Measures of Dispersion Test - 3

variance is unaffected due to change of origin.

The Lorenz curve, developed by Max O Lorenz in 1905,  is a graphical representation of the distribution of the income or of wealth. It is used to  represent inquality of the wealth distribution.

It should be based on all the items so as to be more representative of whole series

A perfectly equal income distribution would be one in which every person has the same income. In this case, the bottom N% of society would always have N% of the income. This can be depicted by the straight line y = x; called the "line of perfect equality."

Since variance is a relative measures which is free from the units in which the values have been expressed, they can be compared even across different groups having different units of measurement.

Change in variance is the square of change in scale.

It is a absolute measure of dispersion

Measures of dispersion attempt to describe the spread of the data values in a data set. Dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

Relative measures of dispersion, are also known as coefficients of dispersion, are obtained as ratios or percentages. The coefficient of variation (CV) is the ratio of the standard deviation to the mean (average).

Quartile deviation or Semi-interquartile range is one-half the difference between the first and the third quartiles. (QD = Q3-Q1/2)

We find the mean of the square of the differences. The number that we get  is called the variance. The standard deviation can be calculated by taking the square root of the variance. Formula is :

The Standard Deviation is a measure of how spread out numbers are. Its symbol is σ (the greek letter sigma). The formula is easy: it is the square root of the Variance. So the square of σ is variance.

Absolute measure of dispersion include range, quartile deviation, standard deviation, variance and relative measure of dispersion include coefficient of range, coefficient of quartile deviation, coefficient of standard deviation, coefficient of variance.

These measures give us an idea about the amount of dispersion in a set of observations. They give the answers in the same units as the units of the original observations. When the observations are in kilograms, the absolute measure is also in kilograms.

Variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value.The variance is the square of the standard deviation, the second central moment of a distribution, and the co variance of the random variable with itself.

It is very hard for a person to calculate square root and even square of deviations. Formula of standard deviation is very hard to remember.

In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartile. IQR = Q₃ −  Q₁

sum of squared deviations of observations = ∑(x_i-μ)²

μ=  ∑X_i/N = (3+4+5+6+7)/5= 25/5= 5

(x_i-μ)= -2, -1, 0, 1, 2

 ∑(x_i-μ)²​​​​​​​= 4+1+0+1+4= 10

It is easy to calculate because of a simple formula R= L-S where L is largest observation and S is smallest observation.

It is based on the lower quartile Q1 and the upper quartile Q3. The difference Q3−Q1 is called the inter quartile range. The difference  Q3−Q1 divided by 2 is called semi-inter-quartile range or the quartile deviation.

As it is not much affected by extreme situations.

Standard Deviation is the positive square root of the mean of squared deviations from mean. So if there are five values x1, x2, x3, x4 and x5, first their mean is calculated. Then deviations of the values from mean are calculated. These deviations are then squared. The mean of these squared deviations is the variance. Positive square root of the variance is the standard deviation.

Its a step deviation method to calculate standard deviation.

Standard deviation is square root of variance which is also knows as mean square deviation.

As the variance is mean of sum of square deviation so unit will be in squared cm.