Measures of Central Tendency Variability · •The coefficient of variation is based on the size of...
Transcript of Measures of Central Tendency Variability · •The coefficient of variation is based on the size of...
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4.1
Measures of Central Tendency
Measures of Variability
1
• Summarizes what is average or
typical of a distribution
• Summarizes how scores are
scattered around the center of
the distribution
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4.1
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The difference between the highest and lowest scores in a distribution
• Provides a crude measure of variation
The Range
R H L
range
highest score in a distribution
lowest score in a distribution
R
H
L
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4.1
3
The difference between the score at the first quartile and the score at the third quartile
• The higher the IQR, the more spread out the data points; in contrast, the smaller the IQR, the more bunched up the data points are around the mean
• Best used with other measurements such as the median and total range to build a complete picture of a data set
The Inter-Quartile Range
3 1IQR Q Q
inter-quartile range
1 the score value at or below which 25% of the cases fall
3 the score value at or below which 75% of the cases fall
IQR
Q
Q
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4
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4.2
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We need a measure of variability that takes into account every score
• Deviation: the distance of any given raw score from the mean
• Squaring deviations eliminates the minus signs
• Summing the squared deviations and dividing by N gives us the average of the squared deviations
The Variance
2
2X X
sN
2
2
variance
sum of the squared deviations from the mean
total number of scores
s
X X
N
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4.2
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With the variance, the unit of measurement is squared
• It is difficult to interpret squared units
• We can remove the squared units by taking the square root of both sides of the equation
• This will give us the standard deviation
The Standard Deviation
2
X Xs
N
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4.2
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There is an easier way to calculate the variance and standard deviation
• Using raw scores
The Raw-Score Formulas
2
2 2X
s XN
2
2X
s XN
2
2
variance
standard deviation
total number of scores
mean squared
s
s
N
X
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Example 4.3
Obtaining the variance and standard deviation from a simple frequency distribution
X f fX fX2
31 1 31 961
30 1 30 900
29 1 29 841
28 0 0 0
27 2 54 1,458
26 3 78 2,028
25 1 25 625
24 1 24 576
23 2 46 1,058
22 2 44 968
21 2 42 882
20 3 60 1,200
19 4 76 1,444
18 2 36 648
575 13,589
22
22
2
22
57523
25
(23) 529
13,589529 543.56 529 14.56
25
14.56 3.82
fXX
N
X
fXs X
N
fXs X
N
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4.4
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The standard deviation converts the variance to units we can understand
But, how do we interpret this new score?
• The standard deviation represents the average variability in a distribution
– It is the average deviations from the mean
• The greater the variability, the larger the standard deviation
The Meaning of the Standard Deviation
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4.5
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Used to compare the variability for two or more characteristics that have been measured in different units
• The coefficient of variation is based on the size of the standard deviation
• Its value is independent of the unit of the measurement
The Coefficient of Variation
100s
CVX
coefficient of variation
standard deviation
mean
CV
s
X
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Arithmetic Mean
• Obtained by dividing the numerical values of observations by the sum of observations.
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Calculating mean from grouped frequencies
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Calculate the midpoint for every group
Multiply the midpoint with that group’s frequency
Add the results together
Divide the sum by number of total cases
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16820/340=49.47
Grades X f fx
0 - 10 5 3 15
10 – 20 15 12 180
20 – 30 25 35 875
30 – 40 35 45 1575
40 – 50 45 110 4950
50 – 60 55 45 2475
60 – 70 65 35 2275
70 – 80 75 30 2250
80 – 90 85 15 1275
90 - 100 95 10 960
Total 340 16820
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Median
The value which divides a serie of numbers in two when this series is ordered from lowest to highest.
In ungrouped data,
If n is odd (n+1)/2th value
If n is even the mean of n/2. and (n+2). values
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12,24,13,46,23,15,17
• 12,13,15,17,23,24,46 n=7
• (n+1) / 2 = 4
• Med=17
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12,24,13,46,23,15
• 12,13,15,23,24,46 n=6
• n/ 2 = 3 , (n+2) /2=4
• 3. eleman=15 4.eleman=23 a.o(15+23)/2=19
• Med=19
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Median in grouped data
• Cumulative frequencies are calculated to determine which group contains the median
• cf= 340
• 340/2=170.th value is the median. The group containing this value is the group of the median.
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Median in grouped data
l : lower limit of the median group
F : number of values lower than l
f : frequency of the median group
i : range of the median group
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Grades X f C.f
0 -10 5 3 3
10 -20 15 12 15
20 – 30 25 35 50
30 – 40 35 45 95
40 – 50 45 110 205
50 – 60 55 45 250
60 - 70 65 35 285
70 - 80 75 30 315
80 - 90 85 15 330
90 - 100 95 10 340
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Mode
• To find out the mode, the following formula is used after determining the group with the highest frequency.
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Variance
• Sum of squares of deviation from mean
• General indicator for variability
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Students IQ
Score
Student1 127 5,08 25,80 130,92 10,91
Student2 118 -3,92 15,36
Student3 125 3,08 9,48
Student4 120 -1,92 3,68
Student5 119 -2,92 8,52
Student6 125 3,08 9,48
Student7 123 1,08 1,16
Student8 120 -1,92 3,68
Student9 128 6,08 36,96
Student10 119 -2,92 8,52
Student11 120 -1,92 3,68
Student12 120 -1,92 3,68
Student13 121 -0,92 0,84
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Standart Deviation
Most prominent value representing the ‘spread’ of a data set.
Most commonly used measure of variability
• LowValues are close to the mean
• High Values are farther away from the mean
Represented with a ‘s’ in short.
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S=√359.31=18.96
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The Coefficient of Variation
100s
CVX