# Connexions

You are here: Home » Content » Elementary Statistics » The Central Limit Theorem for Sums

• Student Welcome Letter

• 13. Tables

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Endorsed by (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
• College Open Textbooks

This module is included inLens: Community College Open Textbook Collaborative
By: CC Open Textbook CollaborativeAs a part of collection: "Collaborative Statistics"

"Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"

Click the "College Open Textbooks" link to see all content they endorse.

Click the tag icon to display tags associated with this content.

• JVLA Endorsed

This module is included inLens: Jesuit Virtual Learning Academy Endorsed Material
By: Jesuit Virtual Learning AcademyAs a part of collection: "Collaborative Statistics"

"This is a robust collection (textbook) approved by the College Board as a resource for the teaching of AP Statistics. "

Click the "JVLA Endorsed" link to see all content they endorse.

• WebAssign

This module is included inLens: WebAssign The Independent Online Homework and Assessment Solution
By: WebAssignAs a part of collection: "Collaborative Statistics"

"Online homework and assessment available from WebAssign."

Click the "WebAssign" link to see all content they endorse.

Click the tag icon to display tags associated with this content.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• OrangeGrove

This module is included inLens: Florida Orange Grove Textbooks
By: Florida Orange GroveAs a part of collection: "Collaborative Statistics"

Click the "OrangeGrove" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

• Bookshare

This module is included inLens: Bookshare's Lens
By: Bookshare - A Benetech InitiativeAs a part of collection: "Collaborative Statistics"

"DAISY and BRF versions of this collection are available."

Click the "Bookshare" link to see all content affiliated with them.

• Featured Content

This module is included inLens: Connexions Featured Content
By: ConnexionsAs a part of collection: "Collaborative Statistics"

"Collaborative Statistics was written by two faculty members at De Anza College in Cupertino, California. This book is intended for introductory statistics courses being taken by students at two- […]"

Click the "Featured Content" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

#### Also in these lenses

• statistics

This module is included inLens: Statistics
By: Brylie OxleyAs a part of collection: "Collaborative Statistics"

Click the "statistics" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

• Lucy Van Pelt

This module is included inLens: Lucy's Lens
By: Tahiya MaromeAs a part of collection: "Collaborative Statistics"

"Part of the Books featured on Community College Open Textbook Project"

Click the "Lucy Van Pelt" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

• Educational Technology Lens

This module is included inLens: Educational Technology
By: Steve WilhiteAs a part of collection: "Collaborative Statistics"

Click the "Educational Technology Lens" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

• Statistics

This module is included inLens: Mathieu Plourde's Lens
By: Mathieu PlourdeAs a part of collection: "Collaborative Statistics"

Click the "Statistics" link to see all content selected in this lens.

• statf12

This module is included inLens: Statistics Fall 2012
By: Alex KolesnikAs a part of collection: "Collaborative Statistics"

Click the "statf12" link to see all content selected in this lens.

• UTEP

This module is included inLens: Amy Wagler's Lens
By: Amy WaglerAs a part of collection: "Collaborative Statistics"

Click the "UTEP" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

• Make Textbooks Affordable

This module is included inLens: Make Textbooks Affordable
By: Nicole AllenAs a part of collection: "Collaborative Statistics"

Click the "Make Textbooks Affordable" link to see all content selected in this lens.

• BUS204 Homework

This module is included inLens: Saylor BUS 204 Homework
By: David BourgeoisAs a part of collection: "Collaborative Statistics"

"Homework for Discrete Variables/Probability. "

Click the "BUS204 Homework" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

• crowe

This module is included in aLens by: Chris RoweAs a part of collection: "Collaborative Statistics"

Click the "crowe" link to see all content selected in this lens.

• Bio 502 at CSUDH

This module is included inLens: Bio 502
By: Terrence McGlynnAs a part of collection: "Collaborative Statistics"

"This is the course textbook for Biology 502 at CSU Dominguez Hills"

Click the "Bio 502 at CSUDH" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Textbook):

Textbook by: Kathy Chu, Ph.D.. E-mail the author

# The Central Limit Theorem for Sums

Suppose XX is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:

• a. μ X = μ X = the mean of XX
• b. σ X = σ X = the standard deviation of XX
If you draw random samples of size nn, then as nn increases, the random variable ΣXΣX which consists of sums tends to be normally distributed and

Σ XΣX ~ N ( n μ X , n σ X ) N(n μ X , n σ X )

The Central Limit Theorem for Sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution) which approaches a normal distribution as the sample size increases. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.

The random variable Σ XΣX has the following z-score associated with it:

• a. ΣxΣx is one sum.
• b. z = Σ x - n μ X n σ X z= Σ x - n μ X n σ X
• a. n μ X = n μ X = the mean of ΣXΣX
• b. n σ X = n σ X = standard deviation of ΣXΣX

## Example 1

An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population.

### Problem 1

• a. Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7500.
• b. Find the sum that is 1.5 standard deviations above the mean of the sums.

#### Solution

Let XX = one value from the original unknown population. The probability question asks you to find a probability for the sum (or total of) 80 values.

ΣXΣX = the sum or total of 80 values. Since μ X = 90 μ X =90, σ X = 15 σ X =15, and n = 80 n=80, then

Σ XΣX ~ N ( 80 90 , 80 15 ) N(8090, 80 15)

• •. mean of the sums = n μ X = ( 80 ) ( 90 ) = 7200 n μ X =(80)(90)=7200
• •. standard deviation of the sums = n σ X = 80 15 n σ X = 80 15
• •. sum of 80 values = Σx = 7500 Σx=7500

• a: Find P ( Σx > 7500 ) P ( Σx 7500 )

P ( Σx > 7500 ) = 0.0127 P ( Σx 7500 )=0.0127

normalcdf(lower value, upper value, mean of sums, stdev of sums)

The parameter list is abbreviated (lower, upper, n μ X , n σ X n μ X , n σ X )

normalcdf(7500,1E99, 80 90 , 80 15 ) = 0.0127 8090, 80 15)=0.0127

Reminder: 1E99 = 10 99 1E99= 10 99 . Press the EE key for E.

• b: Find ΣxΣx where zz = 1.5:

ΣxΣx = n μ X n μ X + z n σ X z n σ X = (80)(90) + (1.5)( 80 80 ) (15) = 7401.2

## Glossary

Central Limit Theorem:
Given a random variable (RV) with known mean μμ and known standard deviation σσ. We are sampling with size n and we are interested in two new RVs - the sample mean, X¯X¯, and the sample sum, ΣXΣX. If the size nn of the sample is sufficiently large, then X¯X¯ size 12{ { bar {X}}} {} N μ σ n N μ σ n and ΣXΣX size 12{X} {}N ( , n σ )N(,nσ). If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n σ n , is called the standard error of the mean.
Normal Distribution:
A continuous random variable (RV) with pdf f(x)=1σe(xμ)2/2f(x)=1σe(xμ)2/2 size 12{ ital "pdf"= { {1} over {σ sqrt {2π} } } e rSup { size 8{ - $$x - μ$$ rSup { size 6{2} } /2σ rSup { size 6{2} } } } } {}, where μμ is the mean of the distribution and σσ is the standard deviation. Notation: XX ~ N μ σ N μ σ . If μ=0μ=0 and σ=1σ=1, the RV is called the standard normal distribution.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks