Central Limit Theorem
Module Objectives
At the end of this module the student will be
able to:
Describe the basic meaning of the Central Limit Theorem
Explain (mathematically) how the means of samples from various distributions are similar to and different from their parent distribution
Inferential Statistics - 2 Major Topics
Parameter Estimation
interested in
estimating the parameter
mean tensile strength of
steel
Scope:
-Point Estimation
Central Limit Theorem
-Interval Estimation
Confidence Intervals
Hypothesis Testing
. Engineer interested in
concluding if temperature t1
resulting in higher yield than
temperature t2
Scope:
Hypothesis formulation
Discrete Data Testing
Continuous Data Testing
ANOVA
Regression
Central Limit Theorem
The Central Limit Theorem and Confidence Intervals are the fundamental tools used in making (inferential) statistical decisions.
The CLT is the basic concept of inferential statistics.
It allows us to make inferences about a population characteristics based upon sample data.
Confidence Intervals are derived from the Central Limit Theorem.
Introductory Exercise
Follow the instructions listed at the bottom of the slide:
generate 100 rows by 10
columns of random data
plot the I-chart with the data
plot the Xbar chart with the
means of the rows
compare the control limits
of the charts
plot & compare the
histograms for the data and
the means of the rows
Use Calc > Random Data > Normal Distribution to generate the data
Use Calc > Row Statistics (select mean) to create C11 as Xbar
Use Manip > Stack> Stack Columns, stack C1-C10 data in Column C12, label as Stack
Use Stat > Control Charts > Individuals to obtain the I-chart
Use Stat > Control Charts > Individuals to obtain the Xbar chart
Individual vs Xbar Charts
Why the difference? After all, it’s the same data ?
How do the control limits compare ?
Original Histogram vs Sampling Histogram
The new histogram shows the sampling distribution of the mean
Note the difference between the two histograms
(Use Graph > Histogram to plot Stack and Xbar data)
- generate 1000 random Binomial data, with Trials = 20, p =
- plot the histograms (original and means of subgroup 10)
What About Sampling on Binomial Distribution?
Definition of the Central Limit Theorem: Part 1
If random samples of size n are taken from a distribution with mean and
standard deviation , then the sample means will form a distribution with
the same mean but with a smaller standard deviation given by
Population Distribution Sampling Distribution
x -
-
-
-
* The concept holds for both normally and non-normally distributed data
Definition of The Standard Error of the Mean
Standard error of the mean (SEMean) is the standard
deviation of the distribution of means
= Standard Error of the Mean
= Standard Deviation for the Individual Scores
n = Sample Size for the mean
This formula shows that a “sample mean” is less variable than
a single individual observation by a factor of the square-root of
the sample size (n)
The SEMean tells us that the distribution of sample means has
less variance than the original population for all n > 1
Population of
Individuals
Population of
Sample Average
Note:
If is unknown, the sample standard deviation s
may be substituted into the previous equation. Then
the estimated standard deviation of
Practical Application
We usually rely on one reading per part from a measurement system.
This reading is used to estimate the “true” quality of our characteristic.
We can reduce measurement system error using the CLT by taking averages
of two or more readings* on the same part.
% Contribution of MS =
* Repeated measurements should reflect the entire measurement system.
The precision of our measurement system then is increased by a factor of the square-root of the sample size (number of repeat measurements).
This is NOT an excuse to avoid fixing the gage!
Standard Error and Sample Size
N=4
N=25
Minitab Demonstration
Let’s create some simulated data to test the CLT
Create 100 rows and 9 columns of data from a normal distribution
Mean = 70 and a standard deviation = 9
Store the average of the first nine columns in C10
Question: what is the expected standard deviation for column C10?
Calc > Random Data > Normal Calc > Row Statistics
Demonstration: Graphical Results
What is the expected Std Deviation for C10 ?
Demonstration: Numerical Results
The mean of the distribution
of sample means is very close
to the population mean (70)
The standard deviation of the
distribution of sample means
is the population standard
deviation reduced by the
square root of the sample size
The distribution of sample
means approximates a normal
distribution
Stat> Basic Statistics > Display Descriptive Statistics
Exercise: CLT with Non-Normal Distributions
Using a Chi2 population of individuals (df=2)
Recall Minitab
File > New > Worksheet
Calc > Random Data > Chi-Square >
Generate 250 Rows
Store in C1 - C9
Degrees of Freedom = 2
Calc > Row Statistics >
Mean C1 - C9
Store results into C10
Graph > Histogram >
C9 & C10 on the same scale
Stat > Basic Stats > Display Descriptive Stats >
Exercise Results
Descriptive Statistics
Variable N Mean Median TrMean StDev SE Mean
C1 250
C2 250
C3 250
C4 250
C5 250
C6 250
C7 250
C8 250
C9 250
Mean 250
Mean = Degree of Freedom
Variance = 2*(Degree of Freedom)
2
Note : For Chi-square distribution :
Exercise Results
Central Limit Theorem: Part 2
As n increases, the distribution of means
becomes more normally distributed for
any distribution.
Sampling Distribution of for Various Sample Sizes
n = 1
n = 5
n = 30
n = 2
Summary
The central limit theorem allows us to assume that the distribution
of sample means from any distribution will approximate the
normal distribution if “n”is sufficiently high (n > 30 for unknown
distributions).
For a normal distribution, the central limit theorem also allows us
to assume that the distributions of sample means are themselves
normal, regardless of sample size.
The standard error of the mean shows that as sample size
increases, the standard deviation of the sample means decreases.
The standard error will help us calculate confidence intervals.
Exercise (Homework)
Choose 2 out of the 3 distributions below. Rerun the CLT-with-
Non-normal-distribution exercise with n equal to 9, 16 and 36.
-Predict the standard error of the mean for each of the respective
sample sizes.
-Also generate histograms and descriptive statistics to verify
your predictions.
-What happens as sample size increases?
Normal distribution (choose any mean and standard deviation)
c2 distribution (choose degrees of freedom < 10)
Exponential distribution (choose any mean)
Module Objectives
At the end of this module the student will be able to:
Describe the basic meaning of the Central Limit Theorem
Explain (mathematically) how the means of samples from various distributions are similar to and different from their parent distribution
