lecture (2) presentationof hydrological data. presentation of hydrological data presentation of...
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Lecture (2)Lecture (2)
Presentation Presentation of of
Hydrological DataHydrological Data
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
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1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5
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
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
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
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
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1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5
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
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
Frequency PolygonFrequency Polygon
- If the adjacent points are connected by straight lines, a frequency polygon is obtained.
0
1
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1 2 3 4 5 6 7 8 9 10 11 12
0
1
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6
1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5
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
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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
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
Histogram ShapesHistogram Shapes
0 1 2 3 4 5 6-3 -2 -1 0 1 2 3
Histogram
0
2
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6
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4 4.25 4.5 5 5.25 5.5 5.75 6 More
Bin
Fre
quency
Excel ApplicationExcel Application
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
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
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
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
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)
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
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