Quartile deviation. • Coefficient of quartile deviation. Let us begin the concept of
DISPERSION. Just as variable series differ with respect to their location on the ...
• • • • • •
LECTURE # 27 Measures of Dispersion Concept of dispersion Absolute and relative measures of dispersion Range Coefficient of dispersion Quartile deviation Coefficient of quartile deviation
Let us begin the concept of DISPERSION. Just as variable series differ with respect to their location on the horizontal axis (having different ‘average’ values); similarly, they differ in terms of the amount of variability which they exhibit. Let us understand this point with the help of an example: EXAMPLE: In a technical college, it may well be the case that the ages of a group of first-year students are quite consistent, e.g. 17, 18, 18, 19, 18, 19, 19, 18, 17, 18 and 18 years. A class of evening students undertaking a course of study in their spare time may show just the opposite situation, e.g. 35, 23, 19, 48, 32, 24, 29, 37, 58, 18, 21 and 30. It is very clear from this example that the variation that exists between the various values of a data-set is of substantial importance. We obviously need to be aware of the amount of variability present in a data-set if we are to come to useful conclusions about the situation under review. This is perhaps best seen from studying the two frequency distributions given below:
EXAMPLE: The sizes of the classes in two comprehensive schools in different areas are as follows:
Number of Pupils 10 – 14 15 – 19 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 - 49
Number of Classes Area A Area B 0 3 13 24 17 3 0 0 60
5 8 10 12 14 5 3 3 60
If the arithmetic mean size of class is calculated, we discover that the answer is identical: 27.33 pupils in both areas. Average class-size of each school
X = 27.33
Even though these two distributions share a common average, it can readily be seen that they are entirely DIFFERENT. And the graphs of the two distributions (given below) clearly indicate this fact.
Number of Classes
25
Area A
20 15 10
Area B
5 0 4–
9 10
–1
4 15
4 4 9 4 9 4 9 9 –1 0 –2 5 –2 0 –3 5 –3 0 –4 4 -4 0 –5 4 5 4 3 3 2 2
Number of Pupils The question which must be posed and answered is ‘In what way can these two situations be distinguished?’ We need a measure of variability or DISPERSION to accompany the relevant measure of position or ‘average’ used. The word ‘relevant’ is important here for we shall find one measure of dispersion which expresses the scatter of values round the arithmetic mean, another the scatter of values round the median, and so forth. Each measure of dispersion is associated with a particular ‘average’. Absolute versus Relative Measures of Dispersion: There are two types of measurements of dispersion: absolute and relative. An absolute measure of dispersion is one that measures the dispersion in terms of the same units or in the square of units, as the units of the data. For example, if the units of the data are rupees, meters, kilograms, etc., the units of the measures of dispersion will also be rupees, meters, kilograms, etc. On the other hand, relative measure of dispersion is one that is expressed in the form of a ratio, co-efficient of percentage and is independent of the units of measurement. A relative measure of dispersion is useful for comparison of data of different nature. A measure of central tendency together with a measure of dispersion gives an adequate description of data. We will be discussing FOUR measures of dispersion i.e. the range, the quartile deviation, the mean deviation, and the standard deviation. RANGE: The range is defined as the difference between the two extreme values of a data-set, i.e. R = Xm – X0 where Xm represents the highest value and X0 the lowest. Evidently, the calculation of the range is a simple question of MENTAL arithmetic. The simplicity of the concept does not necessarily invalidate it, but in general it gives no idea of the DISTRIBUTION of the observations between the two ends of the series. For this reason it is used principally as a supplementary aid in the description of variable data, in conjunction with other measures of dispersion. When the data are grouped into a frequency distribution, the range
is estimated by finding the difference between the upper boundary of the highest class and the lower boundary of the lowest class. We now consider the graphical representation of the range:
f
X X0
Xm Rang
Obviously, the greater the difference between the largest and the smallest values, the greater will be the range. As stated earlier, the range is a simple concept and is easy to compute. However, because of the fact that it is computed from only the two extreme values in a data-set, it has two serious disadvantages. 1. It ignores all the INFORMATION available from the intermediate observations. 2. It might give a MISLEADING picture of the spread in the data. From THIS point of view, it is an unsatisfactory measure of dispersion. However, it is APPROPRIATELY used in statistical quality control charts of manufactured products, daily temperatures, stock prices, etc. It is interesting to note that the range can also be viewed in the following way: It is twice of the arithmetic mean of the deviations of the smallest and largest values round the mid-range i.e.
(Midrange − X 0 ) + (X m − Midrange ) 2 Midrange − X 0 + X m − Midrange = 2 X − X0 = m 2
Because of what has been just explained, the range can be regarded as that measure of dispersion which is associated with the mid-range. As such, the range may be employed to indicate dispersion when the mid-range has been adopted as the most appropriate average. The range is an absolute measure of dispersion. Its relative measure is known as the COEFFICIENT OF DISPERSION, and is defined by the relation given below:
Coefficient of Dispersion:
=
1 2
(Range )
Mid − Range Xm − X0 X − X0 2 = = m X m + X0 Xm + X0 2 This is a pure (i.e. dimensionless) number and is used for the purposes of COMPARISON. (This is so because a pure number can be compared with another pure number.) For example, if the coefficient of dispersion for one data-set comes out to be 0.6 whereas the coefficient of dispersion for another data-set comes out to be 0.4, then it is obvious that there is greater amount of dispersion in the first data-set as compared with the second. QUARTILE DEVIATION: The quartile deviation is defined as half of the difference between the third and first quartiles i.e.
Q.D. =
Q3 − Q1 2
It is also known as semi-interquartile range. Let us now consider the graphical representation of the quartile deviation:
f
X Q1
Inter-quartile
Q3
Quartile Deviation (Semi Inter-quartile Range) Although simple to compute, it is NOT an extremely satisfactory measure of dispersion because it takes into account the spread of only two values of the variable round the median, and this gives no idea of the rest of the dispersion within the distribution. The quartile deviation has an attractive feature that the range “Median + Q.D.” contains approximately 50% of the data. This is illustrated in the figure given below:
f
50%
Median-Q.D.
Median Median+Q.D.
X
Let us now apply the concept of quartile deviation to the following example: EXAMPLE:
The shareholding structure of two companies is given below:
1st quartile Median 3rd quartile
Company X 60 shares 185 shares 270 shares
Company Y 165 shares 185 shares 210 shares
The quartile deviation for company X is
270 − 60 = 105 2
Shares
For company Y, it is
210 − 165 = 22 2
Shares
A comparison of the above two results indicate that there is a considerable concentration of shareholders about the MEDIAN number of shares in company Y, whereas in company X, there does not exist this kind of a concentration around the median. (In company X, there is approximately the SAME numbers of small, medium and large shareholders.) From the above example, it is obvious that the larger the quartile deviation, the greater is the scatter of values within the series. The quartile deviation is superior to range as it is not affected by extremely large or small observations. It is simple to understand and easy to calculate. The mean deviation can also be viewed in another way: It is the arithmetic mean of the deviations of the first and third quartiles round the median i.e.
(M − Q1 ) + (Q3 − M ) 2 M − Q1 + Q3 − M = 2 Q − Q1 = 3 2 Because of what has been just explained, the quartile deviation is regarded as that measure of dispersion which is associated with the median. As such, the quartile deviation should always be employed to indicate dispersion when the median has been adopted as the most appropriate average. The quartile deviation is also an absolute measure of dispersion. Its relative measure called the CO-EFFICIENT OF QUARTILE DEVIATION or of Semi-Inter-quartile Range, is defined by the relation: Coefficient of Quartile Deviation:
Quartile Deviation Mid − Quartile Range Q3 − Q1 Q − Q1 2 = 3 = , Q3 + Q1 Q3 + Q1 2 =
The Coefficient of Quartile Deviation is a pure number and is used for COMPARING the variation in two or more sets of data. The next two measures of dispersion to be discussed are the Mean Deviation and the Standard Deviation. In this regard, the first thing to note is that, whereas the range as well as the quartile deviation are two such measures of dispersion which are NOT based on all the values, the mean deviation and the standard deviation are two such measures of dispersion that involve each and every data-value in their computation. The range measures the dispersion of the data-set around the mid-range, whereas the quartile deviation measures the dispersion of the data-set around the median. How are we to decide upon the amount of dispersion round the arithmetic mean? It would seem reasonable to compute the DISTANCE of each observed value in the series from the arithmetic mean of the series. But the problem is that the sum of the deviations of the values from the mean is ZERO! (No matter what the amount of dispersion in a data-set is, this quantity will always be zero, and hence it cannot be used to measure the dispersion in the data-set.) Then, the question arises, ‘HOW will we be able to measure the dispersion present in our dataset?’ In an attempt to answer this question, we might look at the numerical differences between the mean and the data values WITHOUT considering whether these are positive or negative. By ignoring the sign of the deviations we will achieve a NON-ZERO sum, and averaging these absolute differences, again, we obtain a non-zero quantity which can be used as a measure of dispersion. (The larger this quantity, the greater is the dispersion in the data-set). This quantity is known as the MEAN DEVIATION. Let us denote these absolute differences by ‘modulus of d’ or ‘mod d’. Then, the mean deviation is given by MEAN DEVIATION:
M.D. =
∑| d | n
As the absolute deviations of the observations from their mean are being averaged, therefore the complete name of this measure is Mean Absolute Deviation --- but generally, it is simply called “Mean Deviation”. In the next lecture, this concept will be discussed in detail. (The case of raw data as well as the case of grouped data will be considered.)Next, we will discuss the most important and the most widely used measure of dispersion i.e. the Standard Deviation.
