for undergraduate

1. FFGC 6513
BIOSTATISTICS
Jamil Ahmad PhD
Fakulti Pendidikan ● Faculty Of Education

2. VARIABLES, MEASUREMENT SCALES
AND STATISTICS

3. Testing, Measurement and Evaluation
What is the difference between testing, measurement ,
evaluation and assessment?

4. Test Measurement Data Data Analysis Evaluation
An instrument
or activity used
to accumulate
data on a
person’s ability
(either
cognitive,
psychomotor
(skill), or
affective)
The translation
of behavior into
a numerical or
verbal
descriptor which
is then recorded
in written form.
 If the test
collects
quantitative
data, the
score is a
number.
 If the test
collects
qualitative
data, the
score may
be a phrase
or word.
The systematic
assignment of
numerical values
(quantitative) or
verbal descriptors
(qualitative) to the
characteristics of
objects or
individuals.
The process of
making
judgments
about the
results of
measurement
in terms of the
purpose of the
measurement.
Data Analysis is
the process of
systematically
applying
statistical and/or
logical
techniques to
describe and
illustrate,
condense and
recap, and
evaluate data..
Statistics is a
mathematical
body of science
that pertains to
the collection,
analysis,
interpretation or
explanation, and
presentation of
data
 Face validity
 Content validity
 Construct validity
 Concurrent
validity
 Predictive validity
Reliability is
the degree to
which an
assessment
tool produces
stable and
consistent
An Instrument
must be
Validity and
Reliable
Validity is the
extend to
which a test
measures
what it is
intended to
measure
 Descriptive
Statistics
 Inferential
Statistics
 Test-retest Reliability
 Split-Half Reliability
 Internal Consistency Reliability

5.  Assessment in education is the process of
gathering, analysis, interpreting, recording, and
using information about pupils’responses to an
educational task.
PENTAKSIRAN (ASSESMENT)

6. VARIABLE
A variable is any factor, trait, or condition that can exist in
different amounts or types
FOUR TYPES OF VARIABLES
 Independent Variable
 Dependent Variable
 Controlled Variable
 Extraneous Variable

7. Independent Variable
 The variable being manipulated or changed
Dependent Variable
 The observed result of the independent variable being manipulated.
 The event studied and expected to change whenever the independent
variable is altered.
Controlled Variable
 They are the variables that are kept constant to prevent their influence on
the effect of the independent variable on the dependent.
Extraneous Variable
 Variable that might affect the relationship between the independent and
dependent variables
VARIABLE

8.  Independent variables answer the question "What do I change?“
 Dependent variables answer the question "What do I observe?“
 Controlled variables answer the question "What do I keep the
same?"
 Extraneous variables answer the question "What uninteresting
variables might mediate the effect of the IV on the DV

9. Question
Independent
Variable
(What I change)
Dependent
Variables
(What I observe)
Controlled Variables
(What I keep the same)
Does heating
water allow it to
dissolve more
sugar?
Temperature of
the water
measured in
degrees Celsius
Amount of sugar
that dissolves
completely,
measured in grams
 Stirring
 Type of sugar
"More stirring might also increase the amount of sugar that
dissolves, and different sugars might dissolve in different
amounts, so to ensure a fair test I want to keep these variables
the same for each cup of water."
Does fertilizer
make a plant
grow bigger?
Amount of
fertilizer,
measured in
grams
Growth of the plant,
measured by its
height
Growth of the plant,
measured by the
number of leaves
 Same type of fertilizer
 Same pot size for each plant
 Same plant type in each pot
 Same type and amount of soil in each pot
 Same amount of water and light
 Make measurements of growth for each plant at the same
time
"The many variables above can each change how fast a plant
grows, so to ensure a fair test of the fertilizer, each of them
must be kept the same for every pot."

10. TWO TYPES OF VARIABLES
 Discrete Variable
A discrete variable, restricted to certain values, usually consists of whole
numbers, such as the family size
 Continuous Variable
A continuous variable may take on an infinite number of intermediate
values along a specified interval. Examples are:
 The sugar level in the human body;
 Blood pressure reading;
 Temperature;
VARIABLE

11. The level of measurement refers to the relationship among the
values that are assigned to the attributes for a variable.
There are four types of measurements or levels of measurement
or measurement scales used in statistics:
 Nominal
 Ordinal
 Interval
 Ratio.
LEVELS OF MEASUREMENT

12. Nominal
 variable measured on a "nominal" scale is a variable that does not
really have any evaluative distinction.
 One value is really not any greater than another.
 The numerical values just "name" the attribute uniquely.
 A good example of a nominal variable is sex (or gender).
-Information in a data set on sex is usually coded as 0 or 1, 1
indicating male and 0 indicating female (or the other way around--0
for male, 1 for female).
-1 in this case is an arbitrary value and it is not any greater or better
than 0. There is only a nominal difference between 0 and 1.
 With nominal variables, there is a qualitative difference between
values, not a quantitative one.

13. Nominal Scale
Example:
Jersey numbers in basketball are measures at the nominal level.
A player with number 30 is not more of anything than a player with
number 15, and is certainly not twice whatever number 15 is.

14. Ordinal
 Something measured on an "ordinal" scale does have an evaluative
connotation.
 One value is greater or larger or better than the other.
 Example: ProductA is preferred over product B, and thereforeA
receives a value of 1 and B receives a value of 2.
 Another example: Academic Qualification might be rating on a scale
from 1 to 10, with 10 representing the highest academic qualification.
 With ordinal scales, we only know that 2 is better than 1 or 10 is better
than 9; we do not know by how much. It may vary. The distance
between 1 and 2 maybe shorter than between 9 and 10.

15.  A variable measured on an interval scale gives information about
more or betterness as ordinal scales do, but interval variables have
an equal distance between each value.
 The distance between 1 and 2 is equal to the distance between 9
and 10.
 Temperature using Celsius or Fahrenheit is a good example, there is
the exact same difference between 100 degrees and 90 as there is
between 42 and 32.
Interval

16.  Something measured on a ratio scale has the same properties that
an interval scale has except.
 With a ratio scaling, there is an absolute zero point.
 Weight is an example, 0 lbs. is a meaningful absence of weight.
Ratio

17. Paras Ukuran Data Sifat
Nominal Pengkelasan objek atau orang dan sebagainya kepada kategori yang
diskrit mengikut sifat kualitatif.
Ukuran paling asas.
Contoh: Pembolehubah jantina dikategori kepada lelaki dan perempuan.
Ordinal Pengkelasan objek atau orang dan sebagainya mengikut urutan
keutamaan atau rank
Contoh: Tahap kelulusan akademik, persepsi.
Jeda/Sela (interval) Mempunyai urutan atau rank serta wujud perbezaan antara jeda tetapi
tiada mutlak kosong
Contoh: Markat pelajar, IQ, sikap, minat.
Nisbah (ratio) Mempunyai urutan atau rank serta wujud perbezaan antara jeda dan
mempunyai mutlak kosong
Contoh: Tinggi, Umur, Berat

19. Why is Level of Measurement Important?
 First, knowing the level of measurement helps you decide how to
interpret the data from that variable.
-When you know that a measure is nominal, then you know that the
numerical values are just short codes for the longer names.
 Second, knowing the level of measurement helps you decide what
statistical analysis is appropriate on the values that were assigned.
- If a measure is nominal, then you know that you would never average
the data values or do a t-test on the data.

20. P. Ukuran Data Sifat Tujuan Ujian Yg Sesuai
Nominal Pengkelasan mengikut
kategori
Perkaitan pembolehubah Khi Kuasa Dua
Ordinal Mengikut urutan
keutamaan atau rank
Perkaitan pembolehubah Speraman rho
Mann-Whitney
Wilcoxon

21. P. Ukuran
Data
Sifat Tujuan Ujian Yg Sesuai
Jeda
(interval)
Mempunyai urutan atau
rank serta wujud
perbezaan antara jeda
tetapi tiada mutlak
kosong
 Perbezaan Min satu pembolehubah bersandar
berdasarkan satu pembolehubah tak bersandar (2
kategori)
 Perbezaan Min satu pembolehubah bersandar
berdasarkan satu pembolehubah tak bersandar (3
kategori atau lebih)
 Perbezaan Min satu pembolehubah bersandar
berdasarkan dua pembolehubah tak bersandar
 Perbezaan Min bagi dua atau lebih pembolehubah
bersandar berdasarkan dua atau lebih pembolehubah
tak bersandar
 Korelasi antara pembolehubah
 Sumbangan pembolehubah tak bersandar ke atas
pembolehubah bersandar
Ujian-t
ANOVA Satu-Hala
ANOVA Dua-Hala
MANOVA
Korelasi Pearson
Regrasi Berganda
Nisbah
(ratio)
Mempunyai urutan/
rank serta wujud
perbezaan antara jeda
dan mutlak kosong
Sama seperti untuk skala Jeda Sama seperti untuk
skala Jeda

22. What is Statistic?

23. Each individual may be different. If you try to understand a group by remembering the
qualities of each member, you become overwhelmed and fail to understand the group.
20-Oct-22
Class A: IQs of 13
Students
102 115 128 109
131 89 98 106
140 119 93 97
110
Class B: IQs of 13
Students
127 162 131 103
96 111 80 109
93 87 120 105
109
Which Group is Smarter?

24. 20-Oct-22
24
Which group is smarter now?
Class A: Average IQ Class B: Average IQ
110.54 110.23
They’re roughly the same!
With a summary descriptive statistic, it is much easier to
answer our question.

25. Math Science
ALI 12 96
AHMAD 45 60
RMASAMY 60 80
T.SENG 80 70
SALMAH 22 80
SWEELAN 76 54
KUMARY 55 56
AHCHONG 43 64
AHMOI 67 33
JASTINA 62 73
JASTINI 40 59
JASTISHA 35 83
XXX 25 34
XXX 46 76
XXX 23 83
XXX 32 43
XXX 90 65
XXX 45 64
XXX 85 73
XXX 67 85
XXX 45 90
20-Oct-22
25

26. 20-Oct-22
26
ALI
AHMAD
RMASAMY
T.SENG
SALMAH
SWEELAN
KUMARY
AHCHONG
AHMOI
JASTINA
JASTINI
JASTISHA
XXX
XXX
XXX
XXX
XXX
XXX
XXX
XXX
XXX
MIN(PURA
TA)
Math
12
45
60
80
22
76
55
43
67
62
40
35
25
46
23
32
90
45
85
67
45
50.75
Science
91
60
74
70
72
54
56
64
44
73
59
83
42
76
83
43
65
64
73
85
90
71.0
MATH
MIN = 50.75
SKOR MINIMUM = 12
SKOR MAKSIMUM = 90
JULAT = 90-12 = 78
SAINS
MIN = 71.0
SKOR MINIMUM = 42
SKOR MAKSIMUM = 91
JULAT = 91-42= 49

27. What is Statistic?
 A statistic is a numerical representation of information.
 Statistics is a mathematical science involving the collection,
interpretation, analysis, and presentation of data.
 Whenever we quantify or apply numbers to data in order to
organize, summarize, or better understand the information, we
are using statistical methods.

28. Two major branches of statistics
There are two major branches of statistics, each with
specific goals and specific formulas.
1. Descriptive statistics
2. Inferential Statistics

29. Descriptive Statistics
What is descriptive statistic?
 Describing what is or what the data shows/ describe what's going on in
our data.
 Descriptive Statistics are used to present quantitative descriptions in a
manageable form.

30. 20-Oct-22
30
Descriptive Statistics
Summarizing Data:
🞑 Central Tendency (or Groups’ “Middle Values”)
 Mean
 Median
 Mode
🞑 Variation (or Summary of Differences Within Groups)
 Range
 Variance
 Standard Deviation
Organize Data
• Tables
• Frequency Distributions
• Relative Frequency Distributions
• Graphs
• Bar Chart , Histogram, Polygon

31. Statistik deskriptif digunakan
untuk mengumpul data,
menyusunnya dan
mempersembahkan data itu
supaya data yang banyak dapat
disimpulkan dengan
menggunakan indeks seperti
kekerapan, peratusan, min,
mod, median, varians dan
sisihan piawai.
Statistik inferensi digunakan untuk
membuat sesuatu anggaran tentang
indeks populasi dengan
menggunakan satu indeks statistik
daripada sampel yang representatif.
Dengan menggunakan indeks
statistik daripada sampel kita boleh
membuat satu kesimpulan tentang
sifat sesuatu populasi. Statistik
yang digunakan bergantung kepada
paras ukuran data. Contoh; Ujian-t,
ANOVA, Korelasi, Regresi dan lain-
lain.
STATISTIK DALAM PENYELIDIKAN SAINS SOSIAL
STATISTIK DESKRIPTIF STATSITIK INFERENSI

32. INFERENTIAL SATISTICS
INFERENTIAL STATISTICS is defined as the branch of statistics that is
used to make inferences about the characteristics of a populations
based on sample data.
 The goal is to go beyond the data at hand and make inferences
about population parameters.
 With inferential statistics, you are trying to reach conclusions that
extend beyond the immediate data alone.
 Inferential statistics allow one to draw conclusions about the
unknown parameters of a population based on statistics which
describe a sample from that population.

33. Important information about samples
 For qualitative research, we are looking at specific situations. It
may not be important to have a representative sample. We often
use nonprobability sampling with qualitative methods (snowball,
purposive, or convenience samples).
 For most types of quantitative research we do want a sample that
is representative of the population. We will want to generalize our
findings from the sample to the population.

34. 34
POPULATION=N=1000
MIN =  ?
Sample=n=200
min = x
Generalize our findings from
the sample to the population
using
INFERENTIAL STATISTICS

35. 20-Oct-22
35
POPULATION AND SAMPLE
POPULATION
The totality to whom/which you wish to
generalize your study finding
Select sample from
population and conduct
study with sample subject
Generalized or ‘project
back” the findings/results
from your sample subjects
back to the population
SAMPLE
The subjects you actually “use” in your
study

36. Population Parameter vs. Sample Statistic
When you collect data from a population or a sample, there are various
measurements and numbers you can calculate from the data.
A PARAMETER is a measure that describes the whole population.
A STATISTIC is a measure that describes the sample.
The goal of quantitative research is to understand characteristics of populations by
finding parameters. In practice, it’s often too difficult, time-consuming or unfeasible
to collect data from every member of a population. Instead, data is collected from
samples.
With inferential statistics, we can use sample statistics to make educated guesses
about population parameters.You can use estimation or hypothesis testing to
estimate how likely it is that a sample statistic differs from the population parameter.

37. For example, we use the Greek letter mu (i.e., µ) to symbolize the population mean
and the Roman/English letter X with a bar over it, (called X bar), to symbolize the
sample mean.
X

38. Sampling Error
A sampling error is the difference between a population parameter and a sample
statistic.
Sampling errors happen even when you use a randomly selected sample. This
is because random samples are not identical to the population in terms of
numerical measures like means and standard deviations.
Because the aim of scientific research is to generalize findings from the sample
to the population, you want the sampling error to be low. You can reduce
sampling error by increasing the sample size.

39. Important information about samples
 For qualitative research, we are looking at specific situations. It
may not be important to have a representative sample. We often
use nonprobability sampling with qualitative methods (snowball,
purposive, or convenience samples).
 For most types of quantitative research we do want a sample that
is representative of the population. We will want to generalize our
findings from the sample to the population.

40. Togeneralize means that we can say that we would
expect to have the same findingsif we studied
everyoneinthe population aswedid when we
looked at the sample(within a certain degree of
probability)

41. In studies in which we will generalize from the sample to the population
 We must have a sample that is similar or the same on specific
dimensions as the population.
 We will want to use inferential statistics to analyze our data so that we
can infer that findings from a sample are the same as those we would
get from the population.
 Theoretically, we must have a normal distribution in order to use
inferential statistics.
 We will use sampling methods in which every respondent has a known
probability of selection (probability sampling)
 The best type of sampling method to use with inferential statistics is
that in which each participant has an equal probability of selection
(random sampling).

42. 42
PROBABILITY SAMPLING
Every member of the
population had a chance
of “making it” into your
sample
NONPROBABILITY
SAMPLING
Not every member of the
population had a chance of
“making it” into your sample
TYPE OF SAMPLING
Simple
Random
Systematic
Random
Sampling Sampling
Random Sampling Sampling
Sampling
Stratified Cluster Convenience Purposive Judgment Snow-
Sampling Sampling balling
Sampling
20-Oct-22

43. SAMPLE SIZE
“How big a sample do I require?”
“How many people should I sample?”
 This is one of the most frequently asked questions by investigators, although
unfortunately there is no single answer!
 In general, the larger the sample size, the more closely your sample data will
match that how many responses will give you sufficient precision at an
affordable cost.
 The aim of the calculation sample size is to determine an adequate sample size
to estimate the population prevalence with a good precision. It can be
calculated using a simple formula as the calculation needs only a few simple
steps. However, the decision to select the appropriate values of parameters
required in the formula is not simple in some situations

44.  Sample size calculation is concerned with how much data we
require to make a correct decision on particular research.
 If we have more data, then our decision will be more accurate and
there will be less error of the parameter estimate.
 This doesn’t necessarily mean that more is always best in sample
size calculation. A statistician with expertise in sample size
calculation will need to apply statistical techniques and formulas in
order to find the correct sample size calculation accurately.
 There are some basics formulas for sample size calculation, although
sample size calculation differs from technique to technique.

45. HOW TO CALCULATE THE SAMPLE SIZE
Basic Formula For a Mean
The required formula is: s = (z / e)²
Where:
s = the sample size
z = a number relating to the degree of confidence you wish to have
in the result.
95% confidence* is most frequently used and accepted.
For 99% confidence, the value of ‘z’ should be 2.58.
For 95% confidence , z is 1.96
For, 90% confidence z is 1.64
For, 80% confidence z is 1.28 for .
e = the error you are prepared to accept, measured as a proportion
of the standard deviation (accuracy)

46. HOW TO CALCULATE THE SAMPLE SIZE
s = (z / e)²
For example, imagine we are estimating mean income, and wish to
know what sample size to aim for in order that we can be 95%
confident in the result.
Calculating a Sample Size
we are prepared to accept an error of 10% of the population standard
deviation (previous research might have shown the standard deviation
of income to be 8000 and we might be prepared to accept an error of
800 (10%)), we would do the following calculation:
s = (1.96 / 0.1)²
Therefore s = 384.16
In other words, 385 people would need to be sampled to meet our
criterion.

47. HOW TO CALCULATE THE SAMPLE SIZE
Formula For a Proportion
Although we are doing the same thing here, the formula is different:
The following simple formula (Daniel, 1999) can be used:
s = z² ( p ( 1-p ))
e²
Where:
s = the sample size
z = the number relating to the degree of confidence you
wish to have in the result
p = an estimate of the proportion of people falling into the
group in which you are interested in the population
e = the proportion of error we are prepared to accept

48. OR
n = Z² P ( 1− P)
d²
=
where n = sample size,
Z = Z statistic for a level of confidence,
P = expected prevalence or proportion
(in proportion of one; if 20%, P = 0.2), and
d = precision
(in proportion of one; if 5%, d = 0.05).

49. HOW TO CALCULATE THE SAMPLE SIZE
 If we assume that we wish to be 99% confident of the result i.e. z =
2.85 and that we will allow for errors in the region of +/- 3% i.e. e =
0.03.
 But in terms of an estimate of the proportion of the population who
would vote for the candidate (p), if a previous survey had been
carried out, we could use the percentage from that survey as an
estimate. However, if this were the first survey, we would assume that
50% (i.e. p = 0.5) of people would vote for candidate X and 50%
would not.
 Choosing 50% will provide the most conservative estimate of sample
size. If the true percentage were 10%, we will still have an accurate
estimate; we will simply have sampled more people than was
absolutely necessary. The reverse situation, not having enough data
to make reliable estimates, is much less desirable.

50. HOW TO CALCULATE THE SAMPLE SIZE
s = z² (p(1-p))
e²
In the example:
s = 2.58² (0.5*0.5) = 6.6564 (0.25) = 1.6641 = 1,849
0.0009
0.03² 0.0009
Therefore s = 1,849
This rather large sample was necessary because we wanted to be 99%
sure of the result and desired and desired a very narrow (+/-3%) margin
of error. It does, however reveal why many political polls tend to
interview between 1,000 and 2,000 people

51. SAIZ SAMPEL
Berapakah saiz sampel yang sesuai dalam sesuatu
kajian?
 Roscoe (1975) mencadangkan sampel yang melebihi 30
dan kurang daripada 500 adalah bertepatan dalam
kebanyakan penyelidikan.
 Manakala Sekaran (1992) menegaskan bahawa saiz sampel
yang terlampau besar, iaitu melebihi 500 boleh
menyebabkan berlakunya kesilapan dalam mentafsir
keputusan penyelidikan. Saiz sampel yang terlampau besar
boleh menghasilkan perkaitan yang signifikan walaupun
perkaitannya terlalu kecil dan tidak praktikal.

52. SAIZ SAMPEL
Halphin (1957), dalam penulisannya tentang berapa bilangan sampel
yang diperlukan bagi setiap sekolah untuk menghasilkan skor indek
tingkah laku pengetua yang memuaskan, yang dimuatkan di dalam
"Manual for the Leader Behavior Description Quentionaire"
menyatakan bahawa;
Experience suggests that a minimum of four respondents per leader is
desirable, and that additional respondents beyond ten do not increase
significantly the stability of the index scores. Six or seven respondents
per leader would be a good standard. Oviously, much depends upon
the particular leader and group in which one may be interested. If the
group is large, then it is posible to select about seven respondents
from the larger group by use of a table of random numbers.
(Halphin 1957: 2)

53. HOW TO CALCULATE THE SAMPLE SIZE
Formula Penentuan Saiz sampel
S = ²NP(1 - P)
d²(N - 1) + ²P(1 - P)
Di mana;
S = Saiz sampel yang diperlukan
N = Saiz populasi
P = Nisbah populasi (dianggarkan sebagai 0.50 untuk
memberikan saiz sampel yang maksimum)
d = Darjah ketepatan dinyatakan sebagai 0.05
² = Nilai chi-square untuk 1 darjah kebebasan pada aras
keyakinan 0.05 (3.841).
Sumber: National Education Assosiation dalam Krejcie dan Morgan (1970).

54. Jadual Penentuan Saiz Sampel oleh Krejcie dan Morgan (1970)
N S N S N S
10 10 220 140 1200 291
15 14 230 144 1300 297
20 19 240 148 1400 302
25 24 250 152 1500 306
30 28 260 155 1600 310
35 32 270 159 1700 313
40 36 280 162 1800 317
45 40 290 165 1900 320
50 44 300 169 2000 322
55 48 320 175 2200 327
60 52 340 181 2400 331
65 56 360 186 2600 335
70 59 380 191 2800 338
75 63 400 196 3000 341
80 66 420 201 3500 346
85 70 440 205 4000 351
90 73 460 210 4500 354
95 76 480 214 5000 357
100 80 500 217 6000 361
110 86 550 226 7000 364
120 92 600 234 8000 367
130 97 650 242 9000 368
140 103 700 248 10000 370
150 108 750 254 15000 375
160 113 800 260 20000 377
170 118 850 265 30000 379
180 123 900 269 40000 380
190 127 950 274 50000 381
200 132 1000 278 75000 382
210 136 1100 285 100000 384

55. INFERENTIAL STATISTICS
PARAMETRIC STATISTICS
 t-Test (Independent Sample)
 T-Test (Paired Sampel)
 One-Way ANOVA
 Two-Way ANOVA
 One-Way MANOVA
 Two-Way MANOVA
 Correlation
 Simple Regression
 Multiple Regression
NON PARAMETRIC STATISTICS
 Mann Whitney Test
 Wilcoxon
 Kruskal-Wallis
 Spearman’s rank
 Chi-Square Test

56. TEST IV DV PURPOSE
T- test 1 IV (2 Cat.) 1 DV (Cont.) Group diff.
One-way ANOVA 1 IV (2+ Cat.) 1 DV (Cont.) Group diff.
Two-Way ANOVA 2 IVs (2+ Cat.) 1 DV (Cont.) Group diff
One-way MANOVA 1 IV (2+ Cat.) 2+ DVs (Cont.) Group diff.
Two-way MANOVA 1 IV (2+ Cat.) 2+ DVs (Cont.) Group diff.
Correlation 1 IV (Cont.) 1 DV (Cont.) Relationship
Simple Regression 1 IV (Cont.) 1 DV (Cont.) Relation/Prediction
Multiple Regression 2+ IVs (Cont.) 1 DV (Cont.) Relation/Prediction

57. 57

58. Distribution of Sample Means
&
Central Limit Theorem
Jamil Ahmad PhD
Fakulti Pendidikan UKM
Fakulti Pendidikan ● Faculty Of Education

59. 20-Oct-22
59
Sampling Distributions
One of the most important concepts in inferential
statistics is that of the sampling distribution. That's
because the use of a sampling distributions is what
allows us to make "probability" statements in
inferential statistics.

60. 20-Oct-22
60
Sampling Distributions
What is sampling distribution ?

61. 20-Oct-22
61
POPULATION: 2, 4, 6, 8
SAMPEL

62. 62
SAMPEL MIN PROBALITY
20-Oct-22
2,2
2,4
2,6
2,8
4,2
4,4
4,6
4,8
6,2
6,4
6,6
6,8
8,2
8,4
8,6
8,8
POPULATION: 2, 4, 6, 8

63. 63
SAMPEL MIN PROBALITY
20-Oct-22
2,2 2
2,4 3
2,6 4
2,8 5
4,2 3
4,4 4
4,6 5
4,8 6
6,2 4
6,4 5
6,6 6
6,8 7
8,2 5
8,4 6
8,6 7
8,8 8
POPULATION: 2, 4, 6, 8

64. 64
SAMPEL MIN PROBALITY
20-Oct-22
2,2 2 1/16
2,4 3 1/16
2,6 4 1/16
2,8 5 1/16
4,2 3 1/16
4,4 4 1/16
4,6 5 1/16
4,8 6 1/16
6,2 4 1/16
6,4 5 1/16
6,6 6 1/16
6,8 7 1/16
8,2 5 1/16
8,4 6 1/16
8,6 7 1/16
8,8 8 1/16
POPULATION: 2, 4, 6, 8

65. 20-Oct-22
65
POPULATION: 2, 4, 6, 8
SAMPEL MIN PROBALITY
2,2 2 1/16
2,4 3 1/16
2,6 4 1/16
2,8 5 1/16
4,2 3 1/16
4,4 4 1/16
4,6 5 1/16
4,8 6 1/16
6,2 4 1/16
6,4 5 1/16
6,6 6 1/16
6,8 7 1/16
8,2 5 1/16
8,4 6 1/16
8,6 7 1/16
8,8 8 1/16
Population Mean = ??
Mean of the sampling distribution of the
mean = ??

66. 20-Oct-22
66
POPULATION: 2, 4, 6, 8
SAMPEL MIN PROBALITY
2,2 2 1/16
2,4 3 1/16
2,6 4 1/16
2,8 5 1/16
4,2 3 1/16
4,4 4 1/16
4,6 5 1/16
4,8 6 1/16
6,2 4 1/16
6,4 5 1/16
6,6 6 1/16
6,8 7 1/16
8,2 5 1/16
8,4 6 1/16
8,6 7 1/16
8,8 8 1/16
Population Mean = (2+4+6+8)/4 = 5
Mean of the sampling distribution of the mean
= (2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8)/16
= 80/16
= 5
Mean of the sampling distribution of the
mean is equal to the population mean!
That tells you that repeated sampling will,
over the long run, produce the correct
mean.

67. 20-Oct-22
SAMPEL MIN PROBALITY
Table of Frequency Distribution
2,2 2 1/16
2,4 3 1/16
MIN FREQ PROBALITY
2,6 4 1/16
2,8 5 1/16 2 1 1/16
4,2 3 1/16 3 2 2/16
4,4 4 1/16 4 3 3/16
4,6 5 1/16 5 4 4/16
4,8 6 1/16
6 3 3/16
6,2 4 1/16
7 2 2/16
6,4 5 1/16
8 1 1/16
6,6 6 1/16
6,8 7 1/16
8,2 5 1/16
8,4 6 1/16
8,6 7 1/16
8,8 8 1/16
67

68. 68
MIN FREQ PROBALITY
2 1 1/16
3 2 2/16
4 3 3/16
5 4 4/16
6 3 3/16
7 2 2/16
8 1 1/16
•The sampling distribution of the mean is
normally distributed (as long as your sample
size is about 30 or more for your sampling).
20-Oct-22
Table of Frequency Distribution
Histogram

69. 20-Oct-22
69

70. 20-Oct-22
70
What is Sampling Distributions?
 A sampling distribution is defined as "The theoretical probability distribution of
the values of a statistic that results when all possible random samples of a
particular size are drawn from a population.
 A one specific type of sampling distribution is called the sampling distribution
of the mean.
If you wanted to generate this distribution, you would randomly select a sample,
calculate the mean, randomly select another sample, calculate the mean, and
continue this process until you have calculated the means for all possible
samples. This process will give you a lot of means, and you can construct a line
graph to depict your sampling distribution of the mean

71.  The sampling distribution of the mean is normally distributed
(as long as your sample size is about 30 or more for your
sampling).
 Also, note that the mean of the sampling distribution of the
mean is equal to the population mean!
 That tells you that repeated sampling will, over the long run,
produce the correct mean. The spread or variance shows you
that sample means will tend to be somewhat different from the
true population mean in most particular samples.
20-Oct-22
71

72.  The computer program that a researcher uses (e.g., SPSS and SAS)
uses the appropriate sampling distribution for you.
 The computer program will look at the type of statistical analysis you
select (and also consider certain additional information that you have
provided, such as the sample size in your study), and then the
statistical program selects the appropriate samplin20g-Odcti-s22tribution.
72
It is important to understand that researchers do not actually empirically
construct sampling distributions! When conducting research,
researchers typically select only one sample from the population of
interest; they do not collect all possible samples.

73. Important concepts about sampling distributions:
 If a sample is representative of the population, the mean (on a variable
of interest) for the sample and the population should be the same.
 However, there will be some variation in the value of sample means due
to random or sampling error. This refers to things you can’t necessarily
control in a study or when you collect a sample.
 The amount of variation that exists among sample means from a
population is called the standard error of the mean.
 Standard error decreases as sample size increases.

74. 20-Oct-22
74
STANDARD ERROR
The standard deviation of a sampling distribution is called
the standard error.
In other words, the standard error is just a special kind of
standard deviation .
•The smaller the standard error, the less the amount of
variability present in a sampling distribution.

75. Distribution of Sample Means
1. The mean of a sampling distribution is identical to mean of
raw scores in the population (µ)
2. If the population is Normal, the distribution of sample
means is also Normal
3. If the population is not Normal, the distribution of sample
means approaches Normal distribution as the size of
sample on which it is based gets larger Central
Limit
Theorem

76.  Unlike descriptive statistics, which are used to describe the
characteristics (i.e. distribution, central tendency, and dispersion) of a
single variable, inferential statistics are used to make inferences about
the larger population based on the sample.
 Since a sample is a small subset of the larger population, the inferences
are necessarily error prone.
 That is, we cannot say with 100% confidence that the characteristics of
the sample accurately reflect the characteristics of the larger population.
 Hence, only qualified inferences can be made, within a degree of
certainty, which is often expressed in terms of probability (e.g., 90% or
95% probability that the sample reflects the population).
20-Oct-22
76
Confidence Level and Significance Level

78. 20-Oct-22
78