This document discusses the distribution of sample means and introduces three key principles: 1) There will usually be a difference between sample statistics and the true population mean due to random selection. 2) Larger sample sizes produce more accurate estimates of the population mean. 3) Greater variability in the population variable leads to greater differences between sample statistics and the population mean. It also discusses how the distribution of sample means approaches a normal distribution as sample size increases, and how this distribution can be used to determine probabilities regarding sample means.

1. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
1
Chapter 7
THE DISTRIBUTION
OF SAMPLE MEANS

2. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
SAMPLING
?

3. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Prinsip-Prinsip Sampling
 Pada kebanyakan kasus dimana pengambilan
sampel dilakukan terjadi perbedaan antara
statistik sampel dan rata-rata populasi, yang
dianggap disebabkan oleh pemilihan unit
dalam sampel
 Contoh:
Usia A = 18 tahun, B = 20 tahun, C = 23 tahun,
D = 25 tahun. Usia rata-rata A, B, C, D adalah
21,5 tahun.
3

4. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Prinsip-Prinsip Sampling
 Jika kita ingin mengambil dua individu
untuk memperkirakan usia rata-rata dari
empat individu.
 4
C2= 6  AB, AC, AD, BC, BD, CD
4
nCr =
n!
r! (n-r)!

5. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
n = 2
A = 18 B = 20 C = 23 D = 25
SAMPLE M μ M - μ
AB 19 21,5 -2,5
AC 20,5 21,5 -1,5
AD 21,5 21,5 0
BC 21,5 21,5 0
BD 22,5 21,5 1,5
CD 24 21,5 2,5
5

6. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Prinsip-Prinsip Sampling
 Dari ke-enam kemungkinan kombinasi
sampel, hanya dua yang tidak terdapat
perbedaan antara statistik sampel dan rata-
rata populasi.
 Perbedaan ini dianggap disebabkan sampel
dan diketahui sebagai sampling error.
6

7. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Principle-ONE
In majority of cases of sampling there will
be a difference between the sample
statistics and the true population mean,
which is attributable to selection of the
units in the sample
7

8. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Prinsip-Prinsip Sampling
 Jika kita ingin mengambil tiga individu
untuk memperkirakan usia rata-rata dari
empat individu.
 4
C3= 4  ABC, ABD, ACD, BCD
8

9. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
n = 3
A = 18 B = 20 C = 23 D = 25
SAMPLE M μ M - μ
ABC 20,33 21,5 -1,17
ACD 21 21,5 -0,5
ACD 22 21,5 -0,5
BCD 22,67 21,5 1,17
9

10. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Principle-TWO
The greater sample size, the more accurate
will be estimate of the true population
mean
10

11. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Principle-THREE
The greater difference in the variable under
study in a population for a given sample
size, the greater will be the difference
between the sample statistics and the true
population mean
11

12. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Perbedaan besar variabel yang diteliti pada
populasi, besar pula perbedaan antara statistik
sampel dan rata-rata populasi.
 Contoh:
Usia A = 18 tahun, B = 26 tahun, C = 32 tahun
dan D = 40 tahun.
 Dengan prosedur yang sama, diketahui
rentang perbedaan jauh berbeda dengan
contoh-contoh sebelumnya.
 Hal ini dianggap disebabkan perbedaan usia
yang besar dalam populasi (heterogen)
12

13. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Faktor-faktor yang mempengaruhi
kesimpulan yang ditarik dari sampel
Prinsip-prinsip di atas menunjukkan terdapat dua
faktor yang dapat mempengaruhi tingkat
keyakinan tentang kesimpulan yang ditarik dari
sampel.
1. Ukuran sampel
Temuan yang didasarkan sampel yang besar
lebih dapat diyakini dibandingkan dengan yang
didasarkan dengan sampel yang lebih kecil. Sesuai
prinsip, semakin besar ukuran sampel semakin
akurat temuannya.
13

14. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
Faktor-faktor yang mempengaruhi
kesimpulan yang ditarik dari sampel
Prinsip-prinsip di atas menunjukkan terdapat
dua faktor yang dapat mempengaruhi tingkat
keyakinan tentang kesimpulan yang ditarik dari
sampel.
2. Besarnya variasi populasi
Variasi besar dalam karakteristik populasi,
besar pula ketidakyakinannya (semakin besar
standar deviasi, semakin tinggi standard error).
14

15. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
15
THE DISTRIBUTION OF SAMPLE MEANS
 Two separate samples probably will be
different even though they are taken from the
same population
 The sample will have different individual,
different scores, different means, and so on
 The distribution of sample means is the
collection of sample means for all the possible
random samples of a particular size (n) that
can be obtained from a population

16. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
16
COMBINATION
 Consider a population that consist of 5 scores:
3, 4, 5, 6, and 7
 Mean population = ?
 Construct the distribution of sample means for
n = 1, n = 2, n = 3, n = 4, n = 5
nCr =
n!
r! (n-r)!

17. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
17
SAMPLING DISTRIBUTION
 … is a distribution of statistics obtained by selecting
all the possible samples of a specific size from a
population
CENTRAL LIMIT THEOREM
 For any population with mean μ and standard
deviation σ, the distribution of sample means for
sample size n will have a mean of μ and a standard
deviation of σ/√n and will approach a normal
distribution as n approaches infinity

18. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
18
The STANDARD ERROR OF MEAN
 The value we will be working with is the
standard deviation for the distribution of
sample means, and it called the σM
 Remember the sampling error
 There typically will be some error between
the sample and the population
 The σM measures exactly how much
difference should be expected on average
between sample mean M and the population
mean μ

19. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
19
The MAGNITUDE of THE σM
 Determined by two factors:
○The size of the sample, and
○The standard deviation of the population from
which the sample is selected

20. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
20
 A population of scores is normal with μ = 100
and σ = 15
○ Describe the distribution of sample means for
samples size n = 25 and n =100
LEARNING CHECK

21. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
21
PROBABILITY AND THE DISTRIBUTION
OF SAMPLE MEANS
 The primary use of the standard distribution
of sample means is to find the probability
associated with any specific sample
 Because the distribution of sample means
present the entire set of all possible Ms, we can
use proportions of this distribution to
determine probabilities

22. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
22
EXAMPLE
 The population of scores on the SAT forms a
normal distribution with μ = 500 and σ = 100.
If you take a random sample of n = 16
students, what is the probability that sample
mean will be greater that M = 540?
σM =
σ
√n
= 25 z =
M - μ
σM
= 1.6
z = 1.6  Area C  p = .0548

23. © aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
23
 The population of scores on the SAT forms a
normal distribution with μ = 500 and σ = 100.
We are going to determine the exact range of
values that is expected for sample mean 95%
of the time for sample of n = 25 students
See Example 7.3 on Gravetter’s book page 207
LEARNING CHECK