Q.8. What do you mean by dispersion? Enlist the important measures of dispersion. (2005-06)
Or Define dispersion. Why standard deviation is most widely used measure of dispersion? (2014-15
Ans. Dispersion and Measures of Dispersion: Also known as spread or variation, dispersion 15 defined as measure of extent to which the items vary from some central value. The degree to wn numerical data tends to spread about value is called measure of dispersion of the data. It measures on the degree but not the direction of the variation. Different sets of data may have the same measure central tendency but differ greatly in terms of variation. A good measure of dispersion should possess the same properties as that of a good measure of central tendency.
Some Important Measures of Dispersion
Some well known measures of dispersion which provide a numerical index of the variability of the given data are:
1. Range: It is the simplest method of studying dispersion which represents the difference between the value of the extremes-the largest value and the smallest value.
R=L-S [L = Largest value, S = Smallest value]
2. Mean Deviation: It is a more sensitive measure of dispersion that has the advantage of giving equal weight to the deviation of every value from the mean or median.
M.D. – = Ef|xi – x/N
[x = Mean, N = No. of observations]
3. Quartile Deviation: It is easy to calculate and understand and is unreliable if there are gaps in the data around the quartiles. It is superior to the range as a rough measure of dispersion.
Q.D. – Q3 – Q1/2
[Q3 = 3rd quartile,Q, = 1st quartile]
4. Standard Deviation (6): This is the most widely used and important measure of dispersion, where in computing the average deviation, the signs are ignored. S.D. overcomes this problem by squaring the deviations which makes them all positive.
a = E(x – x)2/2
[X = Mean, N= No. of observations]
Q.9. Distinguish between measures of central tendency and dispersion. Illustrate with the help of examples.(2006-07)
Ans. Measures of central tendency refers to a single value that attempts to describe a set of data by identifying the central position within that set of data. It is a typical value around which other figures congregate. For example, the monthly income of 5 persons is 132, 140, 144, 136 and 148.
Arithmetic mean can be calculated as,
Thus, statistical measures indicate the location or position of a control value to describe the central tendency of the entire data.
Measures of dispersion can be defined as the degree to which observations are clustered around a line of best fit. It gives the variability of data and depicts the spread of data. These measures are independent of units of measurement. The measure of dispersion include various methods like range, mean deviation, standard deviation, etc. and can be explained as under:
For example, the net profit of a business concern in thousands of rupees is given below:
Q.10. ‘Measures of central tendency, dispersion and skewness are complementary to one another in understanding a frequency distribution.’ Elucidate. (2005-06)
Ans. Measures of central tendency or averages, represent a whole series and its value always lies in between the minimum and maximum values and generally it is located in the centre or middle of the Frequency distribution containing its major characteristics. Condensation of data is necessary in statistical analysis because a large number of big figures are not only confusing for the mind but are also lifficult to analyse also.
Dispersion or spread is the degree of the scatter or variation of the variable about a central va Dispersion not only gives a general impression about the variability of a series, but also a precise meas of this variation. In a precise study of dispersion, the deviations of size of items from a measure of cent tendency are found out and then these deviations are found out and after that these deviations averaged to give a single figure representing the dispersion of the series.
The empirical relations between various averages/measures of central tendency and measures dispersion hold good only in asymmetrical distribution. A normal curve is a bell shaped frequency cury in which the values on either side of a measure of central tendency are symmetrical. Skewness is opposite of symmetry and its presence tells us that a particular distribution is not symmetrical or in other words it is skewed. Thus, measures of central tendency tell us about the concentration of items around the central value and measures of skewness tell us whether the dispersal of items from average is symmetrical.
Fig. Distribution of data.
Taken together, all the three aspects will give us a more comprehensive idea about the given series.
Skewness = Mean – Mode
Coefficient of skewness = Mean – Mode/Standard deviation
Thus, from the above discussion we can say that ‘Measures of central tendency, dispersion and skewness are complementary to one another in understanding a frequency distribution.’
Q.11. Define kurtosis.
Ans. Kurtosis: The peakedness of the frequency distribution is another characteristic which might be measured. The arithmetic mean of two or more series may be the same but dispersion of their items may be different. Kurtosis defines whether a distribution is more flat or more peaked than the normal distribution, it studies the concentration of items about the central part of the series. If the item concentrate too much at the centre, the curve becomes leptokurtic and if the concentration in the centre is comparatively little, the curve becomes platykurtic.
The coefficientß, is used as kurtosis. The measures of kurtosis is given as:
Kurtosis = B2 – 3
Where the results are:
1. If kurtosis is zero, the curve is mesokurtic, i.e. normal curve.
2. If kurtosis is a positive value, i.e. B,>3 the curve is
leptokurtic and curve is more peaked.
3. If kurtosis is a negative value, i.e. B, <3, the curve is platykurtic. Kurtosis is not much used so far as sociological studies are concerned. It is mainly used in biological studies.
Q.12. Distinguish between skewness and kurtosis.(2015-16)
Or What is meant by skewness? How is it measured?(2007-08, 11-12)
Or Write the differences between skewness and kurtosis.(2012-13)
Ans. Skewness: The measures of central tendency and variation do not reveal all the characteristics of a given set of data. For example, two distributions may have the same mean and standard deviation but may differ widely in the shape of their distribution. Either the distribution of data is symmetrical or it is not. If the distribution of data is not symmetrical, it is called asymmetrical or skewed. Thus, skewness refers to the lack of symmetry in distribution.
Measuring Skewness: The measure of skewness tells us the direction of dispersion about the centre of the distribution. tstanding the tendency for Skewness refers to the lack of symmetry in distribution.
A simple method of finding the direction of skewness is to points into a certain consider the tails of a frequency polygon.The concept of skewness is clear from the figures showing symmetrical, positively skewed and negatively skewed distributions,
Fig. Distribution of data.
It is clear that data are symmetrical when the spread of the frequencies is same on both sides of the middle point of frequency polygon.
Thus, Mean = Median = Mode
When the distribution is not symmetrical, it is said to be skewed-positively skewed or negatively skewed. In such a situation either,
Mean > Median > Mode or Mean < Median < Mode
Skewness is measured using the formula:
Skewness = (3 * (mean – median))/Standard deviation. Where standard deviation shows how far the numbers spread out from the mean and median.
Kurtosis: Refer to Sec-B, Q.11.