Descriptive Statistics PPT

13. The Methodology of Descriptive StatisticsThe purpose of a descriptive statistical investigation is clarify a number of characteristics of a given variable measures in time or as a cross section

A descriptive statistical analysis consists of: Setting up a histogram (or a time series plot) Calculating descriptive statistics

Measures of location and position

Inspection for outliers Classification of the distribution of the examined data set

The range of statistical techniques utilized have not provided us with anything more than we would have got by taking the [...] variables and looking at their graphs

Statistics EUS & Negot Chinese 1

In statistics, we consider the following types of data:

Cross-section:Many sectors/categories/regions at a given point in time

Time series:One sector/category/regions over a period of time e.g. a year

Panel:A combination of times series and cross section

Census:Statistics provided through a questionnaire


24. Histogram A histogram displays classification into intervals of a

quantitative variable The horizontal axis (x-axis) is the interval scale The vertical axis (y-axis) is used to display the frequency

Data set with 20 observations of incomes in 1,000 DKK

Ranked


9 6 12 10 13 15 16 14 14 16 17 16 24 21 22 18 19 18 20 17

6 9 10 12 13 14 14 15 16 16 16 17 17 18 18 19 20 21 22 24

How can the data set be divided into some efficient categories or groups?

Ad hoc method:

More mathematical approach: 2k=n where k is the number of categories


Below 5 6 to 10 11 to 15 16 to 20 21 or more TotalNumber 20Frequency 0 3 5 9 3 20

Relative % 0 0.15 0.25 0.45 0.15 1.00

Cumulative % 0 0.15 0.40 0.85 1.00

10.5 to 15 16 to 15 15 to 19.5 19.5 to 24 TotalObservations 20Frequency 3 5 8 4 20Relative % 0.15 0.25 0.40 0.20 1.00Cumulative % 0.15 0.40 0.80 1.00

3Statistics EUS & Negot Chinese 5

0

1

2

3

4

5

6

7

8

9

10

Under 5 5 to 10 11 to15 16 to 20 Over 20

Frequency

Interval (1,000 DKK)

Monthly Income

Construction of a Histogram by use of Excel


4A Special Histogram

Age: 0 to 4 5 to 14 15 to 29 30 to 49 50 to 69 70 or more Total Persons, mill 116.60 196.90 350.50 283.10 147.90 36.80 1131.90Persons, % 10.30 17.40 30.97 25.01 13.07 3.25 100.00 Units of 5 years 1 2 3 4 4 [4]* 18 % units of 5 years 10.30 8.70 10.32 6.25 3.27 0.81 *=assumed Using data from the first part of the table the following graph can be drawn:

0,005,00

10,0015,0020,0025,0030,0035,00

0 to 4 5 to 14 15 to 29 30 to 49 50 to 69 70 or more

Percent

Age

Population China 1990


0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85plus

PopulationChina1.7.1990


Age, year 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 >85 Person,% 10.3 8.7 8.7 10.3 10.3 10.3 6.25 6.25 6.25 6.25 3.3 3.3 3.3 3.3 0.8 0.8 0.8 0.8 0.8

55. Measures of LocationMost frequent or typical observation

Sample mean (MB page 26)Modus or Mode (MB page 32)Median (MB page 31)Geometric Mean (MB page 57)Relation among the mean, mode and medianQuartiles and Perentiles


The meanUses information from all observations

Man

From the example:

Grouped data set:


6Example of Grouped data set on GradesExam in the course International Economics that was held in February 2011 at the BA-int study in Flensburg

Grouped mean:

Modus or ModeThis is the most common observed observation (highest frequency)

Income data example mode = 16Grouped data examplemode = 7


Grades of passed (7-point DK scale) 2 4 7 10 12 Total Frequency 10 26 33 19 4 92

MedianThe middlemost observation:

Median = 0.50(n + 1) ordered position0.50(20+1) = 10.5 ordered observation = 16

Example with grades: At the 46.5 ordered obs. = 7

Important measure because it is not sensitive with regard to outliers


Data 6 9 10 12 13 14 14 15 16 16 16 17 17 18 18 19 20 21 22 24 Frequency .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 Cumulative .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7Sum function:


Dealing with symmetry


8Summing upSymmetry: M0 = Md = Skewed to the right: M0 < Md < (bulk of data left)Skewed to the left: < Md < M0 (bulk of data right)

Income data set: = 15.85 < M0 = 16 and Md = 16 data is skewed to the left

Grade data set: = 6.45 < Mo = 7 and Md = 7 data is skewed to the left


Quartiles and Percentiles

Quartile = q(n+1) ordered position

Percentile = p(n+1) ordered position

5-point summary:1st decil is 0.10-percentileLower quartile is 0.25-percentile (called Q1)Median is 0.50-percentileUpper quartile is 0.75-percentile (called Q3)9th decil is 0.90-percentile


9Example

10: (20+1)(10/100) = 2.10 observations appears at = 9.1025: (20+1)(25/100) = 5.25 observations appears at = 13.7550: (20+1)(50/100) = 10.50 observations appears at = 16.0075: (20+1)(75/100) = 15.75 observations appears at = 18.2590: (20+1)(90/100) = 18.90 observations appears at = 21.90


Data 6 9 10 12 13 14 14 15 16 16 16 17 17 18 18 19 20 21 22 24 Frequency .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 Cumulative .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Geometric (multiplicative) MeanDefined as:

The geometric mean is always smaller than the arithmic mean

Example:


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6. Measures of DispersionRange, inter quartile range, decil range and Box-plotVariance and standard deviationCoefficient of variationSkewness and kurtosis

Range = maximum minimumQuartile range = Q3 Q1 = 50 % of obs.Decil range = D9 D1 = 80 % of obs.


Box-plotA Box-plot is used in order to identify outliersOutlier: obs. more than 3 times the IRQ away from Q1 and Q3 Suspected outlier: obs. more than 1.5 (but less than 3) IRQ away from Q1 and Q3

For our little data set we get

(supected) Outlier Q1 Median Q3

BoxPlot

0 5 10 15 20 25 30


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Lower inner fence:Q1 1.5IQR = 13.75 1.5(4.5) = 7.00

Lower outer fence:Q1 3.0IQR = 13.75 3.0(4.5) = 0.25

Upper inner fence:Q3 + 1.5IQR = 18.25 + 1.5(4.5) = 25.50

Upper outer fence:Q3 + 3.0IQR = 18.25 + 3.0(4.5) = 32.25


Variance and Standard DeviationMake use of all observations

or

Example on data set for incomes


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Grouped data set

Example:


The Coefficient of Variation:Gives the relative dispersionRecommended for comparisons of different data sets

If the distribution has large variation (is very flat) then CVtakes a large value.If the distribution has small variation (is very steep) then CVtakes a small value.


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Some examples:

SK > 0: RightSK = 0: SymmetrySK < 0: Left

KU large: DensityKU low: Uniform


7. Descriptive statistics on a Computer or Calculator

Use of ExcelUse of MegastatUse of pocket calculator


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8. Descriptive Statistics in a Grouped Data Sets


More complex data set for the distribution of income, Denmark Disposal house hold incomes, Denmark, 1987 i

Interval for incomes 1,000 DKK

Number of households,

1,000

Mean income 1,000 DKK

Income mass Mio. DKK

Deviation

Square

fi xi fixi (xi ) (xi )2 fi(xi )2 1 2 3 4 5 6 7 8

0 50

100 150 200 250 300 400

- 49.9 - 99.9

- 149.9 - 199.9 - 249.9

299.9 399.9

-

146 590 414 323 325 210 139 55

36.9 73.2

123.7 175.1 225.9 273.6 340.6 548.3

5,387 43,202 51,224 56,568 73,435 57,446 47,339 30,156

-128.7 -92.4 -41.9

9.5 60.3

108.0 175.0 382.7

16563.69 8537.76 1755.61

90.25 3636.09

11664.00 30625.00

146459.29

2418298 5036983 726822 29151

1181729 2449440 4256875 8055261

Sum 2,202 364,757 24154559 Source: Statistics Denmark, Annual Statistical Review, 1994, page 220-221.

Mean and Standard Deviation


Mean and Standard Deviation There are 8 categories i.e. k = 8. By insertion in the formulas:

Mean: 6.165648,165202,2757,3641 DKK

nxfk

i ii

Standard deviation: 73.104202,2

559,154,24)(

1

2

n

xfk

iii

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Histogram, Quartiles, Median and Box-plotConsider the relative and cumulative distribution of data


Disponible husstandsindkomster, Danmark, 1987 i

Interval for incomes 1,000 DKK

Number of households,

1,000

Number of households

frequency, %

Cumulative frequency, %

fi fi/n1 2 3 4 5 6 7 8

0 50

100 150 200 250 300 400

- 49.9 - 99.9

- 149.9 - 199.9 - 249.9

299.9 399.9

-

146 590 414 323 325 210 139 55

6.6 26.8 18.8 14.7 14.8

9.5 6.3 2.5

6.6 33.4 52.2 66.9 81.7 91.2 97.5

100.0

Sum 2,202 100.0 Source: Statistics Denmark, Annual Statistical Review, 1994, page 220-221

Histogram

Distribution Income, Denmark, 1987

0,00

5,00

10,00

15,00

20,00

25,00

30,00

0 - 49 50 - 99 100 -149

150 -199

200 -249

250 -299

300 -349

350 -399

Above400

%


16

Sum Function


How to do the interpolation

We use a formula for example given as:

Value = End value interval """"

pctpercentinwidthTotalfractiletorelativelongtoo interval width in value

Illustration: Frequency % 52.2 50 33.4 100 ? 149 income (1,000 DKK)


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Median: 149,144851,5000,150000,508.18

)502.52(000,150 Similarly for the other quartiles and deciles:

Lower quartile: 328,84000,508.26

)254.33(000,100 (Q1)

Upper quartile: 365,227000,508.14

)757.81(000,250 (Q3)

Lower decile: 343,56000,508.26

)104.33(000,100

Upper decile: 684,293000,505.9

)902.91(000,300


Inter Quartile Range (IQR): (Q3Q1) = 227,365 84,328 = 143,037 Lower inner fence: Q1 1.5IQR = 84,328 1.5(143,037) = 130,228 Lower outer fence: Q1 3.0IQR = 84,328 3.0(143,037) = 344,783 Upper inner fence: Q3 + 1,5IQR = 227,365 + 1.5(143,037) = 441,921 Upper outer fence: Q3 + 3.0IQR = 227,365 + 3.0(143,037) = 656,476 Box-plot

300 200 100 0 100 200 300 400 500 600

LOF = 345 LIF = 130 Q1=84 M=144 Q3=227 UIF = 442 UOF = 656


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9. Descriptive Statistics an Example of Outliers

Outliers are extremes

Outliers make distributions non-normal Outliers changes the mean, standard deviation and skewness

However, the median remains constant


Basic Max=34 Max=44 Max=54 Mean 15.85 16.35 16.85 17.35 Increases Standard Error 1.00 1.29 1,69 2.13 Median 16 16 16 16 Constant!! Modus / Mode 16 16 16 16 Standard deviation 4.46 5.79 7.56 9.52 Sample variance 19.92 33.50 57.08 90.66 Kurtosis 0.12 3.88 8.99 12.55 Skewness -0.35 1.19 2.43 3.16 Increases Range 18 28 38 48 Minimum 6 6 6 6 Maximum 24 34 44 54 Sum 317 327 337 347 Observations 20 20 20 20 Confidence interval(95 %) 2.09 2.71 3.54 4.46 Increases


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Descriptive Statistics PPT