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Lecture (2) Lecture (2) Presentation Presentation of of Hydrological Data Hydrological Data

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Page 1: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Lecture (2)Lecture (2)

Presentation Presentation of of

Hydrological DataHydrological Data

Page 2: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Presentation of Hydrological DataPresentation of Hydrological Data

Tabular form:

Graphical form:

day depth1 4.51012 4.72933 4.23944 4.5235 4.41996 4.31687 3.01478 2.82149 2.8085

10 4.136311 2.615112 2.228413 3.646414 4.04615 4.52316 4.04617 4.63918 3.324119 3.027620 2.718221 3.117922 2.821423 2.408824 2.434625 2.228426 1.893227 3.852728 3.9329 4.033130 3.839831 3.1436

Depth

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30 35

0

1

2

3

4

5

6

1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5

Page 3: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Histogram for Grouped Data

we divide a grouped data with many values into several class intervals and give the corresponding "frequency" of the class.. The number of data members that fall in a class interval is called the class frequency and the relative and percentage frequencies are computed as following formulas

Page 4: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

How to Make the Histogram

11

11

11

111

11

11

11

11

11

11

111

11

11

11

1

1 2 2 2 3 4 1 1 3 5 2 5 0 01.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5

Class width=0.25 m

- Divide the data into number of classes (13) with certain class width. day depth

1 4.51012 4.72933 4.23944 4.5235 4.41996 4.31687 3.01478 2.82149 2.8085

10 4.136311 2.615112 2.228413 3.646414 4.04615 4.52316 4.04617 4.63918 3.324119 3.027620 2.718221 3.117922 2.821423 2.408824 2.434625 2.228426 1.893227 3.852728 3.9329 4.033130 3.839831 3.1436

Page 5: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Frequency Diagram-Histogram Frequency Diagram-Histogram (cont.)(cont.)

=Class width=0.25 m

- The frequency is computed by counting the number of observations falling in each class, such number is called the absolute class frequency.

max min

#( , )j j j

x xx

kf x x x

= # of classesk

x

Page 6: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Draw a Histogram

To draw a histogram, we mark the class intervals on the horizontal axis. On each interval, we erect a vertical rectangle whose area represents the absolute or relative frequency

• The total area of a histogram is 1

0

1

2

3

4

5

6

1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5

Page 7: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Empirical rules for the number of Empirical rules for the number of

classesclasses USGS: Choose class intervals that provides 20-30 well-distributed points on the curve.

It is recommended (Spiegel, 1972): the number of classes should be between 5 and 20 depending of the sample size.

A rough guide (Brooks and Carruthers, 1953): the number of classes should not exceed five times the logarithm of the number of observations.

In principle, class limits or boundaries can be chosen arbitrarily.

5log( )k N

Page 8: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Relative Frequency DiagramRelative Frequency Diagram- The relative frequency is computed by dividing the absolute frequency by the number of observations, n.

1

1

100 100 100

j jr j j k

jj

j jr j j k

jj

f ff

nf

f ff

nf

the absolute frequency of the jth class. the relative frequency of of jth class. r jfjf

Absolute Class frequency Relative frequency

Total number of the sample data

Page 9: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Frequency PolygonFrequency Polygon

- If the adjacent points are connected by straight lines, a frequency polygon is obtained.

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5

Page 10: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Cumulative (Mass) Frequency Cumulative (Mass) Frequency Diagram Diagram

The cumulative frequency of variates not exceeding a given value is the sum of all frequencies less than or equal to the given value.

0

1

2

3

4

5

6

1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5

1

, 1,...,

,

j i

j jj

j

F f i k

if i k F n

0

5

10

15

20

25

30

35

1.75 2

2.25 2.

5

2.75 3

3.25 3.

5

3.75 4

4.25 4.

5

4.75 5

Series1

1.75 1 12 2 3

2.25 2 52.5 2 7

2.75 3 103 4 14

3.25 1 153.5 1 16

3.75 3 194 5 24

4.25 2 264.5 5 31

Page 11: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Relative Cumulative Frequency Relative Cumulative Frequency DiagramDiagram

The cumulative frequency of variates not exceeding a given value is the sum of all frequencies less than or equal to the given value.

1

, 1,...,

, 1

j i

rj rjj

rj

F f i k

if i k F

1.75 1 1 0.032258 0.0322582 2 3 0.064516 0.096774

2.25 2 5 0.064516 0.161292.5 2 7 0.064516 0.225806

2.75 3 10 0.096774 0.3225813 4 14 0.129032 0.451613

3.25 1 15 0.032258 0.4838713.5 1 16 0.032258 0.516129

3.75 3 19 0.096774 0.6129034 5 24 0.16129 0.774194

4.25 2 26 0.064516 0.838714.5 5 31 0.16129 1

4.75 0 31 0 15 0 31 0 1

0

0.2

0.4

0.6

0.8

1

1.2

Series1

Series2

Page 12: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Histogram ShapesHistogram Shapes

0 1 2 3 4 5 6-3 -2 -1 0 1 2 3

Histogram

0

2

4

6

8

10

12

14

16

4 4.25 4.5 5 5.25 5.5 5.75 6 More

Bin

Fre

quency

Page 13: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Excel ApplicationExcel Application

Page 14: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Why use Excel?

• Software more accessible• Previous familiarity with software• Easy to format output• Better charting facilities than some statistical

applications• Access to other key Excel facilities• Easy to use results with other applications

Page 15: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Problems with Excel

• Errors due to rounding, missing data or extreme values

• Not suitable for very large data sets• Some algorithms are numerically unstable - little or

no information about algorithms employed• Analysis ToolPak results are not dynamic and may

vary with results generated by functions

Page 16: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Frequency Histogram

• Use COUNTIF to count how many times an item appears in a listCell =COUNTIF(range, criteria)

• Use FREQUENCY to calculate how often values occur within a rangeCell =FREQUENCY(data_array, bins_array)

• Can also use Histogram tool in Analysis Toolpak

Page 17: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Statistical Functions

• Frequency Histogram• Mean, Median and Mode• Percentiles and Quartiles• Deviation and Squared Deviation about the Mean• Variance and Standard Deviation• Covariance and the Correlation Coefficient

Page 18: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Excel ExampleExcel Example

-0.23919 0.675719 -0.49077 0.183279 0.345529 -1.41717 -1.16886 0.692688 0.716057 -1.85463

-1.22727 -0.28752 -0.81636 -0.32145 -1.20227 -1.92614 0.663723 0.991945 0.159973 0.789211

-0.69544 -0.13502 1.044491 -0.51898 -0.02417 0.657543 0.329369 -1.42255 0.125517 -0.10306

-0.23055 1.201702 0.120641 1.844993 0.281807 -0.7636 -0.14315 1.298498 -1.60356 -1.32574

0.395318 -0.16511 0.750337 2.204276 -0.24336 0.388192 -1.49879 0.722584 -1.55262 -0.76486

-0.18059 -0.20551 -1.70469 0.571409 -0.63799 -0.86915 0.862368 -0.3158 -0.2826 -0.957

-0.65752 -0.01569 0.779566 -1.08243 0.050219 -0.25687 1.331441 -0.18138 0.294738 -0.30852

-0.31469 -0.85465 -0.1821 0.371825 0.991005 0.484973 1.342515 0.255583 0.780924 -0.60423

0.0064 2.158496 0.6065 0.421751 1.127745 0.306608 0.34904 -0.60732 -1.0366 -1.54858

-0.07485 0.584718 0.680966 -0.24889 1.161446 -0.4919 -0.90306 -0.10034 1.666535 2.049359

0.024877 0.843523 1.272567 1.394709 -0.57985 0.818071 -0.33835 -0.15251 0.903302 -0.23452

-1.34101 0.011979 1.362308 0.632874 1.763065 -0.34541 0.350568 0.357853 0.929899 -1.48423

0.067484 -1.39319 -1.24994 0.294193 0.122471 -0.20526 -0.72575 0.381297 -0.05487 -0.33428

1.06594 -0.91221 2.011211 -0.18541 1.081773 -1.76175 0.196777 0.79094 2.081141 -0.61544

0.311525 2.181965 -0.49721 1.236363 -0.26295 -1.48049 0.302778 1.744695 1.027442 -0.40868

-0.15695 0.083377 -1.53829 0.052848 0.93794 0.374716 1.121503 0.713738 0.075938 -0.1719

0.8452 -1.82075 -0.38257 -0.0545 0.537131 1.182016 0.751896 -0.2684 -0.4623 0.065687

0.45135 -0.28096 -0.13209 2.775904 0.098545 -0.94346 0.669611 -1.65344 0.446124 -0.14243

-0.9994 0.270152 -0.0583 0.474012 0.269291 1.152637 -1.90684 -1.24633 -0.21921 1.963428

Data of rain gauge errors (mm)

Page 19: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Excel Example (cont.)Excel Example (cont.)

  Max 2.049062

  Min -3.24332

  # cells 190

  # classes 11

  delta 0.481126

  mid range frequency exp -pdf

0 -3.24332 1 0.010939

1 -2.7622 1 0.010939

2 -2.28107 0 0

3 -1.79995 6 0.065636

4 -1.31882 15 0.164089

5 -0.83769 19 0.207846

6 -0.35657 27 0.29536

7 0.124558 42 0.459449

8 0.605684 35 0.382874

9 1.08681 23 0.251603

10 1.567936 15 0.164089

11 2.049062 6 0.065636

  mean -0.08288

  var 0.914184

Page 20: Lecture (2) Presentationof Hydrological Data. Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

Excel Example (Cont.)Excel Example (Cont.)

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

-3.0 -2.4 -1.8 -1.3 -0.7 -0.1 0.5 1.1 1.7 2.2 2.8 3.4

Series1

Series3