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Summary of Formulas 1. Determining the sample size where: N - the population size e - the margin of error
N n= 1 + Ne 2
n - the sample size 2.
Range = highest observation – lowest observation
3.
k = 1 + 3.3 log n
4.
C = Range ÷ k 5. Mean of ungrouped data
observations X = observations
∑Xi
where:
∑Xi – sum of all n – total number of
n 6.
Mean of grouped data
∑FiXi between the frequency Xmidpoint = classmark n observations
value of the Xi Xunit-deviation = X0 + between the
where:
FiXi – the product and
the
n – total number of
∑FiUi
where: c
n
X 0 – any chosen F iUi – product frequency and
the unit deviation c – class size n – total number of observations
136
7.
Median of grouped data
n
~
X = LMe +
– cfb
2
where: c
fMe
LMe – lower boundary of the Median class cfb – cumulative frequency below one interval fMe – frequency of the median class c – class size n – total number of
observation 8. Mode of grouped data
d1
^
boundary of the X = LMo +
where: c
d1 + d 2
the
LMo – lower
Modal class d 1 – difference between frequency of the modal class and the frequency of the next lower class d 2 – difference between
the
frequency of the modal class and the frequency of the next higher class c – class size 9.
Quartile
(n)(i ) of the Qi = LQi + frequency
– cfb
4
where: LQi – lower boundary c
fQi
Quartile class cfb – cumulative below one interval fQi – frequency of the Quartile class c – class size
10. Decile
137
(n)( i ) the Di = LDi +
– cfb
10
where: L Di – lower boundary of c
Decile class cfb – cumulative frequency below one interval fDi – frequency of the Decile class c – class size n – total number of
fDi
observation 11.
Percentile
( n)(i ) boundary of Pi = LPi + class
100
– cfb
where: L Pi – lower c
fPi
12.
Interquartile Range
13.
IR = Q3 – Q1 Semi-interquartile range or Quartile Deviation
the Percentile cfb – cumulative frequency below one interval fPi – frequency of the percentile class c – class size
QD = ½ ( Q3 – Q1 ) or IR / 2 14.
Mean Deviation of Ungrouped Data
∑X
i
−X
where:
X – represents the
individual values MD = distribution 15.
X – is the mean of the
n
Mean Deviation for Grouped Data
∑F
i
X i −X
where:
Xi – represents the
classmark MD = distribution
X – is the mean of the
n 16.
Fi – frequency
Variance for Ungrouped Data
138
Standard Formula
s2 = 17.
∑( X
i
Alternative Formula n∑X 2 −(∑X )
− X )2
s2 = n( n −1)
n −1
Standard Deviation for Ungrouped Data Standard Formula
∑( X
s=
18.
2
i
Alternative Formula 2 2 sn∑ = X − (∑X )
− X )2
n( n −1)
n −1
Variance for Grouped Data Standard Formula
s
2
=
∑F ( X i
i
− X )2
n −1
Alternative Formula
n∑ Fi X i − (∑ Fi X i ) 2 2
s
2
=
n(n − 1)
Coding Formula
s 19.
n∑FiU i 2 −(∑FiU i ) 2 = n(n −1)
2
c2
Standard Deviation for Grouped Data Standard Formula
∑F ( X
s2 =
i
i
− X )2
n −1
Alternative Formula
n∑ Fi X i − (∑ Fi X i ) 2 2
s2 =
n( n − 1)
Coding Formula
s2
20.
=
n∑FiU i 2 − (∑FiU i ) 2 n(n −1)
c2
Skewness
˜
3(X–X) 139
Sk = s 21.
Kurtosis Ungrouped Σ ( Xi – X ) K=
22.
Grouped 4
Σ Fi( Xi – X ) K=
ns4
4
ns4
Permutation n! nPr =
a.
(n - r)! N!
where:
b. P = repeated object
N – total number of objects n – frequency of each
n1! n2! n3! … nk! c.
23.
( n-1 )
P
( n-1 )
= (n–1)!
Combination n! n
24.
Cr =
r! ( n – r )!
Test Statistics Concerning Means A. Z-test ( used when n ≥ 30 ) 1. Z-test for comparing hypothesized and sample means (X–μ) z=
where: n.
X – sample mean μ – population mean
140
σ
σ – population standard
dev. n – sample size 2. Z-test for comparing 2 sample means a. When the population standard deviation is given X1 – X2 Z =
where: X1 – mean of the first sample X2 – mean of the second
√
sample
σ
1
+
1
σ – population
standard dev.
n1
n2
n1 – size of the first
sample n2 – size of the second sample
b.
When the sample standard deviations are given X1 – X2 Z =
where:
X1 – mean of the first sample X2 – mean of the second
√
sample
S12
+ S22
S1 –
standard dev. of sample1
n1
n2
S2 – standard dev. of
sample2 n1 – size of the first sample n 2 – size of the second sample
B.
T-test ( used if n < 30 and σ is unknown )
1. T-test for comparing hypothesized and sample means and σ is unknown (X–μ) t=
where: n .
X – sample mean μ – population
mean s df = n – 1
s – sample standard dev. n – sample size
141
2. T-test for comparing two independent sample means and σ is unknown. X 1 – X2 t=
√
√
( n1 – 1 ) s12 + ( n2 – 1 ) s22 n1 + n2 – 2
1
+
n1
1 n2
df = n1 + n2 - 2 where:
X1 – mean of the first sample
X2 – mean of the second sample S1 – standard dev. of sample1 S2 – standard dev. of sample2 n1 – size of the first sample n2 – size of the second sample
25.
Pearson Product Moment Coefficient of Correlation ( ungrouped ) Method 1: Computation of Pearson Product Moment Coefficient of Correlation from Ungrouped Data when Deviations are taken from the Actual Means of the Series.
Standard Formula:
rxy =
Alternative Formula:
∑xy
rxy =
nσ xσ y
σx =
∑x
σy =
∑y
∑xy (∑x )( ∑y 2
2
)
2
n
2
n
142
Method 2: Computation of Pearson Product Moment Coefficient of Correlation from Ungrouped Data when Deviations are taken from the Assumed Means of the Series.
∑x' y ' −c c
rxy =
cx =
σx’ =
x
n
y
σx 'σy '
∑x'
cy =
n
∑x' n
2
∑y '
σy’ =
− c2x
n
∑y ' n
2
− c2 y
Method 3: Computation of Pearson Product Moment Coefficient of Correlation from Ungrouped Data when Deviations are expressed as sigma scores.
x
y •
Σ σx
σy
rxy = n
Method 4: Computation of Pearson Product Moment Coefficient of Correlation from Ungrouped Data based on Original Measurement.
Standard Formula:
rxy =
[∑ X
2
∑XY − nXY − n( X ) ] [∑Y 2
2
− n(Y ) 2
]
Alternative Formula:
143
rxy =
[n ∑ X
n∑XY − ∑X ∑Y
][
− (∑X ) 2 n∑Y 2 − (∑Y ) 2
2
]
26. Pearson Product Moment Coefficient of Correlation ( grouped ) Σx’y’ - cx c n rxy =
* note: Σx’y’= Σuu* = Σvv* σx’ σy’
27.
Spearman Rank-Order Coefficient of Correlation where:
6 - constant D - refers to the difference between a subject’s ranks on the two variables N - the number of paired
observations
28.
Regression equation:
Y = a + bx
where a & b are constants and b
( ΣY ) ( ΣX
2
) – ( ΣX ) ( ΣXY )
where:
0
ΣX – sum of all
values of X
a=
ΣY – sum of all values of Y
nΣX
2
– ( ΣX )
2
ΣXY – sum of the
product of X and Y 2
ΣX – sum of the squared values of X n – number of paired
