Post on 16-Apr-2017
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RangeMean Deviation
VarianceStandard Deviation
ange
Range• For a set of data is found by subtracting
the smallest value in the given set of data.• The difference between the highest score
to lowest score in a distribution.• The simplest and the crudest measure of
variation
Importance of Range
Tells you the
distance from the
smallest to largest.
Ungrouped Data R= HS-LS
Whereas: R- RangeHS- Highest scoreLS- Lowest score
Find the range of the two groups of score distribution.
Group A Group B10 (LS) 15 (LS)
12 1615 1617 1725 1726 2328 2530 26
35 (HS) 30 (HS)
Group ARA= HS-LSRA =35-10RA =25
Group BRB= HS-LSRB= 30-15RB= 15
Grouped DataR= HSUB-LSLB
Whereas:R- Range HSUB – Upper boundary of highest
score LSLB – Lower boundary of lowest
score
Find the value of range of the scores of 50 students in Mathematics achievement
testx f
25-32 333-40 741-48 549-56 457-64 1265-72 673-80 881-88 389-97 2
n=50
LS= 25LSLB= 24.5
HS= 97HSUB= 97.5
R= HSUB-LSLB
R=97.5-24.5R= 73
end
Deviation ean
Mean DeviationMeasures the average deviation of the values from the arithmetic mean. It gives equal weight to the deviation of every score in the distribution.
A. Mean Deviation for Ungrouped Data
MD =
Where,
MD = mean deviation valueX = individual score
= sample meanN = number of cases
Steps in Solving Mean Deviation for Ungrouped Data
1. Solve the mean value.2. Subtract the mean value from each
score.3. Take the absolute value of the
difference in step 2.4. Solve the mean deviation using the
formula MD =
Example 1: Find the mean deviation of the scores of 10 students in a Mathematics test. Given the scores: 35, 30, 26, 24, 20, 18, 18, 16, 15, 10
Analysis:The mean deviation of the 10 scores of students is 6.04. This means that on the average, the value deviated from the mean of 212 is 6.04.
B. Mean Deviation for Grouped Data
where ,MD = mean deviation valuef = class frequency
x = class mark or midpoint of each category
x = mean value n = number of cases
Steps in Solving Mean Deviation for Grouped Data
1. Solve for the value of the mean.2. Subtract the mean value from each midpoint or class mark.3. Take the absolute value of each difference.4. Multiply the absolute value and the corresponding class
frequency.5. Find the sum of the result in step 4.6. Solve for the mean deviation using the formula for grouped
data.
Example 2: Find the mean deviation of the given below.
Analysis:The mean deviation of the 40 scores of students is 0.63. This means that on the average, the value deviated from the mean of 33.63 is 10.63.
ariance
VARIANCE is the square of the standard deviation.
In short, having obtained the value of the standard deviation, you can already determine the value of the variance.
VARIANCEOne of the most
important measures of variation.It shows variation
at the mean.
Variance for
Ungrouped Data
How to Calculate the Variance for Ungrouped Data
1. Find the Mean.2. Calculate the difference between
each score and the mean.3. Square the difference between
each score and the mean.
How to Calculate the Variance for Ungrouped Data
4. Add up all the squares of the difference between each score and the mean.
5. Divide the obtained sum by n – 1.
Find the Variance353535353535
210Mean= 35
731149351527210
Mean= 35
Find the Variancex x-ẋ (x-ẋ)2
35 0 0
35 0 0
35 0 0
35 0 0
35 0 0
35 0 0
∑(x-ẋ)2 0
x x-ẋ (x-ẋ)2
73 38 1444
11 -24 576
49 14 196
35 0 0
15 -20 400
27 -8 64
∑(x-ẋ)2 2680
x x-ẋ (x-ẋ)2
35 0 0
35 0 0
35 0 0
35 0 0
35 0 0
35 0 0
∑(x-ẋ)2 0Mean=35
Find the Variance
Variance for Grouped Data
How to Calculate the Variance for Grouped Data
1. Calculate the mean.2. Get the deviations by finding the
difference of each midpoint from the mean.
3. Square the deviations and find its summation.
4. Substitute in the formula.
Find the VarianceClass Limits
(1)
F(2)
Midpoint(3)
FMp(4)
_X
_
Mp - X
_(Mp-X)2
_f( Mp-X)2
28-29 4 28.5 114.0 20.14 8.36 69.89 279.5626-27 9 26.5 238.5 20.14 6.36 40.45 364.0524-25 12 24.5 294.0 20.14 4.36 19.01 228.1222-23 10 22.5 225.0 20.14 2.36 5.57 55.7020-21 17 20.5 348.5 20.14 0.36 0.13 2.2118-19 20 18.5 370.0 20.14 -1.64 2.69 53.8016-17 14 16.5 231.0 20.14 -3.64 13.25 185.5014-15 9 14.5 130.5 20.14 -5.64 31.81 286.2912-13 5 12.5 62.5 20.14 -7.64 58.37 291.85
N= 100
∑fMp= 2,014.0
∑(Mp-X)2= 1,747.08
Find the Variance
TANDARD DEVIATION
Standard Deviation• Is the most important measures of variation.
• It is also known as the square root of the variance.
• It is the average distance of all the scores that deviates from the mean value.
Formula of
Standard Deviation of Ungrouped Data
POPULATION STANDARD DEVIATION
SAMPLE STANDARD DEVIATION
Steps in solving STANDARD DEVIATION of UNGROUPED DATA
1.Solve the mean value.2.Subtract the mean value from each
score.3.Square the difference between the
mean and each score.4.Find the sum of step 3.5.Solve for the population standard
deviation or sample standard deviation using the formula for ungrouped data.
EXAMPLEX X - X (X – X)²
19 4.4 19.3617 2.4 5.7616 1.4 1.9616 1.4 1.9615 0.4 0.1614 -0.6 0.3614 -0.6 0.3613 -1.6 2.5612 -2.6 6.7610 -4.6 21.16
Ʃx= 146 Ʃ(x-x)²= 60.40x= 14.6
POPULATION STANDARD DEVIATION
SAMPLE STANDARD DEVIATION
Formula
of Standar
d Deviatio
n of Grouped Data
POPULATION STANDARD DEVIATION
SAMPLE STANDARD DEVIATION
Steps in solving the STANDARD DEVIATION of GROUP DATA
1. Solve the mean value.2. Subtract the mean value from each midpoint
or class mark.3. Square the difference between the mean
value and midpoint or class mark.4. Multiply the squared difference and the
corresponding class frequency.5. Find the sum of the results in step 4.6. Solve the population standard deviation or
sample standard deviation using the formula for grouped data.
EXAMPLEx f x
15-20 3 17.5 52.5 33.7 -16.2 262.44 787.32
21-26 6 23.5 141 33.7 -10.2 104.04 624.24
27-32 5 29.5 147.5 33.7 -4.2 17.64 88.2
33-38 15 35.5 532.5 33.7 1.8 3.24 48.6
39-44 8 41.5 332 33.7 7.8 60.84 486.72
45-50 3 47.5 142.5 33.7 13.8 190.44 571.32
n=40 = 1348
Ʃf (Xm-X)²= 2606.4
POPULATION STANDARD DEVIATION
SAMPLE STANDARD DEVIATION
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