Statistics

A2.L.1Normal & z-scores01/04

Normal distribution and z-scores.

Step01of 04

The normal distribution is the bell-shaped curve that describes many naturally occurring measurements (heights, test scores, measurement errors). It's symmetric around the mean $μ$ with spread determined by the standard deviation $σ$ .

Empirical rule (68-95-99.7).

Within ±1 SD of mean	→about 68% of data
Within ±2 SD of mean	→about 95%
Within ±3 SD of mean	→about 99.7%

A2.L.1 — Normal distribution and z-scores.

Step 01 of 04

Empirical rule (68-95-99.7).

Within ±1 SD of mean	→about 68% of data
Within ±2 SD of mean	→about 95%
Within ±3 SD of mean	→about 99.7%

Step 02 of 04

Z-score. Tells you how many standard deviations a value lies above or below the mean.

z = \frac{x - μ}{σ}

Positive z = above mean, negative = below. $z = 0$ = exactly at the mean.

Step 03 of 04

Worked example. Test scores are normal with mean 75 and SD 10. A student scores 92.

z = \frac{92 - 75}{10} = 1.7

The student is 1.7 SDs above the mean. Roughly 4-5% of students score this high or higher.

Step 04 of 04

Z-scores let you compare across DIFFERENT distributions. Student A scored 80 on a test with mean 70, SD 5 → $z = 2$ . Student B scored 85 on a test with mean 75, SD 10 → $z = 1$ . Student A did better RELATIVE to peers, even though Student B's raw score is higher.

Key insight

z-score = "how many SDs away from the mean." Lets you compare values across different distributions. Empirical rule (68-95-99.7) gives quick probability estimates without a table.

A2.L.2 — Sample vs. population, sampling methods.

Step 01 of 04

Population = the entire group you want to know about. Sample = the subset you actually measure. Statistics is the art of inferring population properties from sample properties.

The quality of a sample depends on HOW it was collected. Different methods have different bias risks.

Step 02 of 04

Common sampling methods.

Simple random	→every member equally likely — gold standard
Stratified	→split into groups (strata), random sample from each
Systematic	→pick every nth member — easy if list is in random order
Cluster	→randomly pick whole clusters (e.g., entire classes), survey everyone in those
Convenience	→whoever's easiest to reach — typically biased
Voluntary response	→subjects opt in — biased toward strong opinions

Step 03 of 04

Bias and how to spot it.

Selection bias	→sampling method excludes some part of population
Response bias	→wording / setting changes how subjects answer
Nonresponse bias	→subjects who refuse differ systematically from those who respond
Undercoverage	→parts of the population have no chance of being sampled

A perfect statistical analysis on a biased sample is still useless. Sample quality is upstream of every other consideration.

Step 04 of 04

Sample statistics vs. population parameters.

Population mean	→ $μ$ (Greek)
Sample mean	→ $\overset{x}{ˉ}$ (English with bar)
Population SD	→ $σ$
Sample SD	→ $s$
Population proportion	→ $p$
Sample proportion	→ $\overset{p}{^}$

Greek letters = population (usually unknown). English letters = sample (computed from data). The whole point of inferential stats is using the English to estimate the Greek.

Key insight

Sample = subset, population = full group. Method matters MORE than sample size — a small random sample beats a huge convenience sample. Greek letters describe populations; English letters describe samples.

A2.L.3 — Margin of error.

Step 01 of 04

When a sample's $\overset{x}{ˉ}$ or $\overset{p}{^}$ estimates a population $μ$ or $p$ , the estimate has UNCERTAINTY. The margin of error (MOE) reports that uncertainty as a +/- band at a chosen confidence level.

"55% support, ±3% margin at 95% confidence" means: we're 95% confident the true population proportion is between 52% and 58%.

Step 02 of 04

Margin for a proportion (95% confidence).

MOE \approx 1.96 \cdot \frac{p ^ ( 1 - p ^ )}{n}

Quick rule of thumb that's good enough for many estimates: $MOE \approx 1/ n$ (assumes $\overset{p}{^} \approx 0.5$ ).

Step 03 of 04

Worked example. Survey of 1,000 voters: 540 say they'll vote "yes" on a referendum. Estimate the margin.

$\overset{p}{^} = 0.54$ , $n = 1000$ .

MOE = 1.96 \frac{0.54 \cdot 0.46}{1000} \approx 1.96 \cdot 0.0158 \approx 0.031

About ±3.1%. Reported result: "54% yes, ±3% at 95% confidence."

Step 04 of 04

How sample size affects margin. Margin shrinks like $1/ n$ . To HALVE the margin, you need 4× the sample. To divide by 10, 100× the sample.

$n = 100$	→~10% margin
$n = 400$	→~5%
$n = 1, 000$	→~3%
$n = 10, 000$	→~1%

Diminishing returns set in fast — going from 1,000 to 10,000 only cuts margin from 3% to 1%. Most polls cap at ~1,000-2,000.

Key insight

MOE quantifies sampling uncertainty. Roughly

1/ n

at 95% confidence. Halving the margin = 4× the sample. The sample method (skill 2) determines whether MOE even applies — it assumes random sampling.

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A2.L.1 – A2.L.3 · CCSS S.ID.A.4 / S.IC.A.1 / S.IC.B.4·Full curriculum →