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Maths Exercices
 
Statistics

Central Tendency Measures
Measures of Dispersion
Measures of skewness and kurtosis

Range is the difference between the highest value and the lowest value of the given set of observations:

Range = maximum value - minimum value.

The heights of a sample of five people are 180, 183, 190, 179 and 180 cm. Find the range.

Maximum value = 190
Minimum value = 179
Range = 190 - 179 = 11

Properties.

  • It is easy to understand.
  • It is simple to calculate.
  • It does not depend on all observations, and is based on only the largest and the smaller among them.
  • It is highly affected by extreme values.
  • It does not take into account the form of the distribution.
The relative measure of range, also called coeffcicient of range, is defined as:

where:

  • L is the largest value.
  • S is the smallest value.

Let us take two sets of observations. Set A contains marks of five students in Algebra out of 25 marks and group B contains marks of the same student in Geometry out of 100 marks.

Set A:   15, 16, 18, 21, 25
Set B:   35, 35, 40, 45, 55
The values of range and coefficient of range are calculated as:

 

Range

Coefficient of Range

Algebra

25 - 15 = 10

Geometry

55 - 35 = 20

            In set A the range is 10 and in set B the range is 20. Apparently it seems as if there is greater dispersion in set B. But this is not true. The range of 20 in set B is for large observations and the range of 10 in set A is for small observations. Thus 20 and 10 cannot be compared directly. Their base is not the same. Marks in Algebra are out of 25 and marks of Geometry are out of 100. Thus, it makes no sense to compare 10 with 20. When we convert these two values into coefficient of range, we see that coefficient of range for set A is greater than that of set B. Thus there is greater dispersion or variation in set A. The marks of students in Geometry are more stable than their marks in Algebra.


The interquartile range (IQR) is the distance covered by the middle 50 percent of the distribution. The interquatile range is the difference between the third quartile, Q3 and the first quartile, Q1.

IQR = Q3 - Q1

The interquartile range is especialy useful in situations where data users are more interested in values towards the middle and less interested in extremes.


For the following data, find the interquartile range:
1 2 2 3 3 4 4 5 5 6 6 7 8 8 8
IQR = Q3 - Q1 = 7 - 3 = 4

The semi-interquartile range (also called quartile deviation) is half of the quartile range: