Measures of Dispersion Test - 4

The Lorenz curve is a graphical representation of the distribution of income or of wealth. It was developed by Max O Lorenz in 1905 for representing inequality of the wealth distribution.

As it is calculated from sum of squared deviations which can never be negative.

The symbol for a population standard deviation is σ .Population standard deviation looks at the square root of the variance of the set of numbers. It's used to determine a confidence interval for drawing conclusions

Q.D= (Q₃-Q₁)\2= 350\2= 125

Q₃=[3(n+1)\4]^th observation= 575

Q₁=  [(n+1)\4]^th observation= 225

M.D= Σ|X-x̅|/N= 1570/9= 174.444

As it is not much affected by extreme situations.

It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution.

Quartile Deviation (QD) means the semi variation between the upper quartiles (Q3) and lower quartiles (Q1) in a distribution. The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. Standard deviation is a measure of the dispersion of a set of data from its mean. It is calculated as the square root of variance by determining the variation between each data point relative to the mean. If the data points are further from the mean, there is higher deviation within the data set.

It posseses most of the features laid down for an ideal measure of dispersion.

The formula for standard deviation (SD) is:

Step 1: Find the mean.

Step 2: For each data point, find the square of its distance to the mean.

Step 3: Sum the values from Step 2.

Step 4: Divide by the number of data points.

Step 5: Take the square root.

This formula is for individual series where, f=1,for each value of the variable.

we know that

For variance, apply a squared symbol (s² or σ²). The symbol for variance is represented by the Greek symbol sigma squared, which looks like this. The formula of population variance is sigma squared equals the sum of x minus the mean squared divided by n

This is a relative measure of dispersion and is based on the value of the range. It is also called range coefficient of dispersion.Co-efficient of range= L-S/L+S= 7/15= 0.4667

Standard deviation for a given set of observations is defined as root mean square deviation when deviations are taken from mean.

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. Dispersion is contrasted with location or central tendancy, and together they are the most used properties of distributions.

Arrange in ascending order, 384, 391,407, 522, 591, 672, 733, 777, 1490, 2488

N= 10

Quartile deviation = (Q₃-Q₁)/2= (955.25-403)/2= 276.125

Q₃= [3(N+1)/4]^th observation= [3(11)/4]^th observation= [8.25]^th  observation= 8^th observation + 0.25*713=777+178.25= 955.25

Q1= [(N+1)/4]^th observation=  [(11)/4]^th observation= [2.75]^th  observation= 2^nd observation+ 0.75*16= 391+12= 403

Q3= 575

coefficient of QD=[(Q3-Q1)/ (Q3+Q1)]=[350\800]=.44

Here, maximum observation =166 and minimum observation =158.

R= L-S, Where L is largest item and S is smallest item

R=19-(-10)= 19+10= 29

Now, N/4=357/4=89.5

C.f just greater than 89. 5 is 237 

Thus, 500.5-650.5 is quartile class

Range is the simplest measure of dispersion. It is defined as the difference between largest and smallest values in the series.  

R=L-S= 75-60= 15

coefficient of range= L-S/L+S= 15/135= .1111

A fundamental task in many statistical analyses is to estimate a location parameter for the distribution; i.e., to find a typical or central value that best describes the data. Measures of location describe the central tendency of the data. They include the mean, median and mode. A measure of dispersion quantifies how much persons in the group vary from each other and from our measure of central location. Measures of dispersion describe the spread of the data. They include the range, interquartile range, standard deviation and variance.

R= L-S where L is largest value and S is smallest value

R= 11-4= 7

It is a square of standard deviation and square of anything is non negative.

coefficient of variation= σ/x̅, where σ= standard deviation and x̅ is mean

5= σ/10

σ=10*5=50