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NUMERICAL DESCRIPTIVE MEASURESMeeting 3
Course : I0262 – Statistics ProbabilityYear : 2011
Topic : Numerical descriptive Measures
• Measures of central Tendency• Variation and Shape• Numerical descriptive Measures for a
Population• Application in PH Stat Excel
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Learning Outcome
• Identify basic statistics (data, sample, population, symbolism, and definition)
• Interpret the result of identify and the calculation
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DEFINITION
The central tendency is the extent to which all the data values group around a typical or central value.
The variation is the amount of dispersion, or scattering, of values
The shape is the pattern of the distribution of values from the lowest value to the highest value.
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3.1 MEASURES OF CENTRAL TENDENCY
• Mean• Median• Mode• Quartiles
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3.1 MEASURES OF CENTRAL TENDENCY
A. Mean Population
Sample
n = number of sample size xi = observation
N = number of population size
N
xxx
N
x NN
i
i
...21
1
n
xxx
n
xx n
n
i
i
...21
1
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3.1 MEASURES OF CENTRAL TENDENCY
Example : cost of a fast-food hamburger meal
City HamburgerTokyo 5.99London 7.62New York 5.75Sydney 4.45Chicago 4.99San Francisco 5.29Boston 4.39Atlanta 3.70Toronto 4.62Rio de Janeiro 2.99
98,4
10
99.2...62.799.5
10
10
1
i
ixx
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3.1 MEASURES OF CENTRAL TENDENCY
B. Median Odd
Even
n = number of sample sizex= observations that have been sequenced
2/)1( nxmedian
)(2
112/2/ nn xxmedian
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3.1 MEASURES OF CENTRAL TENDENCY
Example : cost of a fast-food hamburger meal
City HamburgerTokyo 5.99London 7.62New York 5.75Sydney 4.45Chicago 4.99San Francisco 5.29Boston 4.39Atlanta 3.70Toronto 4.62Rio de Janeiro 2.99
81.4
)99.462.4(2
1
)(2
1
)(2
1
65
12/102/10
xx
xxmediani Hamburger
1 2.992 3.703 4.394 4.455 4.626 4.997 5.298 5.759 5.99
10 7.62
Data sequence
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3.1 MEASURES OF CENTRAL TENDENCY
C. Mode/Modus Is the value that occurs most often
10
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
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3.1 MEASURES OF CENTRAL TENDENCY
D. QuartilesQuartiles split the ranked data into 4 segments with an equal number of values per segment.
The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the values are greater than the third quartile
25% 25% 25% 25%
1Q 2Q 3Q
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3.1 MEASURES OF CENTRAL TENDENCY
D. Quartiles
• First quartile position: Q1 = (n+1)/4 ranked value
• Second quartile position: Q2 = median
• Third quartile position: Q3 = 3(n+1)/4 ranked value
where n is the number of observed values
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3.1 MEASURES OF CENTRAL TENDENCY
Example : cost of a fast-food hamburger meal
CityHamburg
erTokyo 5.99London 7.62New York 5.75Sydney 4.45Chicago 4.99San Francisco 5.29Boston 4.39Atlanta 3.70Toronto 4.62Rio de Janeiro 2.99
i Hamburger1 2.992 3.703 4.394 4.455 4.626 4.997 5.298 5.759 5.99
10 7.62
Data sequence
• Q1 = (n+1)/4 = 2.75 ~ 3
4.39• Q3 = 3(n+1)/4 = 8.25 ~ 9
5.99
jika Q bukan bernilai integer maka dinaikkan
jika Q bernilai integer maka p adalah rata-rata nilai di posisi Q dan Q+1
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3.2 MEASURES OF VARIATION
Variation measures the spread, or dispersion, of values in a data set. Range Interquartile Range Variance Standard Deviation Coefficient of Variation
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3.2 MEASURES OF VARIATION
A. Range Simplest measure of variation Difference between the largest and the smallest values:
Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 13 - 1 = 12
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3.2 MEASURES OF VARIATIONA. RangeDisantavages of range Ignores the way in which data are distributed
Sensitive to outliers
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 5 - 1 = 4
Range = 120 - 1 = 119
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3.2 MEASURES OF VARIATION
B. Interquartile Range Problems caused by outliers can be eliminated by using the
interquartile range. The IQR can eliminate some high and low values and calculate
the range from the remaining values.
Interquartile range = Q3 – Q1
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3.2 MEASURES OF VARIATIONC. Variance The variance is the average (approximately) of squared
deviations of values from the mean.
Population
Sample
= meann = sample sizeXi = ith value of the variable X
N
i
i
N
x
1
22
n
i
i
n
xxs
1
22
1x
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3.2 MEASURES OF VARIATION
D. Standard Deviation The standard deviation is the root of variance
Population
Sample
N
i
i
N
x
1
22
n
i
i
n
xxss
1
22
1
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3.2 MEASURES OF VARIATION
D. Standard DeviationExample : Computing variance and standard deviation of cost of
a fast-food hamburger meal x 2.99 -1.989 3.9561213.70 -1.279 1.6358414.39 -0.589 0.3469214.45 -0.529 0.2798414.62 -0.359 0.1288814.99 0.011 0.0001215.29 0.311 0.0967215.75 0.771 0.5944415.99 1.011 1.0221217.62 2.641 6.974881
1.671
2xxi xxi
10
98.4
n
x 671.1
11
22
n
i
i
n
xxs
293.1671.1 s
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3.2 MEASURES OF VARIATION
E. Coefficient of Variance The coefficient of variation is the standard deviation divided by
the mean, multiplied by 100. It is always expressed as a percentage. (%) It shows variation relative to mean. The CV can be used to compare two or more sets of data
measured in different units.
100%X
SCV
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3.2 MEASURES OF VARIATIONComparing mean, variance, standard deviation, and CV
Mean = 15.5 s = 3.338 Cv = 0.21511 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5 s = 0.9258Cv = 0,060
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 s = 4.57Cv = 0,295
Data C
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3.3 APPLICATION IN PH STAT EXCELMenu : Data> Data Analysis
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3.3 APPLICATION IN PH STAT EXCELInput Range : Column Hamburger DataClick Summary Statistics
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3.3 APPLICATION IN PH STAT EXCEL
Output
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EXERCISES
1. Calculate the mean, median, quartile, and variance from this data
Motion pictureOpening
Gross SalesTotal Gross
SalesNumber of Theaters
Weeks in Top 60
Coach Carter 29.17 67.25 2574 16Ladies in Lavender 0.15 6.65 119 22Batman Begins 48.75 205.28 3858 18Unleashed 10.9 24.47 1962 8Pretty P 0.06 0.23 24 4Fever P 12.4 42.01 3275 14Harry P & the Goblet of Fire 102.69 287.18 3858 13Monster in Law 23.11 82.89 3424 16White Noise 24.11 55.85 2279 7Mr. & Mrs. Smith 50.34 186.22 3451 21
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THANK YOU