Measures of Variation

Most data sets have a characteristic in which the values are not all the same, or the same; in fact, the degree to which they differ from one another or vary among themselves is of fundamental significance in statistics. Think about the following instances: In a clinic where patients' pulse rates are recorded three times daily, patient A's pulse rate is 70, 78, and 74, while patient B's pulse rate is 70, 91, and 61. Although the two patients had the identical 74-beat-per-minute mean pulse rates, notice the variation in variability. While the heart rate of patient A is constant, the measure of variation measures how much the data in a data collection varies. There are various variants measurements, some of which include:

• Range
• Inter-quartile range
• Quartile Deviation

Range:

Let's look at the example above, where the pulse rate ranges between 70 and 78 in the case of patient A and between 61 and 91 in the case of patient B, to illustrate a straightforward method of evaluating variability. The variance between the two sets of data is shown by these extreme (smallest and largest) numbers, and if we take the distances between the corresponding extremes, we may obtain the same information. Let's therefore create the following definition:

The difference between the highest number and the smallest value in a piece of data is known as the range.

The range of the pulse rates for patients A and B was 78 to 70 = 8 and 91 to 61 = respectively, 8 and 30. The range encompasses all of the sample's values.

The range is regarded as a good indicator of variability when the sample size is relatively small. Therefore, it is frequently employed in quality control, where the variability of raw materials or final goods is continuously monitored. The weather forecast also makes advantage of the range.

Example 1

Problem:	Hari took 7 statistics tests in one marking period. What is the range of her test scores?
	89, 73, 84, 91, 87, 77, 94
Solution:	Ordering the test scores from least to greatest, we get:
	73, 77, 84, 87, 89, 91, 94
	highest - lowest = 94 - 73 = 21
Answer:	The range of these test scores is 21 points.

In the case of calculating the range for grouped data it only depends on data values but not on frequency.

Inter-Quartile Range:

The data in a data collection are divided into quartiles to create the interquartile range (IQR), which is a measure of variation.

An ordered data collection is divided into four equal portions using quartiles. The first, second, and third quartiles—abbreviated Q1, Q2, and Q3, respectively—are the values that separate each portion.

• Q1 is the mid value in the first half of the ordered data set.
• Q2 is the median value in the set.
• Q3 is the mid value in the second half of the ordered data set.

Therefore, interquartile range=Q3-Q1

Example 2:

Find interquartile range from the following frequency distribution table:

Q1=N/4=5 lies on the interval 10-20

Q1=\( L + \frac{\frac{N}{4}- c.f.}{f} \times h \), here i =1

=\(10 + \frac{5- 2}{3} \times 10 \)

=20

Q3=3N/4=15 lies in the interval 40-50

Q3=\( L + \frac{\frac{3N}{4}- c.f.}{f} \times h \), here i =3

=\( 40 + \frac{15- 12}{6} \times 10 \)

=45

IQR=Q3-Q1

=45-20

=25

Quartile Deviation:

It is based on the bottom and upper quartiles of Q1 and Q3, respectively. The interquartile range, as we are all aware, is the difference between Q3 and Q1. When divided by 2, the difference between Q3 and Q1 is known as the quartile deviation or semi-interquartile range. Thus,

Qd=\(\frac{Q3-Q1}{2}\)

It can also be written as

Qd=\(\frac{IQR}{2}\)

Compared to the range, the quartile deviation is a more accurate indicator of absolute dispersion. However, it doesn't take into account the observation on the tails.

Example 3:

Find interquartile range from the following frequency distribution table:

Q1=N/4=5 lies on the interval 10-20

Q1=\( L + \frac{\frac{N}{4}- c.f.}{f} \times h \), here i =1

=\(10 + \frac{5- 2}{3} \times 10 \)

=20

Q3=3N/4=15 lies on the interval 40-50

Q3=\( L + \frac{\frac{3N}{4}- c.f.}{f} \times h \), here i =3

=\( 40 + \frac{15- 12}{6} \times 10 \)

=45

IQR=Q3-Q1

=45-20

=25

Qd=\(\frac{Q3-Q1}{2}\)

=\(\frac{25}{2}\)

=12.5

Additionally, the quartile deviation coefficient can be calculated as follows:

Coefficient of Quartile Deviation=\(\frac{Q3-Q1}{Q3+Q1}\)

Reference:

• Kunda, Surinder. An Introduciton to business statistics. n.d.
• mathgoodies.com/lessons/vol8/range.html
• cims.nyu.edu/~kiryl/Elementary Statistics/Chapter_4.pdf