A useful mean in the physical sciences such as voltage is the quadratic mean QM

A useful mean in the physical sciences such as

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A useful mean in the physical sciences (such as voltage) is the quadratic mean (QM), which is found by taking the square root of the average of the squares of each value. The formula is: QM = r x 2 n The QM is also used in Forestry, e.g. to compute the mean dbh for trees. Lecture Number 3 Data Description October 19, 2016 19 / 51
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Measures of Central Tendecny - The Mean In some situations, the observations may not have equal importance. In such situations, the observations are given differential weights - e.g. when computing a grade point average. Since courses vary in their credit value, the number of credits must be used as weights. The measure of central tendency of such data is known as the weighted mean .It is given by the formula: WM = n i =1 w i x i n i =1 w i where, w i s are the weights and x i s are the data values for variable X . The mean is a useful measure of the central value of a set of measurements, but it is subject to distortion due to the presence of one or more extreme values in the set. Lecture Number 3 Data Description October 19, 2016 20 / 51
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Measures of Central Tendecny - The Mean In these situations, the extreme values (called outliers ) pull the mean in the direction of the outliers to find the balancing point, thus distorting the mean as a measure of the central value. A variation of the mean, called a trimmed mean , drops the highest and lowest extreme values and averages the rest. For example, a 5% trimmed mean drops the highest 5% and the lowest 5% of the measurements and averages the rest. Similarly, a 10% trimmed mean drops the highest and the lowest 10% of the measurements and averages the rest. Consider the following example: Example 4 The numbers of wood ants in seven pitfall traps set overnight in deciduous forest woodland are: 25, 4, 12, 9, 15, 18, 202 Calculate the mean number of ants per trap. Lecture Number 3 Data Description October 19, 2016 21 / 51
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Measures of Central Tendecny - The Mean Solution for Example 4 ¯ x = n i =1 x i n = 287 7 = 40 . 7 ants 40.7 is larger than six observations and almost five times smaller than 202! Since the mean takes into account the values of all observations in a sample; it can be greatly distorted by a single exceptional value! When a few exceptional values distort the mean in this way, a resistant measure of the average, namely the median may be more appropriate. Lecture Number 3 Data Description October 19, 2016 22 / 51
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Measures of Central Tendecny - The Median The second measure of central tendency we consider is the median. Definition of Median The median of a set of measurements is defined to be the middle value when the measurements are arranged from lowest to highest. When the data set is ordered, it is called a data array . Thus, the median is the midpoint of the data array The median either will be a specific value in the data set or will fall between two values, as shown in examples 5 and 6: Example 5 Find the median for the set of measurements 2, 9, 11, 5, 6.
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