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  • The following data was generated from 20 years profit data of Company XY. Management need you to make an economic analysis and require you to calculate

 

  1. Mean
  2. Median
  3. Mode
  4. Range
  5. Q1, D5, P50
  6. Interquartile range
  7. Mean deviation from the mean
  8. Mean deviation from the median

 

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69
24
37
75
86
51
37
65
21
67
46
80
13
87
38
26
81
93
87
54

 

  1. Mean
  2. Median
  3. Mode
  4. Range
  5. D2, P60
  6. Interquartile range
  7. Mean deviation from the mean
  8. Mean deviation from the median

 

Grade # of students
10 2
20 5
30 20
40 18
50 40
60 35
70 41
80 25
90 10
100 4

 

 

3) The following data on income of residents generated from County X

 

Min Max No. of people
1000 2000 28
2000 3000 52
3000 4000 45
4000 5000 35
5000 6000 63
6000 7000 130
7000 8000 110
8000 9000 62
9000 10000 55
10000 11000 38
11000 12000 35
12000 13000 50
13000 14000 55
14000 15000 88
15000 16000 60
16000 17000 35
17000 18000 30
18000 19000 22
19000 20000 7

 

  1. Find the average, median and mode income of the residents
  2. Find interquartile range
  3. Find the mean deviation from the mean and median

MEASURES OF DISPERSION

Various measures of central tendency give us one single figure that represents the entire data. But the average alone can’t adequately describe a set of observations, unless all the observations are the same. It is necessary to describe the variability or dispersion of the observations. Dispersion (also known as scatter, spread or variation) measures the extent to which the items vary with the central value. Since measures of dispersion give an average of the differences of various item from an average, they are also called average of the ‘second order’.

Measures of dispersion are needed for four basic purposes

  1. To determine the reliability of an average.
  2. To serve as a basis for the control of the variability.
  3. To compare two or more series with regard to their variability.
  4. To facilitate the use of other statistical measures.

The following are the important methods of studying variation.

  1. The Range
  2. The interquartile Range and quartile deviation.
  3. Mean deviation or average deviation
  4. Standard deviation
  5. Lorenz curve.
  6. Skewness and Kurtosis

 

Of these the first two, namely the range and quartile deviations are positional measures, because they depend on the values at a particular position in the distribution. The average deviation and standard deviation are called calculated measures of deviation and the Lorenz curve is a graphic method.

  1. RANGE

Range is the simplest method of studying dispersion. It is defined as the difference between the value of smallest item (S) and the value of largest item ( L) included in thedistribution.

 

Range is useful in studying the variations in the prices of stocks, shares and other commodities that are sensitive to price changes from one period to another period.

 

The meteorological department uses the range for weather forecasts since public is interested to know the limits within which the temperature is likely to vary on a particular day.

 

Range = L – S

 

 

 

2) INTER –QUARTILE RANGE OR QUARTILE DEVIATION

 

By eliminating the lowest 25% and the highest 25% of items in a series, we are left with the central 50% which are ordinarily free of extreme values. To obtain a measure of variation, we use the distance between the first and the third quartiles.

 

Inter quartile range is computed by deducting the value of the first quartile from the value of third quartile.

 

Interquartile range =  Q3 – Q1

 

Semi interquartile range or quartile deviation is defined as half of the distance between the

third and first quarter.

 

Quartile Deviation = (Q3 – Q1) / 2

 

  1. Individual observation –

 

You have observed the following data on Company X stock values during the last 9 years.

 

Year 1 2 3 4 5 6 7 8 9
Stock price A 17 43 25 33 19 22 35 41 29
Stock Price B 18 16 14 15 22 19 24 20 16

 

 

Calculate the range, interquartile range, & quartile deviation 

 

 

  1. Continuous series

 

Wages        Number of employees

30 – 32                                    12

32 – 34                                    18

34 – 36                                    16

36 – 38                                    14

38 – 40                                    12

40 – 42                                     8

42 – 44                                     6

 

 

Calculate the range, interquartile range, & quartile deviation 

 

 

 

3) Mean Deviation from the mean and Median

 

The two methods of dispersion discussed above, namely range and quartile deviation, are not based on all observations. They are positional measures of dispersion.  They do not show any scatter of the observations from an average.  The mean deviation and standard deviations are based on all observations.

 

Mean deviation (M.D) is the average difference between the items in a distribution and the median or mean of that series.  Theoretically there is an advantage in taking the deviations from median because, the sum of deviations from the items from median is minimum when signs are ignored. However, in practice, the arithmetic mean is more frequently used in calculating the value of an average deviation and this is the reason why it is more commonly called mean deviation.

Mean Deviation = Σ|x − μ|/N

  • Σ is Sigma, which means to sum up
  • || (the vertical bars) mean Absolute Value, basically to ignore minus signs
  • x is each value
  • μ is the mean
  • N is the number of values

 Computation of M.D – Individual Observations

 

Steps :

  • Compute the mean or median of the series
  • Calculate the deviations of items from mean or median ignoring+ −signs
  • Obtain the total of these observations ie.,∑||
  • Divide the total obtained by the number of observation

 

Calculate M.D. from mean and median for the following data:

100, 150, 200, 250, 360, 490, 500, 600, 671

 

 

Calculation of M.D – Discrete series

In discrete series the formula of calculating mean deviation is :

Steps :

  • Calculate mean or median
  • Take the deviations of the item from mean or median ignoring signs
  • Multiply these deviations by respective frequencies and obtain the total∑||.
  • Divide this total by number of observations.

 

Example :

Calculate mean deviation from the following series

 

x 10 11 12 13 14

f 3 12 18 12 3

Calculation of M.D – Continuous Series

For calculation mean deviation, in continuous series, the procedure remains same as discussed above.  The only difference is that we have to obtain the –mid-point of various classes and take deviations of these points from mean or median.

Calculate M.D from mean for the following data

Class : 2-4 4-6 6-8 8-10

f: 3 4 2 1

 

4) Standard deviation

 

It is the positive square root of the mean of the squared deviations of the values from their mean

 

Characteristics of Standard Deviation:

  • The standard deviation is affected by the values of every observations.
  • The process of squaring the deviations before adding, avoids the algebraic fallacy of disregarding signs.
  • In general it is less affected by fluctuations of sampling than the other measures of dispersion.
  • It has a specific mathematical meaning and could be easily adapted according to nature of algebraic treatment
  • It has great practical utility in sampling and statistical inference.

 

Standard Deviation for individual observation

The standard deviation of a set of n values, denoted by S. The standard deviation for ungrouped data is mathematically represented as follows;

Find the standard deviation for the values 2, 3, 6, 8 and 11.

 

Standard Deviation for discrete series

In case of a frequency distribution with as class marks and as the corresponding class frequencies, the standard deviation shall be calculated by using the following formula;

 

Steps :

Calculate the mean of the series.

Find deviations for various items from mean,

Square the deviations  and multiply with respective frequencies (f),

Total the product and then apply the formula

 

Standard deviation for continuous series

In the continuous series the method of calculating standard deviation is almost same as in a discrete series.  But here, the mid values of class intervals are to be found out.

 

Calculate standard deviation of the following.

 

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70

No. of Students 5 12 30 45 50 37 21

 

Variance

Square of Standard deviation or the average of the squared differences from the Mean

 

 

Coefficient of Variation

Standard deviation is the absolute measure of dispersion.  It is expressed in terms of the units in which the original figures are collected and stated.  The relative measure of standard deviation is known as coefficient of variation.

 

A coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. It is calculated as follows: (standard deviation) / (expected value). The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.

 

Coefficient of Variation = (Standard Deviation / Mean) * 100.

The coefficient of variation allows investors to determine how much volatility, or risk, is assumed in comparison to the amount of return expected from investments. The lower the ratio of standard deviation to mean return, the better risk-return trade-off. Note that if the expected return in the denominator is negative or zero, the coefficient of variation could be misleading.

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