Statistik: Numerical Descriptive Measures

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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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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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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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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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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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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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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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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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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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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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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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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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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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

Statistik: Numerical Descriptive Measures

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