Lecture (3) Description of Central Tendency

Hydrological Records

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

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.

Arithmetic Mean X t X1 X2 Xi Xn

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

Geometric Mean

Geometric Mean (cont.)

Harmonic Mean

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

General Formula

Applications and Limitations

Applications and Limitations (Cont.)

Flow parallel to the layers Flow perpendicular to the layers

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

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.

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.

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.

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.

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.

Graphical Representation X_0.25 X_0.50X_0.75

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.

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

Four Rules of Summation n

Excel Application See Excel

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)

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

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