Interquartile Range and Quartile Deviation using NumPy and SciPy

Quartiles :
A quartile is a type of quantile. The first quartile (Q1), is defined as the middle number between the smallest number and the median of the data set, the second quartile (Q2) – median of the given data set while the third quartile (Q3), is the middle number between the median and the largest value of the data set.

Algorithm to find Quartiles :
Quartiles are calculated by the help of the median. If the number of entries is an even number i.e. of the form 2n, then, first quartile (Q1) is equal to the median of the n smallest entries and the third quartile (Q3) is equal to the median of the n largest entries.

If the number of entries is an odd number i.e. of the form (2n + 1), then

• the first quartile (Q1) is equal to the median of the n smallest entries
• the third quartile (Q3) is equal to the median of the n largest entries
• the second quartile(Q2) is the same as the ordinary median.

Range: It is the difference between the largest value and the smallest value in the given data set.
Interquartile Range :
The interquartile range (IQR), also called as midspread or middle 50%, or technically H-spread is the difference between the third quartile (Q3) and the first quartile (Q1). It covers the center of the distribution and contains 50% of the observations. IQR = Q3 – Q1

Uses :

• The interquartile range has a breakdown point of 25% due to which it is often preferred over the total range.
• The IQR is used to build box plots, simple graphical representations of a probability distribution.
• The IQR can also be used to identify the outliers in the given data set.
• The IQR gives the central tendency of the data.

Decision Making

• The data set has a higher value of interquartile range (IQR) has more variability.
• The data set having a lower value of interquartile range (IQR) is preferable.

Suppose if we have two data sets and their interquartile ranges are IR1 and IR2, and if IR1 > IR2 then the data in IR1 is said to have more variability than the data in IR2 and data in IR2 is preferable.

Example:

• Following are the number of candidates enrolled each day in last 20 days for the course –
  Data Structures & Algorithms-DSA Online 3 at Lazyroar:
  75, 69, 56, 46, 47, 79, 92, 97, 89, 88, 36, 96, 105, 32, 116, 101, 79, 93, 91, 112
• After sorting the above data set:
  32, 36, 46, 47, 56, 69, 75, 79, 79, 88, 89, 91, 92, 93, 96, 97, 101, 105, 112, 116
• Here the total number of terms is 20.
• The second quartile (Q2) or the median of the above data is (88 + 89) / 2 = 88.5
• The first quartile (Q1) is median of first n i.e. 10 terms (or n i.e. 10 smallest values) = 62.5
• The third quartile (Q3) is the median of n i.e. 10 largest values (or last n i.e. 10 values) = 96.5
• Then, IQR = Q3 – Q1 = 96.5 – 62.5 = 34.0

Interquartile range using numpy.median

Output: 34.0

Interquartile range using numpy.percentile

Output: 34.0

Interquartile range using scipy.stats.iqr

Output: 34.0

Quartile Deviation
Quartile deviation is the half of the difference of third quartile (Q3) and first quartile (Q1) i.e. half of the interquartile range (IQR). (Q3 – Q1) / 2 = IQR / 2

Decision making
The data set having higher value of quartile deviation has higher variability.

Quartile deviation using numpy.median

Output: 17.0