The compilation of these Measures of Dispersion Notes makes students exam preparation simpler and organised.
Relative Measures of Dispersion and Lorenz Curve
Today we will be studying here relative measures of dispersion which help us to compare different distributions. Additionally, we will study a graphical measure of dispersion, called the Lorenz curve, which finds extensive use in the representation of the distribution of income, wealth, profit, wages, and so on.
Relative Measures of Dispersion
The absolute measures of dispersion accurately display the dispersion of data in a given series. However, a drawback to them is that we cannot use them to compare dispersion for series that are in different units. The concept of relative measures of dispersion overcomes this limitation.
Such measures express the scattering of data in some relative terms or in percentage. Since they are devoid of a specific unit, the comparison between different series is hence possible. For every absolute measure of dispersion, there is a relative measure.
In other words, we can derive a relative measure by the ratio of absolute variability to the mean value or by the percentage of absolute variability. They are also known as coefficients of dispersion. The relative measures of dispersion which we will study are:
- Coefficient of range
- Coefficient of quartile deviation
- The coefficient of mean deviation
- Coefficient of standard deviation and coefficient of variation
Coefficient of Range
The coefficient of the range is calculated as the ratio of the difference between the highest and the lowest values and the sum of the highest and lowest values of the series. The formula is as follows:
Coefficient of Range = (H – L) ÷ (H + L)
Here, H = The highest value
L = The lowest value
Coefficient of Quartile Deviation
The coefficient of quartile deviation is calculated using the following formula:
Coefficient of Quartile Deviation = (Q3 – Q1) ÷ (Q3 + Q1)
Here, Q3 = The third quartile and
Q1 = The first quartile.
Thus, for the calculation of the coefficient of quartile deviation, we should know how to calculate the first and the third quartiles.
The First and the Third Quartile
Individual Series: We calculate the first and third quartile for individual series using the given formulae:
Q1 = Size of (N + 1)/4 th item and Q3 = Size of 3(N + 1)/4 th item.
Note that here N is the number of observations.
Discrete Series: For discrete series, we calculate the first and third quartile as follows: Initially, a column of cumulative frequency for each observation is constructed. Further, we find out the values of (N + 1)/4 and 3(N + 1)/4 depending on whether we want to calculate Q1 or Q3 respectively.
Here, N is the summation of frequencies. The cumulative frequency corresponding to the observation just greater than the value (N + 1)/4 or 3(N + 1)/4 is termed as the first or third quartile respectively.
Frequency Distribution Series: The calculation of first and third quartile, in this case, commences with the identification of the respective quartile class. For this purpose, we calculate the value of N/4 or 3(N/4) for first and third quartile respectively. Here, N is the summation of frequencies.
The cumulative frequency corresponding to the observation just greater than the value N/4 or 3(N/4) is termed as the first or third quartile class respectively. Lastly, we apply the following formulae:
Q1 = l + h/f [ N/4 – C] and Q3 = l + h/f [3(N/4) – C]
Note that, l is the lower limit of the respective quartile class, ‘h’ is the size of class intervals, f is the frequency corresponding to the respective quartile class and C is the cumulative frequency corresponding to the interval just before the quartile class.
As a matter of fact, Q2 or the second quartile is also termed as the median. It is calculated exactly similarly to Q1 or Q3, except for the usage of the term N/2.
Coefficient of Mean Deviation
As already mentioned, we can derive relative measure by the division of the absolute measure of variability with the corresponding average. On these lines, we can calculate the coefficient of mean deviation about mean, median or mode.
Coefficient of mean deviation about Mean = (mean deviation about mean)/ arithmetic mean
The coefficient of mean deviation about Median = (mean deviation about median)/ median
Coefficient of mean deviation about Mode = (mean deviation about mode)/ mode
Coefficient of Standard Deviation
The coefficient of standard deviation is simply the ratio of the standard deviation of a series to its arithmetic mean. Mathematically:
Coefficient of Standard deviation = σ/Mean
Here, σ = Standard deviation for the series
Coefficient of Variation
The coefficient of variation is 100 times the coefficient of standard deviation. In other words, the coefficient of standard deviation multiplied by 100 results in the coefficient of variation. In essence:
Coefficient of variation = (Coefficient of standard deviation) × 100
Lorenz Curve
Lorenz curve is a type of absolute measure of dispersion. Unlike others, it is a graphical measure of dispersion. Lorenz curve graphically represents the actual curve and a line of equal distribution and exhibits the deviation between these two.
By and large, the deviation of an actual curve from the line of equal distribution is termed the Lorenz coefficient. The larger is the distance of the Lorenz curve from the line of equal distribution, the greater is the Lorenz coefficient along with the degree of scattering and inequality within the distribution.
Constructing Lorenz Curve
Initially, we convert the series into a cumulative frequency series i.e. an individual column for cumulative frequency is constructed. Likewise, another column for the cumulative sun of observations is constructed (mid-values of intervals for frequency distribution series).
Further, these cumulative sums of frequencies and observations are converted into percentages of the respective sum using the formulae below:
The cumulative percentage for observation = (Cumulative sum corresponding to the observation ÷ Total sum of observations) × 100
The cumulative percentage for frequency = (Cumulative sum corresponding to the observation ÷ Total sum of frequency) × 100
Now we plot cumulative frequencies and cumulative items on X-axis and Y-axis respectively. On both axes, values start from 0 to 100. Next, we need to draw the line of equal distribution. This is a straight line with an inclination of 45° to both axes, joining the origin to the point (100, 100)
Lastly, we plot the values of cumulative sums, which represent the Lorenz curve. A large gap of the Lorenz curve from the line of the equal distribution represents a large variability in the series.
Example:
Question:
Calculate the coefficient of range for the following series.
Answer:
H = Highest value = 22
L = Lowest value = 10
Coefficient of Range = (22 – 10)/(22 + 10)
= 12/32
= 0.37