This document provides an introduction to regression analysis. It discusses correlation as a technique to measure relationships between two variables. Regression allows using the value of one variable to predict the value of another when a consistent relationship exists. The goal of regression is to find the equation of the best fitting straight line for a set of data. This line can be expressed as an equation relating the total cost (Y) variable to the number of hours (X) variable. The best fitting line minimizes the sum of the squared distances between the data points and the line. This process results in a regression equation that can be used to predict Y values given X values. However, predictions should only be made within the range of the original data.

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CORRELATION
• … is a statistical technique that used to
measure and describe a relationship between
two variables
• Usually the two variables are simply observed
as they exist naturally in environment, there is
no attempt to control or manipulate the
variables

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INTRODUCTION TO
REGRESSION

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PREVIEW
• We noted that one common application of
correlation is for purposes of prediction
• Whenever there is a consistent relationship
between two variables, it is possible to use the
value of one variable to predict the value of
another
• The general statistical process of finding and
using a prediction equation is known as
REGRESSION

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INTRODUCTION
• Our goal in this section is to develop a
procedure that identifies and defines the
straight line that provide the best fit for any
specific set of data
• You should realize that this straight line does
not have to be drawn on a graph; it can be
presented in a simple equation
• Thus, our goal is to find the equation for the
line that best describe s the relation for a set of
X and Y data

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LINEAR EQUATION
• In general, a linear relationship between two
variables X and Y can be expressed by the
equation
Y = bX + a
where a and b are fixed constant
• In general linear equation, the value of b is
called the slope
• The slope determines how much the Y
variable will change when X is increase by one
point

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EXAMPLE
• A local tennis club charges a fee of Rp 20.000
per hour plus an annual membership of fee of
Rp 100.000
• With this information the total cost of playing
tennis can be computed using a linear
equation that describe the relationship
between the total cost (Y) and the number of
hours (X)
Y = 20.000(hour) + 100.00

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H
Total Cost Y
1 2 3 4 5 6 7 8
240
220
200
180
160
140
120
100
0

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THE LEAST-SQUARED ERROR
• To determine how well a line fits the data
points, the first step is to define
mathematically distance between the line and
each data point
• For every X value in the data, the linear
equation will determine a Y value on the line

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IQ
IPK
90 95 100 105 110 115 120 125 130 135
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0
Data
Point
distance

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THE LEAST-SQUARED ERROR
• To determine how well a line fits the data
points, the fist step is to define mathematically
distance between the line and each data point
• For every X value in the data, the linear
equation will determine a Y value on the line
• This value is the predicted Y an is called Ŷ (‘y
hat’)
• The distance between this predicted value and
the actual Y value in the data is determined by
distance = Y - Ŷ

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THE LEAST-SQUARED ERROR
• Notice that we are simply are measuring the
vertical distance between the actual data point
(Y) and the predicted point on the line
• The distance measure the error between the
line and the actual data
• Because some of these distance will be positive
and some will be negative, the next sep is to
square each distance in order to obtain a
uniformity positive measure of error
total squared error = Σ(Y - Ŷ)2

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IQ
IPK
90 95 100 105 110 115 120 125 130 135
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0
Data
Point
distance
Ŷ = bX + a

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The EQUATION
Ŷ = bX + a
b =
SP
SSX
a = MY - bMX

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EXAMPLE
X Y X – MX Y – MY (X-MX)(Y-MY) (X-MX)2
7
4
6
3
5
11
3
5
4
7
2
-1
1
-2
0
5
-3
-1
-2
1
10
3
-1
4
0
4
1
1
4
0
b =
SP
SSX
= 1,6 a = MY – bMX = -2Ŷ = 1,6X - 2

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X
Y
1 2 3 4 5 6 7
11
10
9
8
7
6
5
4
3
2
1
0
Ŷ = 1,6X - 2

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TWO CAUTIONS SHOULD BE CONSIDERED
• The predicted value is not perfect (unless r =
+1.00 or -1.00)
it should be clear that the data points do not fit
perfectly on the line
• The regression equation should not be used to
make prediction for X values that fall outside
the range values covered by the original data

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MULTIPLE REGRESSION WITH SOME
PREDICTOR VARIABLES
Ŷ = b1X1 + b2X2 + b3X3 + … + a

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Dasar Pemikiran
• Dalam pengukuran psikologis, kita hanya dapat
memperkirakan besarnya atribut yang hendak diukur.
• Dua kali pengukuran dalam atribut yang sama pada
subyek A bisa memberikan hasil yang berbeda 
berapa skor A yang sesungguhnya dalam atribut ini?
• Dengan demikian, sebuah hasil pengukuran (skor A)
tidak dapat memberikan gambaran yang
sesungguhnya /akurat dari atribut tertentu pada A
(Spearman).
• Dengan kata lain, setiap pengukuran akan selalu
mengandung “error” yang disebut sebagai error of
measurement.
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STANDARD ERROR OF MEASUREMENT
• Koefisien reliabilitas menunjukkan konsistensi
antara beberapa hasil pengukuran pada
subjek yang sama.
• Bahwa setiap pengukuran berharap dapat
mengetahui true score seseorang.
• Bagaimana memperkirakan true score
seseorang?

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STANDARD ERROR OF MEASUREMENT
• Bila subyek dites berulang kali (n kali) dengan
tes yang sama, maka distribusi skor tes akan
menyebar menurut kurva normal.
• Mean dari distribusi skor tes adalah estimated
true score.
• Standard deviation dari penyebaran skor tes
disebut standard error of measurement.

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STANDARD ERROR OF MEASUREMENT
SE= standard error of measurement (SEM)
SX= standar deviasi obtained score
rXX= koefisien reliabilitas
Besarnya SEM menunjukkan indeks rata-rata
jumlah error dalam skor tes tersebut.
xxXE rSS −= 1

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STANDARD ERROR OF MEASUREMENT
• Diasumsikan bahwa random error of
measurement berdistribusi secara normal.
• Dengan tingkat kepercayaan tertentu, rentang
true score seseorang dapat diperkirakan dari
nilai SEM.
• Tingkat kepercayaan 68% (LOC 68%):
 True Score = X ± 1 SEM
• Tingkat kepercayaan 95% (LOC 95%):
 True Score = X ± 1,96 SEM

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MEMAKNAI SKOR TES
• Informasi mengenai koefisien reliabilitas
penting jika kita ingin mengetahui kualitas tes,
tapi jika ingin memaknai skor individu maka
perlu mengetahui SEM.
• Info bahwa koefisien reliabilitas tes PQN =
0,64 tidak dapat membantu menafsirkan hasil
tes yang diperoleh Nina (skor 45) dan Nani
(skornya 54).

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MEMAKNAI SKOR TES
• Di lain pihak info bahwa SEM = 6 (didapat bila SX
= 10) maka kita dapat menyimpulkan hal-hal
berikut:
1. Ada 68,26% kemungkinan skor Nina (X = 45)
berkisar antara 39 – 51 dan ada 99%
kemungkinan kisarannya 45 ± 2,58 x 6.
2. Ada 68,26% kemungkinan skor Nani (X = 54) ada
diantara 48 – 60 dan ada 99% kemungkinan
kisarannya 54 ± 2,58 x 6
 Jadi masih ada kemungkinan bahwa tidak ada
beda antara Nani dan Nina.

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MEMAKNAI SKOR TES (dengan 1 SEM)
45
Nina
54
Nani
5139 6048
X