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Lecture (2) Presentationof Hydrological Data

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Presentation of Hydrological Data Presentation of Hydrological Data Tabular form: Graphical form:

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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

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How to Make the Histogram Class width=0.25 m - Divide the data into number of classes (13) with certain class width.

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Frequency Diagram-Histogram (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. = # of classes

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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

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Empirical rules for the number of classes 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.

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

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

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Cumulative (Mass) Frequency 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.

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Relative Cumulative Frequency 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.

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Histogram Shapes

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Excel Application

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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

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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

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Frequency Histogram Use COUNTIF to count how many times an item appears in a list Cell =COUNTIF(range, criteria) Use FREQUENCY to calculate how often values occur within a range Cell =FREQUENCY(data_array, bins_array) Can also use Histogram tool in Analysis Toolpak

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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

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Excel Example -0.239190.675719-0.490770.1832790.345529-1.41717-1.168860.6926880.716057-1.85463 -1.22727-0.28752-0.81636-0.32145-1.20227-1.926140.6637230.9919450.1599730.789211 -0.69544-0.135021.044491-0.51898-0.024170.6575430.329369-1.422550.125517-0.10306 -0.230551.2017020.1206411.8449930.281807-0.7636-0.143151.298498-1.60356-1.32574 0.395318-0.165110.7503372.204276-0.243360.388192-1.498790.722584-1.55262-0.76486 -0.18059-0.20551-1.704690.571409-0.63799-0.869150.862368-0.3158-0.2826-0.957 -0.65752-0.015690.779566-1.082430.050219-0.256871.331441-0.181380.294738-0.30852 -0.31469-0.85465-0.18210.3718250.9910050.4849731.3425150.2555830.780924-0.60423 0.00642.1584960.60650.4217511.1277450.3066080.34904-0.60732-1.0366-1.54858 -0.074850.5847180.680966-0.248891.161446-0.4919-0.90306-0.100341.6665352.049359 0.0248770.8435231.2725671.394709-0.579850.818071-0.33835-0.152510.903302-0.23452 -1.341010.0119791.3623080.6328741.763065-0.345410.3505680.3578530.929899-1.48423 0.067484-1.39319-1.249940.2941930.122471-0.20526-0.725750.381297-0.05487-0.33428 1.06594-0.912212.011211-0.185411.081773-1.761750.1967770.790942.081141-0.61544 0.3115252.181965-0.497211.236363-0.26295-1.480490.3027781.7446951.027442-0.40868 -0.156950.083377-1.538290.0528480.937940.3747161.1215030.7137380.075938-0.1719 0.8452-1.82075-0.38257-0.05450.5371311.1820160.751896-0.2684-0.46230.065687 0.45135-0.28096-0.132092.7759040.098545-0.943460.669611-1.653440.446124-0.14243 -0.99940.270152-0.05830.4740120.2692911.152637-1.90684-1.24633-0.219211.963428 Data of rain gauge errors (mm)

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Excel Example (cont.) Max2.049062 Min-3.24332 # cells190 # classes11 delta0.481126 mid rangefrequencyexp -pdf 0-3.2433210.010939 1-2.762210.010939 2-2.2810700 3-1.7999560.065636 4-1.31882150.164089 5-0.83769190.207846 6-0.35657270.29536 70.124558420.459449 80.605684350.382874 91.08681230.251603 101.567936150.164089 112.04906260.065636 mean-0.08288 var0.914184

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Excel Example (Cont.)

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