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Lecture (3) Description of Central Tendency

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Hydrological Records

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Population vs. Sample Notation PopulationVsSample World PeopleArabs Infinite Record (i.e. very long) selected year

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Different Types of Means or Averages Arithmetic Geometric Harmonic Quadratic Consider a sample of n observations, X1, X2, …, Xi, …, Xn which can be grouped into k classes with class marks x1,x2,…, xi,…, xk with corresponding absolute frequencies, f1,f2,…,fi,…,fk.

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Arithmetic Mean X t X1 X2 Xi Xn

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The Short Cut Method Assume the mean is =xj Calculate the deviation from the assumed mean, (xi-xj)

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Geometric Mean

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Geometric Mean (cont.)

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Harmonic Mean

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Quadratic Mean (Mean Square Value) X t X1 X2 Xi Xn

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General Formula

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Applications and Limitations

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Applications and Limitations (Cont.)

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Flow parallel to the layers Flow perpendicular to the layers

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Applications and Limitations (Cont.) Quadratic mean describes dispersion, spread or scatter around the mean, and is known as the standard deviation from the mean.

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Median Any value M for which at least 50% of all observations are at or above M and at least 50% are at or below M.

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Median Estimation Order all observations from smallest to largest. If the number of observations is odd, it is the “middle” object, namely the [(n+1)/2]th observation. For n = 61, it is the 31 st If the number of observations is even then, to get a unique value, take the average of the (n/2)th and the (n/2 +1)th observation. For = 60, it is the average of the 30 th and the 31 st observation.

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The median has “nice” properties Easy to understand (½ data above, ½ data below) Resistant measure of central tendency (location) not affected by extreme (unusual) observations.

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Percentiles and Quartiles In the cumulative distribution diagram, the range is from 0 to 100%. If this range is divided into a hundred equal parts. The projection of these parts on the x-axis are percentiles and denoted by, X_0.01, X_0.02,…, X_0.99.

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Percentiles, Quartiles and Median (Cont.) The 25 th and 75 th percentiles correspond to the first and third quartiles. Median (Xm): it is the second quartile, X_0.50, divides the set of observations into two numerically equal groups. Median: geometrically is the value that divides the frequency histogram into two parts having equal areas.

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Graphical Representation X_0.25 X_0.50X_0.75

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Mode The mode is the variate that corresponds to the largest ordinate of a frequency curve. Frequency distributions can be described as: Uni-modal, bi-model, multi-model: if it has one, two or more modes.

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Mode in a Histogram 1.Mode(s) 2.Median 3.Mean

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Four Rules of Summation n

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Excel Application See Excel

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Mean, Median, Mode Use AVERAGE or AVERAGEA to calculate the arithmetic mean Cell =AVERAGE(number1, number2, etc.) Use MEDIAN to return the middle number Cell =MEDIAN(number1, number2, etc) Use MODE to return the most common value Cell =MODE(number1, number2, etc)

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Geometric Mean Use GEOMEAN to calculate the geometric mean Cell =GEOMEAN (number1, number2, etc.)

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Percentiles and Quartiles Use PERCENTILE to return the kth percentile of a data set Cell =PERCENTILE(array, percentile) –Percentile argument is a value between 0 and 1 Use QUARTILE to return the given quartile of a data set Cell =QUARTILE(array, quart) –Quart is 1, 2, 3 or 4 –IQR = Q3-Q1 May return different values to statistical package

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