₀

α (significance level) = 0.05 (5%) is the most common threshold. The p-value is the probability of seeing data this extreme if H₀ is true.

Test Statistic — One-Sample t-test

t = x̄ − μ₀ s √n

x̄ = sample mean, μ₀ = hypothesized mean, s = sample std dev, n = sample size.

Error Type	What happened
Type I (α)	Reject H₀ when it was true (false positive)
Type II (β)	Fail to reject H₀ when H₁ was true (false negative)
Power (1−β)	Probability of correctly rejecting a false H₀

Choosing the Right Test

1 group vs known value:
  → one-sample t-test

2 independent groups:
  → independent t-test

Same group, before/after:
  → paired t-test

3+ groups:
  → ANOVA

2 categorical variables:
  → Chi-square test

Relationship (numeric vs numeric):
  → Pearson correlation
  → Linear regression

p-value ≠ probability H₀ is true. It's the probability of getting data this extreme assuming H₀ is true. Statistical significance doesn't always mean practical importance — always check effect size too.

Confidence Intervals

CI · margin of error

95% Confidence Interval for Mean

CI = x̄ ± z_α/2 · σ √n

z_α/2 = 1.96 for 95% CI · 2.576 for 99% CI · 1.645 for 90% CI

t-based CI (unknown σ, small n)

CI = x̄ ± t_{α/2, df} · s √n

df = degrees of freedom = n − 1. Use t-distribution when n < 30 or σ unknown.

Worked Example

Sample: n=25, x̄=50, s=10, 95% CI
t* = 2.064  (df=24, two-tailed)
SE = 10 / √25 = 2.0
CI = 50 ± 2.064 × 2.0
   = 50 ± 4.13
   = [45.87, 54.13]

Interpretation: We are 95% confident
the true mean lies in [45.87, 54.13]

Correlation & Covariance

Pearson · Spearman · r

Pearson Correlation Coefficient

r = Cov(X,Y) σ_X · σ_Y

Range: −1 to +1. r = +1 perfect positive · r = 0 no linear · r = −1 perfect negative.

\|r\| value	Strength
0.00 – 0.19	Very weak
0.20 – 0.39	Weak
0.40 – 0.59	Moderate
0.60 – 0.79	Strong
0.80 – 1.00	Very strong

Correlation ≠ Causation. Two variables can be strongly correlated without one causing the other — always check for confounding variables.

Sampling & Central Limit Theorem

CLT · sampling error · SEM

Standard Error of the Mean (SEM)

SE = σ √n

How much sample means vary around the population mean. Decreases as n grows — collect more data for more precise estimates.

Central Limit Theorem (CLT)

For large n (≥ 30):
X̄ ~ N(μ, σ²/n)

Sample means follow a normal distribution regardless of the population's shape — if sample size is large enough. Foundation of most inferential statistics.

Sampling Type	How
Simple Random	Every unit has equal probability
Stratified	Divide into strata, sample each
Cluster	Randomly select groups, sample all
Systematic	Every k-th unit in list
Convenience	Easiest to reach (biased!)

Effect Size & Statistical Power

Cohen's d · power · sample size

Cohen's d (Effect Size)

d = x̄₁ − x̄₂ s_pooled

d = 0.2 small · d = 0.5 medium · d = 0.8 large (Cohen's conventions).

Concept	Meaning
α (alpha)	Significance level — prob of Type I error. Usually 0.05.
β (beta)	Prob of Type II error. Usually set to 0.20.
Power (1−β)	Prob of detecting a true effect. Usually ≥ 0.80.
Sample size ↑	Reduces SE, increases power, narrows CI.

Symbols Quick Reference

notation guide

Symbol	Meaning	Pop / Sample
μ (mu)	Mean	Population
x̄ (x-bar)	Mean	Sample
σ (sigma)	Standard deviation	Population
s	Standard deviation	Sample
σ²	Variance	Population
s²	Variance	Sample
N / n	Size	Pop / Sample
α (alpha)	Significance level	—
H₀ / H₁	Null / Alternative hypothesis	—
r	Pearson correlation	Sample
ρ (rho)	Pearson correlation	Population
SE	Standard Error of Mean	—

Statistics Basics Mastery Checklist

sheet 1 complete

Descriptive Stats	Key point
Calculate mean/median/mode	Σxᵢ/n
Compute sample variance	Σ(xᵢ−x̄)² / n−1
Interpret std deviation	avg distance from mean
Detect outliers with IQR	Q1−1.5·IQR rule

Inference	Key point
Interpret z-score	(x − μ) / σ
State H₀ and H₁	null vs alternative
Make p-value decision	p ≤ α → reject H₀
Build a 95% CI	x̄ ± 1.96 · SE

Correlation & Power	Key point
Pearson r range	−1 to +1
Correlation ≠ causation	check confounders
Effect size (Cohen's d)	0.2 / 0.5 / 0.8
Desired power	1−β ≥ 0.80

Next up → Sheet 2: Probability · Bayes' theorem · distributions · expected value · conditional probability · Bayesian vs frequentist

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