Termo-accionamiento de las series LCE-Az y LCE-BAz

(a) Advantages

(i) It uses all the observations.

(ii) It is closely related to the most commonly used measure of location, i.e. the mean.

(iii) It is easy to manipulate arithmetically.

(b) Disadvantages

(i) It is rather complicated to define and calculate. (ii) Its value can be distorted by extreme values.

D. COEFFICIENT OF VARIATION

The standard deviation is an absolute measure of dispersion and is expressed in the units in which the observations are measured. The coefficient of variation is a relative measure of dispersion, i.e. it is independent of the units in which the standard deviation is expressed. The coefficient of variation is calculated by expressing the standard deviation as a

percentage of the mean:

coefficient of variation (CV) 100% x

σ 

 .

By using the coefficient of variation you can compare dispersions of various distributions, such as heights measured in centimetres with a distribution of weights measured in kilograms.

Example:

Compare the dispersion of the monthly rainfall for Town A with the dispersion of employees' heights.

80 Measures of Dispersion

Town A: σ1.47inches x 4.89inches therefore: 100% 30.06% 4.89 1.47 CV   Employees' heights: σ7.24cm x 175.875cm therefore: 100% 4.12% 175.875 7.24 CV  

This shows that the rainfall distribution is much more variable than the employees' heights distribution.

E. SKEWNESS

When the items in a distribution are dispersed equally on each side of the mean, we say that the distribution is symmetrical. Figure 5.2 shows two symmetrical distributions.

Figure 5.2: Symmetrical distributions

When the items are not symmetrically dispersed on each side of the mean, we say that the distribution is skew or asymmetric. Two skew distributions are shown in Figure 5.3. A

distribution which has a tail drawn out to the right, as in Figure 5.3 (a), is said to be positively

Measures of Dispersion 81

Figure 5.3: Skewed distributions

Two distributions may have the same mean and the same standard deviation but they may be differently skewed. This will be obvious if you look at one of the skew distributions and then look at the same one through from the other side of the paper! What, then, does skewness tell us? It tells us that we are to expect a few unusually high values in a positively skew distribution or a few unusually low values in a negatively skew distribution.

If a distribution is symmetrical, the mean, mode and the median all occur at the same point, i.e. right in the middle. But in a skew distribution the mean and the median lie somewhere along the side with the "tail", although the mode is still at the point where the curve is

highest. The more skew the distribution, the greater the distance from the mode to the mean and the median, but these two are always in the same order; working outwards from the mode, the median comes first and then the mean, as in Figure 5.4:

Figure 5.4: Measures of location in skew distributions

For most distributions, except for those with very long tails, the following relationship holds approximately:

82 Measures of Dispersion

The more skew the distribution, the more spread out are these three measures of location, and so we can use the amount of this spread to measure the amount of skewness. The most usual way of doing this is to calculate:

Pearson's first coefficient of skewness =

deviation standard

mode - mean

However, the mode is not always easy to find and so we use the equivalent formula: Pearson's second coefficient of skewness =

deviation standard

median) -

3(mean

(See the next study unit for more details of Pearson.)

You are expected to use one of these formulae when an examiner asks for the skewness (or coefficient of skewness) of a distribution. When you do the calculation, remember to get the correct sign (+ or) when subtracting the mode or median from the mean and then you will get negative answers for negatively skew distributions, and positive answers for positively skew distributions. The value of the coefficient of skewness is between3 and +3, although values below1 and above +1 are rare and indicate very skewed distributions.

Examples of variates (variables) with positive skew distributions include size of incomes of a large group of workers, size of households, length of service in an organisation, and age of a workforce. Negative skew distributions occur less frequently. One such example is the age at death for the adult population of the UK.

SUMMARY

The definitions and formulae introduced in this study unit are very important in statistical analysis and interpretation and you are likely to get questions on them in your examination. You should learn all the formal definitions and the formulae thoroughly. Make sure you know when each of the formulae should be used, and that you can distinguish between those formulae which are only a statement of the definition in symbols and those which are used to calculate the measures.

Remember that even if you are allowed to use a pocket calculator, you must show all the steps in a calculation.

Examination questions will often ask you to comment on the results you have obtained. You will be able to make sensible comments if you have studied the use and the advantages and disadvantages of each of the measures.

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Study Unit 6