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Other Measures of Central Tendency
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As we have already seen, the mean, or expectation, of a random variable X provides a measure of central tendency for the values of a distribution. Although the mean is used most, two other measures of central tendency are also employed. These are the mode and the median. 1. MODE. The mode of a discrete random variable is that value which occurs most often or, in other words, has the greatest probability of occurring. Sometimes we have two, three, or more values that have relatively large probabilities of occurrence. In such cases, we say that the distribution is bimodal, trimodal, or multimodal, respectively. The mode of a continuous random variable X is the value (or values) of X where the probability density function has a relative maximum.
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1 1 2. MEDIAN. The median is that value x for which P(X x) 2 and P(X x) 2. In the case of a con1 tinuous distribution we have P(X x) 2 P(X x), and the median separates the density curve into two parts having equal areas of 1 > 2 each. In the case of a discrete distribution a unique median may not exist (see Problem 3.34).
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CHAPTER 3 Mathematical Expectation
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Percentiles
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It is often convenient to subdivide the area under a density curve by use of ordinates so that the area to the left of the ordinate is some percentage of the total unit area. The values corresponding to such areas are called percentile values, or briefly percentiles. Thus, for example, the area to the left of the ordinate at xa in Fig. 3-2 is a. For instance, the area to the left of x0.10 would be 0.10, or 10%, and x0.10 would be called the 10th percentile (also called the first decile). The median would be the 50th percentile (or fifth decile).
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Fig. 3-2
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Other Measures of Dispersion
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Just as there are various measures of central tendency besides the mean, there are various measures of dispersion or scatter of a random variable besides the variance or standard deviation. Some of the most common are the following. 1. SEMI-INTERQUARTILE RANGE. If x0.25 and x0.75 represent the 25th and 75th percentile values, the 1 difference x0.75 x0.25 is called the interquartile range and 2 (x0.75 x0.25) is the semi-interquartile range. 2. MEAN DEVIATION. The mean deviation (M.D.) of a random variable X is defined as the expectation of u X m u , i.e., assuming convergence, M.D.(X) M.D.(X) E [u X E [u X mu] mu] a ux
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mu f(x) m u f (x) dx
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(discrete variable) (continuous variable)
(61) (62)
Skewness and Kurtosis
1. SKEWNESS. Often a distribution is not symmetric about any value but instead has one of its tails longer than the other. If the longer tail occurs to the right, as in Fig. 3-3, the distribution is said to be skewed to the right, while if the longer tail occurs to the left, as in Fig. 3-4, it is said to be skewed to the left. Measures describing this asymmetry are called coefficients of skewness, or briefly skewness. One such measure is given by a3 E[(X s3 m)3] m3 s3 (63)
The measure s3 will be positive or negative according to whether the distribution is skewed to the right or left, respectively. For a symmetric distribution, s3 0.
Fig. 3-3
Fig. 3-4
Fig. 3-5
2. KURTOSIS. In some cases a distribution may have its values concentrated near the mean so that the distribution has a large peak as indicated by the solid curve of Fig. 3-5. In other cases the distribution may be
CHAPTER 3 Mathematical Expectation
relatively flat as in the dashed curve of Fig. 3-5. Measures of the degree of peakedness of a distribution are called coefficients of kurtosis, or briefly kurtosis. A measure often used is given by a4 E[(X s4 m)4] m4 s4 (64)
This is usually compared with the normal curve (see 4), which has a coefficient of kurtosis equal to 3. See also Problem 3.41.
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