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Calculated Shard: 78 (from laksa174)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa078.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://blog.ryanliu.io/AP-Statistics/Chapter-7\')), getAhrefsUnparsedNoserviceFromURL(\'https://blog.ryanliu.io/AP-Statistics/Chapter-7\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-7","crawl_time":1775843063,"first_indexed_time":1727731403,"http_code":200,"src_unparsed":"io,ryanliu!blog,\/AP-Statistics\/Chapter-7 s443","src_root_hash":"1681979500600740878","history_drop_reason":null,"meta_title":"Chapter 7 – The Central Limit Theorem","meta_descriptions":["The central limit theorem (CLT) states that the sample mean converges to a standard normal distribution given a large enough sample. The size of the sample n that is large enough depends on the size of the original population."],"attrs_boilerpipe_text":"The\ncentral limit theorem (CLT)\nstates that the sample mean converges to a standard normal distribution given a large enough sample.\nThe size of the sample\nthat is large enough depends on the size of the original population. Sampling must be done\nwith replacement\n.\nThe Central Limit Theorem for Sample Means\nSuppose\nis a random variable with\nany\ndistribution. Taking a sufficiently large (\n) number of samples\n, we find that the distribution of sample means\nof all samples tends to be\nnormally distributed\n.\nIf\nthen\nand\n(\nthe law of large numbers\n)\nthen\n(notice that\nis in the denominator): bigger samples vary less\n: the mean of the sample means is the population mean\n: the stdev of the sample means is the population stdev divided by\nwhere\nby the law of large numbers\nRecall that\nby the law of large numbers","attrs_markdown":"# [😼 ryanliu.io](https:\/\/blog.ryanliu.io\/)\nSearch\n\nSearch\n\n# Explorer\n- - AP Macroeconomics\n    - [Chapter 1 – Introduction](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-1)\n    - [Chapter 2 – Scarcity](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-2)\n    - [Chapter 3 – Supply, Demand, and Equilibrium](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-3)\n    - [Chapter 4 – Labour and Financial Markets](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-4)\n    - [Chapter 5 – Elasticity](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-5)\n    - [Chapter 6 – Macroeconomics](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-6)\n    - [Chapter 7 – Economic Growth](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-7)\n    - [Chapter 8 – Unemployment](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-8)\n    - [Chapter 10 – International Trade and Capital Flows](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-10)\n    - [Chapter 11 – Aggregate Demand and Aggregate Supply](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-11)\n    - [Chapter 12 – Keynesian Economics](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-12)\n    - [Chapter 13 – Neoclassical Economics](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-13)\n    - [Chapter 14 – Money and Banking](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-14)\n    - [Chapter 15 – Monetary Policy](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-15)\n    - [Chapter 16 – Exchange Rates and International Capital Flows](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-16)\n    - [Chapter 17 – Government Budgets and Fiscal Policy](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-17)\n    - [Chapter 18 – Government Borrowing](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-18)\n    - [Inflation](https:\/\/blog.ryanliu.io\/AP-Macroeconomics\/Chapter-9)\n  - AP Psychology\n    - [Chapter 2 – Neuroscience and Biological Foundations](https:\/\/blog.ryanliu.io\/AP-Psychology\/Chapter-2)\n    - [Chapter 5](https:\/\/blog.ryanliu.io\/AP-Psychology\/Chapter-5)\n    - [Chapter 5 Test](https:\/\/blog.ryanliu.io\/AP-Psychology\/Chapter-5-Test)\n    - [Chapter 6 – Learning](https:\/\/blog.ryanliu.io\/AP-Psychology\/Chapter-6)\n    - [Chapter 8 – Thinking, Language, and Intelligence](https:\/\/blog.ryanliu.io\/AP-Psychology\/Chapter-8)\n    - [Chapter 11 – Sex and Gender](https:\/\/blog.ryanliu.io\/AP-Psychology\/Chapter-11)\n  - AP Statistics\n    - [Chapter 1 – Sampling and Data](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-1)\n    - [Chapter 2 – Descriptive Statistics](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-2)\n    - [Chapter 3 – Probability](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-3)\n    - [Chapter 4 – Discrete Random Variables](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-4)\n    - [Chapter 5 – Continuous Random Variables](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-5)\n    - [Chapter 6 – The Normal Distribution](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-6)\n    - [Chapter 7 – The Central Limit Theorem](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-7)\n    - [Chapter 12 – Linear Regression and Correlation](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-12)\n    - [INFERENCE](https:\/\/blog.ryanliu.io\/AP-Statistics\/INFERENCE)\n    - [MCQ Review](https:\/\/blog.ryanliu.io\/AP-Statistics\/MCQ-Review)\n    - [Unit 1 – Exploring One-Variable Data](https:\/\/blog.ryanliu.io\/AP-Statistics\/Unit-1)\n    - [Unit 6 – Inference for Categorical Data: Proportions](https:\/\/blog.ryanliu.io\/AP-Statistics\/Unit-6)\n  - Blog\n    - [APSC 160 Midterm 2](https:\/\/blog.ryanliu.io\/Blog\/APSC-160-Midterm-2)\n    - [Demonstrating the Monty Hall Problem in Python](https:\/\/blog.ryanliu.io\/Blog\/Demonstrating-the-Monty-Hall-Problem-in-Python)\n    - [MT2 Practice Question](https:\/\/blog.ryanliu.io\/Blog\/MT2-Practice-Question)\n    - [Remove Element](https:\/\/blog.ryanliu.io\/Blog\/2024-06-09)\n    - [Thoughts About the Future of Tech and Industry](https:\/\/blog.ryanliu.io\/Blog\/2024-02-04)\n  - Calculus 12\n    - Chapter 1 – Prerequisites for Calculus\n      - [1\\.1 – Lines](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-1\/1.1)\n      - [1\\.4 – Parametric Equations](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-1\/1.4)\n    - Chapter 2 – Limits and Continuity\n      - [2\\.1 – Rates of Change and Limits](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-2\/2.1)\n      - [2\\.2 – Limits Involving Infinity](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-2\/2.2)\n      - [2\\.3 – Continuity](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-2\/2.3)\n      - [2\\.4 – Rates of Change and Tangent Lines](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-2\/2.4)\n      - [Chapter 2 Review](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-2\/Review)\n    - Chapter 3 – Derivatives\n      - [3\\.1 – Derivative of a Function](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.1)\n      - [3\\.2 – Differentiability](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.2)\n      - [3\\.3 – Rules for Differentiation](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.3)\n      - [3\\.4 – Rates of Change](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.4)\n      - [3\\.5 – Trigonometric Derivatives](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.5)\n      - [3\\.6 – Chain Rule](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.6)\n      - [3\\.7 – Implicit Differentiation](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/3.7)\n      - [Chapter 3 – Test 1](https:\/\/blog.ryanliu.io\/Calculus-12\/Chapter-3\/Test-1)\n    - [Related Rates](https:\/\/blog.ryanliu.io\/Calculus-12\/Related-Rates)\n    - [Riemann Sums](https:\/\/blog.ryanliu.io\/Calculus-12\/Riemann-Sums)\n    - [Substitution Techniques for Integration](https:\/\/blog.ryanliu.io\/Calculus-12\/Substitution-Techniques-for-Integration)\n    - [The Chain Rule](https:\/\/blog.ryanliu.io\/Calculus-12\/The-Chain-Rule)\n    - [Volumes of Rotation](https:\/\/blog.ryanliu.io\/Calculus-12\/Volumes-of-Rotation)\n\n[Home](https:\/\/blog.ryanliu.io\/)\n\n❯\n\n[AP Statistics](https:\/\/blog.ryanliu.io\/AP-Statistics\/)\n\n❯\n\n[Chapter 7 – The Central Limit Theorem](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-7)\n# Chapter 7 – The Central Limit Theorem\nOct 30, 20241 min read\n\nThe **central limit theorem (CLT)** states that the sample mean converges to a standard normal distribution given a large enough sample.\n\nThe size of the sample n that is large enough depends on the size of the original population. Sampling must be done [with replacement](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-1#863003).\n\n## The Central Limit Theorem for Sample Means\nSuppose X is a random variable with *any* distribution. Taking a sufficiently large (n≥30) number of samples n, we find that the distribution of sample means xˉ of all samples tends to be [normally distributed](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-6).\n\n- If n↑ then xˉ→μ and μxˉ→μ ([the law of large numbers](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-3#37f29d))\n  - n↑ then σ↓ (notice that n is in the denominator): bigger samples vary less\n- μxˉ\\=μ: the mean of the sample means is the population mean\n- σxˉ\\= n  σ : the stdev of the sample means is the population stdev divided by n \n  - σxˉ2\\=nσ2\n- z\\=σxˉxˉ−μxˉ where σxˉ\\= n  σ \n  - z\\=\n    \n    σ\/\n    \n    n\n    \n    \n    \n    xˉ−μ\n    \n    \n    by the law of large numbers\n- xˉ∼ N(μ, n  σ  )\n  - Recall that μ\\=μxˉ by the law of large numbers\n\n### Graph View\n### Backlinks\n- [Chapter 1 – Sampling and Data](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-1)\n- [Unit 6 – Inference for Categorical Data: Proportions](https:\/\/blog.ryanliu.io\/AP-Statistics\/Unit-6)\n- [AP Statistics](https:\/\/blog.ryanliu.io\/AP-Statistics\/)\n\n***\nCreated by Ryan Liu with [Quartz](https:\/\/quartz.jzhao.xyz\/) © 2024\n\n- [Instagram](https:\/\/instagram.com\/rryanlliu)","attrs_readable_markdown":"The **central limit theorem (CLT)** states that the sample mean converges to a standard normal distribution given a large enough sample.\n\nThe size of the sample  that is large enough depends on the size of the original population. Sampling must be done [with replacement](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-1#863003).\n\n## The Central Limit Theorem for Sample Means\nSuppose  is a random variable with *any* distribution. Taking a sufficiently large () number of samples , we find that the distribution of sample means  of all samples tends to be [normally distributed](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-6).\n\n- If  then  and  ([the law of large numbers](https:\/\/blog.ryanliu.io\/AP-Statistics\/Chapter-3#37f29d))\n  - then  (notice that  is in the denominator): bigger samples vary less\n- : the mean of the sample means is the population mean\n- : the stdev of the sample means is the population stdev divided by\n- where\n  - by the law of large numbers\n- - Recall that  by the law of large numbers","meta_canonical":null}

3. Robots.txt Check

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4. Spam/Ban Check

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Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

18 hours ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://blog.ryanliu.io/AP-Statistics/Chapter-7
Last Crawled	2026-04-10 17:44:23 (18 hours ago)
First Indexed	2024-09-30 21:23:23 (1 year ago)
HTTP Status Code	200
Meta Title	Chapter 7 – The Central Limit Theorem
Meta Description	The central limit theorem (CLT) states that the sample mean converges to a standard normal distribution given a large enough sample. The size of the sample n that is large enough depends on the size of the original population.
Meta Canonical	null
Boilerpipe Text	The central limit theorem (CLT) states that the sample mean converges to a standard normal distribution given a large enough sample. The size of the sample that is large enough depends on the size of the original population. Sampling must be done with replacement . The Central Limit Theorem for Sample Means Suppose is a random variable with any distribution. Taking a sufficiently large ( ) number of samples , we find that the distribution of sample means of all samples tends to be normally distributed . If then and ( the law of large numbers ) then (notice that is in the denominator): bigger samples vary less : the mean of the sample means is the population mean : the stdev of the sample means is the population stdev divided by where by the law of large numbers Recall that by the law of large numbers
Markdown	# [😼 ryanliu.io](https://blog.ryanliu.io/) Search Search # Explorer - - AP Macroeconomics - [Chapter 1 – Introduction](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-1) - [Chapter 2 – Scarcity](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-2) - [Chapter 3 – Supply, Demand, and Equilibrium](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-3) - [Chapter 4 – Labour and Financial Markets](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-4) - [Chapter 5 – Elasticity](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-5) - [Chapter 6 – Macroeconomics](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-6) - [Chapter 7 – Economic Growth](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-7) - [Chapter 8 – Unemployment](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-8) - [Chapter 10 – International Trade and Capital Flows](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-10) - [Chapter 11 – Aggregate Demand and Aggregate Supply](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-11) - [Chapter 12 – Keynesian Economics](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-12) - [Chapter 13 – Neoclassical Economics](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-13) - [Chapter 14 – Money and Banking](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-14) - [Chapter 15 – Monetary Policy](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-15) - [Chapter 16 – Exchange Rates and International Capital Flows](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-16) - [Chapter 17 – Government Budgets and Fiscal Policy](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-17) - [Chapter 18 – Government Borrowing](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-18) - [Inflation](https://blog.ryanliu.io/AP-Macroeconomics/Chapter-9) - AP Psychology - [Chapter 2 – Neuroscience and Biological Foundations](https://blog.ryanliu.io/AP-Psychology/Chapter-2) - [Chapter 5](https://blog.ryanliu.io/AP-Psychology/Chapter-5) - [Chapter 5 Test](https://blog.ryanliu.io/AP-Psychology/Chapter-5-Test) - [Chapter 6 – Learning](https://blog.ryanliu.io/AP-Psychology/Chapter-6) - [Chapter 8 – Thinking, Language, and Intelligence](https://blog.ryanliu.io/AP-Psychology/Chapter-8) - [Chapter 11 – Sex and Gender](https://blog.ryanliu.io/AP-Psychology/Chapter-11) - AP Statistics - [Chapter 1 – Sampling and Data](https://blog.ryanliu.io/AP-Statistics/Chapter-1) - [Chapter 2 – Descriptive Statistics](https://blog.ryanliu.io/AP-Statistics/Chapter-2) - [Chapter 3 – Probability](https://blog.ryanliu.io/AP-Statistics/Chapter-3) - [Chapter 4 – Discrete Random Variables](https://blog.ryanliu.io/AP-Statistics/Chapter-4) - [Chapter 5 – Continuous Random Variables](https://blog.ryanliu.io/AP-Statistics/Chapter-5) - [Chapter 6 – The Normal Distribution](https://blog.ryanliu.io/AP-Statistics/Chapter-6) - [Chapter 7 – The Central Limit Theorem](https://blog.ryanliu.io/AP-Statistics/Chapter-7) - [Chapter 12 – Linear Regression and Correlation](https://blog.ryanliu.io/AP-Statistics/Chapter-12) - [INFERENCE](https://blog.ryanliu.io/AP-Statistics/INFERENCE) - [MCQ Review](https://blog.ryanliu.io/AP-Statistics/MCQ-Review) - [Unit 1 – Exploring One-Variable Data](https://blog.ryanliu.io/AP-Statistics/Unit-1) - [Unit 6 – Inference for Categorical Data: Proportions](https://blog.ryanliu.io/AP-Statistics/Unit-6) - Blog - [APSC 160 Midterm 2](https://blog.ryanliu.io/Blog/APSC-160-Midterm-2) - [Demonstrating the Monty Hall Problem in Python](https://blog.ryanliu.io/Blog/Demonstrating-the-Monty-Hall-Problem-in-Python) - [MT2 Practice Question](https://blog.ryanliu.io/Blog/MT2-Practice-Question) - [Remove Element](https://blog.ryanliu.io/Blog/2024-06-09) - [Thoughts About the Future of Tech and Industry](https://blog.ryanliu.io/Blog/2024-02-04) - Calculus 12 - Chapter 1 – Prerequisites for Calculus - [1\.1 – Lines](https://blog.ryanliu.io/Calculus-12/Chapter-1/1.1) - [1\.4 – Parametric Equations](https://blog.ryanliu.io/Calculus-12/Chapter-1/1.4) - Chapter 2 – Limits and Continuity - [2\.1 – Rates of Change and Limits](https://blog.ryanliu.io/Calculus-12/Chapter-2/2.1) - [2\.2 – Limits Involving Infinity](https://blog.ryanliu.io/Calculus-12/Chapter-2/2.2) - [2\.3 – Continuity](https://blog.ryanliu.io/Calculus-12/Chapter-2/2.3) - [2\.4 – Rates of Change and Tangent Lines](https://blog.ryanliu.io/Calculus-12/Chapter-2/2.4) - [Chapter 2 Review](https://blog.ryanliu.io/Calculus-12/Chapter-2/Review) - Chapter 3 – Derivatives - [3\.1 – Derivative of a Function](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.1) - [3\.2 – Differentiability](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.2) - [3\.3 – Rules for Differentiation](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.3) - [3\.4 – Rates of Change](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.4) - [3\.5 – Trigonometric Derivatives](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.5) - [3\.6 – Chain Rule](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.6) - [3\.7 – Implicit Differentiation](https://blog.ryanliu.io/Calculus-12/Chapter-3/3.7) - [Chapter 3 – Test 1](https://blog.ryanliu.io/Calculus-12/Chapter-3/Test-1) - [Related Rates](https://blog.ryanliu.io/Calculus-12/Related-Rates) - [Riemann Sums](https://blog.ryanliu.io/Calculus-12/Riemann-Sums) - [Substitution Techniques for Integration](https://blog.ryanliu.io/Calculus-12/Substitution-Techniques-for-Integration) - [The Chain Rule](https://blog.ryanliu.io/Calculus-12/The-Chain-Rule) - [Volumes of Rotation](https://blog.ryanliu.io/Calculus-12/Volumes-of-Rotation) [Home](https://blog.ryanliu.io/) ❯ [AP Statistics](https://blog.ryanliu.io/AP-Statistics/) ❯ [Chapter 7 – The Central Limit Theorem](https://blog.ryanliu.io/AP-Statistics/Chapter-7) # Chapter 7 – The Central Limit Theorem Oct 30, 20241 min read The central limit theorem (CLT) states that the sample mean converges to a standard normal distribution given a large enough sample. The size of the sample n that is large enough depends on the size of the original population. Sampling must be done [with replacement](https://blog.ryanliu.io/AP-Statistics/Chapter-1#863003). ## The Central Limit Theorem for Sample Means Suppose X is a random variable with any distribution. Taking a sufficiently large (n≥30) number of samples n, we find that the distribution of sample means xˉ of all samples tends to be [normally distributed](https://blog.ryanliu.io/AP-Statistics/Chapter-6). - If n↑ then xˉ→μ and μxˉ→μ ([the law of large numbers](https://blog.ryanliu.io/AP-Statistics/Chapter-3#37f29d)) - n↑ then σ↓ (notice that n is in the denominator): bigger samples vary less - μxˉ\=μ: the mean of the sample means is the population mean - σxˉ\= n σ : the stdev of the sample means is the population stdev divided by n - σxˉ2\=nσ2 - z\=σxˉxˉ−μxˉ where σxˉ\= n σ - z\= σ/ n xˉ−μ by the law of large numbers - xˉ∼ N(μ, n σ ) - Recall that μ\=μxˉ by the law of large numbers ### Graph View ### Backlinks - [Chapter 1 – Sampling and Data](https://blog.ryanliu.io/AP-Statistics/Chapter-1) - [Unit 6 – Inference for Categorical Data: Proportions](https://blog.ryanliu.io/AP-Statistics/Unit-6) - [AP Statistics](https://blog.ryanliu.io/AP-Statistics/) *** Created by Ryan Liu with [Quartz](https://quartz.jzhao.xyz/) © 2024 - [Instagram](https://instagram.com/rryanlliu)
Readable Markdown	The central limit theorem (CLT) states that the sample mean converges to a standard normal distribution given a large enough sample. The size of the sample that is large enough depends on the size of the original population. Sampling must be done [with replacement](https://blog.ryanliu.io/AP-Statistics/Chapter-1#863003). ## The Central Limit Theorem for Sample Means Suppose is a random variable with any distribution. Taking a sufficiently large () number of samples , we find that the distribution of sample means of all samples tends to be [normally distributed](https://blog.ryanliu.io/AP-Statistics/Chapter-6). - If then and ([the law of large numbers](https://blog.ryanliu.io/AP-Statistics/Chapter-3#37f29d)) - then (notice that is in the denominator): bigger samples vary less - : the mean of the sample means is the population mean - : the stdev of the sample means is the population stdev divided by - where - by the law of large numbers - - Recall that by the law of large numbers
Shard	78 (laksa)
Root Hash	1681979500600740878
Unparsed URL	io,ryanliu!blog,/AP-Statistics/Chapter-7 s443