Measures of Variability

विचलनशीलता  का मान

major measures of variability, include

• range (प्रसार )
• interquartile range(अंतर चतुर्थांक प्रसार )
• quartile deviation(चतुर्थांक विचलन )
• standard deviation(माध्य विचलन )
• variance(प्रसरण )
• mean deviation(मानक विचलन )

Like us on Facebook https://www.facebook.com/focussgurgaon

A.  Range

The range for a set of data items is the difference between the largest and smallest

values. Although the range is the easiest of the numerical measures of variability

to compute, it is not widely used because it is based on only two of the items in the

data set and thus is influenced too much by extreme data values.

B. Quartile Deviation and interquartile Deviation:

• 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.

(1)The Inter-quartile Range (IQR): Range of the middle 50% of the distribution 

IQR =Q3 – Q1 

(2) Quartile Deviation (Q.D) = IQR/2 

Quartiles split the distribution into “quarters” (“fourths”) 

Q1 = value at 25%ile 

Q2 = value at 50%ile (median) 

Q3 = value at 75%ile 

Q4 = value at 100%ile 

To calculate:

(a) Arrange all scores from lowest to highest


(b) Find the median value location


(c) Find the quartile (Q1 and Q3) locations & quartiles


(d) Take the difference between Q1 and Q3


(e) Quartile Deviation (Q.D) = (Q3 – Q1 )/2

C. Mean Deviation

The mean deviation is the first measure of dispersion that we will use that actually uses each data value in its computation. It is the mean of the distances between each value and the mean. It gives us an idea of how spread out from the center the set of values is.

Here's the formula.

D. Variance

The Variance is defined as: The average of the squared differences from the Mean.

To calculate the variance follow these steps:

Work out the Mean (the simple average of the numbers)

Then for each number: subtract the Mean and square the result (the squared difference).

Then work out the average of those squared differences.

Deviation (d) = (X - m)

X	d =X-m	d
1	1 – 2	-1
0	0 – 2	-2
6	6 – 2	+4
1	1 - 2	-1

m = 2              Sd = 0

B/c the Sd is ALWAYS ZERO, the average of the deviations will also always be 0

We get around this problem by squaring the deviations

Variance: called a “mean square;” ave of the squared deviations around the mean

Population variance   

                                                        Step 1          Step 2

X	d =X-m	d	d2
1	1 – 2	-1	1
0	0 – 2	-2	4
6	6 – 2	+4	16
1	1 – 2	-1	1

Step 3:  Sd² = (1 + 4 + 16 + 1) = 22

Step 4: 22 / 4 = 5.5

        s² = 5.5

Large values of s² indicate more variability around m

E. Standard Deviation

The standard deviation (σ) is simply the (positive) square root of the variance.

Where 

x represents each value in the population, 

μ is the mean value of the population, 

Σ is the summation (or total), and 

N is the number of values in the population.