Sampling Distributions & Standard Error

Lesson 7

Populations & Samples 

Research goals



Learn about population



Characteristics that widely apply



Impossible/impractical to directly study



Research methods



Study representative sample

 

Introduce sampling error

X

 

~

Sampling Error 

X

  

Difference between sample statistic and population parameter



result of choosing random sample



Many potential samples



Sampling Distributions 

Samples from a single population



Repeatedly draw random samples



Every possible combination



Calculate a test statistic (e.g., t test)



One-sample:

X

 

or



Independent samples: Results

  

and

s

~

X

1 

X

sampling distribution

2

The Distribution of Sample Means 

Distribution of means for many samples from a single population



Repeatedly draw random samples



Calculate

X and s



Sampling variation (or sampling error)



will differ from population



different shape

 

similar mean larger sample



closer to



~

#1 Samples: n=10 #2 #3 #4

Law of Large Numbers 

Large sample size (n)



give better estimates of parameters



i.e., better fit

 :

X

becomes a better estimate of   :

s

becomes a better estimate of s  : * if

n

 30 , good estimate  : * if

n

 6 , moderately good estimate

Parameters: Distribution of

X



Results in narrower distribution



Has



and

s 

Find exact values



take all possible samples



or apply Central Limit Theorem ~

Central Limit Theorem 

1.

 of distributi on of

X

  

2.

s of distributi on of

X

called s

X

 standard s

n

error of the mean ( s X

or

s

X



s n

) 

APA style: SE



Also SEM ~

Central Limit Theorem 

3. As sample size (n) increases



the sampling distribution of means approaches a normal distribution



even if parent population not normal distribution of variable (or X)



Very Important! In n ≥ 6, then…



probabilities from standard normal distribution useful



Because we study samples ~

f

Distributions:

X i

vs

X

 s

= 100 = 15 n = 9

s

M

 s

n

 15 9  5

70 85 90 95 IQ Score 100 105 110 115 130 mean IQ Score

Standard Error of the Mean: Magnitude 

Small standard error



better fit



sample means close m

 

More representative sample Depends on n and

s 

large sample size & small

s 

little control

s 

can increase sample size increase value of denominator ~

Using the distribution of X 

Use samples to describe populations

 

is it representative of population?

 

Sample means normally distributed



Use z table



find area under curve



only slight difference in z formula ~

Conducting an experiment 

Same as randomly selecting...

 one

X

from distributi on of

X



For a sample size n



with mean =

 

& standard error

s

X

 s

n

Calculating z scores for sample means  

z



X

s 

X



for raw scores z



X

s  

How close is

X

to  ?



means are normally distributed



Use area under curve



between mean and 1 standard error above the mean



34%



Same rules as any normal distribution



compute z score ~

Distribution of Sample Means is Normal

f

.34

.34

.02

-2 .14

.14

-1 0 1 2 standard error of mean

( s

X

)

.02

z scores & Distribution of X 

What are z scores that define

 

p in left & right tails = .025 + .025



Look up z scores



Left tail = - 1.96; right tail = + 1.96

f

Distribution of Sample Means is Normal

Boundaries for middle 95% (or .95) of sample means?

for middle 99% (or .95) of sample means?

-2 -2.58

-1.96

-1 0 z scores

( s

X

)

1 2 +1.96

+2.58

Sample Mean

X

Using z scores

X



z

s

X

 

Table: large/smaller portion column

z score area under curve or proportion Or probability or percentage

z



X

s 

X



Table: z column