Statistics with Python (original) (raw)

**Statistics, in general, is the method of collection of data, tabulation, and interpretation of numerical data. It is an area of applied mathematics concerned with data collection analysis, interpretation, and presentation. With statistics, we can see how data can be used to solve complex problems.

In this tutorial, we will learn about solving statistical problems with Python and will also learn the concept behind it. Let's start by understanding some concepts that will be useful throughout the article.

**Note: We will be covering descriptive statistics with the help of the statistics module provided by Python.

Understanding the Descriptive Statistics

In layman's terms, descriptive statistics generally means describing the data with the help of some representative methods like charts, tables, Excel files, etc. The data is described in such a way that it can express some meaningful information that can also be used to find some future trends. Describing and summarizing a single variable is called **univariate analysis. Describing a statistical relationship between two variables is called **bivariate analysis. Describing the statistical relationship between multiple variables is called **multivariate analysis.

**There are two types of Descriptive Statistics:

Types of Descriptive Statistics

Measure of Central Tendency

The measure of central tendency is a single value that attempts to describe the whole set of data. There are three main features of central tendency:

The measure of Central Tendency

Mean

It is the sum of observations divided by the total number of observations. It is also defined as average which is the sum divided by count.

Mean (\overline{x}) = \frac{\sum{x}}{n}

The **mean()function returns the mean or average of the data passed in its arguments. If the passed argument is empty, **StatisticsError is raised.

**Example: Python code to calculate mean

Python `

Python code to demonstrate the working of

mean()

importing statistics to handle statistical

operations

import statistics

initializing list

li = [1, 2, 3, 3, 2, 2, 2, 1]

using mean() to calculate average of list

elements

print ("The average of list values is : ",end="") print (statistics.mean(li))

`

**Output:

The average of list values is : 2

Median

It is the middle value of the data set. It splits the data into two halves. If the number of elements in the data set is odd then the center element is the median and if it is even then the median would be the average of two central elements. it first sorts the data i=and then performs the median operation

**For Odd Numbers:

\frac{n+1}{2}

**For Even Numbers:

{\frac{n}{2} + (\frac{n}{2}+1)}\over{2}

The **median() function is used to calculate the median, i.e middle element of data. If the passed argument is empty, **StatisticsError is raised.

**Example: Python code to calculate Median

Python `

Python code to demonstrate the

working of median() on various

range of data-sets

importing the statistics module

from statistics import median

Importing fractions module as fr

from fractions import Fraction as fr

tuple of positive integer numbers

data1 = (2, 3, 4, 5, 7, 9, 11)

tuple of floating point values

data2 = (2.4, 5.1, 6.7, 8.9)

tuple of fractional numbers

data3 = (fr(1, 2), fr(44, 12), fr(10, 3), fr(2, 3))

tuple of a set of negative integers

data4 = (-5, -1, -12, -19, -3)

tuple of set of positive

and negative integers

data5 = (-1, -2, -3, -4, 4, 3, 2, 1)

Printing the median of above datasets

print("Median of data-set 1 is % s" % (median(data1))) print("Median of data-set 2 is % s" % (median(data2))) print("Median of data-set 3 is % s" % (median(data3))) print("Median of data-set 4 is % s" % (median(data4))) print("Median of data-set 5 is % s" % (median(data5)))

`

**Output:

Median of data-set 1 is 5
Median of data-set 2 is 5.9
Median of data-set 3 is 2
Median of data-set 4 is -5
Median of data-set 5 is 0.0

Median Low

The **median_low() function returns the median of data in case of odd number of elements, but in case of even number of elements, returns the lower of two middle elements. If the passed argument is empty, **StatisticsError is raised

**Example: Python code to calculate Median Low

Python `

Python code to demonstrate the

working of median_low()

importing the statistics module

import statistics

simple list of a set of integers

set1 = [1, 3, 3, 4, 5, 7]

Print median of the data-set

Median value may or may not

lie within the data-set

print("Median of the set is % s" % (statistics.median(set1)))

Print low median of the data-set

print("Low Median of the set is % s " % (statistics.median_low(set1)))

`

**Output:

Median of the set is 3.5
Low Median of the set is 3

Median High

The **median_high() function returns the median of data in case of odd number of elements, but in case of even number of elements, returns the higher of two middle elements. If passed argument is empty, **StatisticsError is raised.

**Example: Python code to calculate Median High

Python `

Working of median_high() and median() to

demonstrate the difference between them.

importing the statistics module

import statistics

simple list of a set of integers

set1 = [1, 3, 3, 4, 5, 7]

Print median of the data-set

Median value may or may not

lie within the data-set

print("Median of the set is %s" % (statistics.median(set1)))

Print high median of the data-set

print("High Median of the set is %s " % (statistics.median_high(set1)))

`

**Output:

Median of the set is 3.5
High Median of the set is 4

Mode

It is the value that has the highest frequency in the given data set. The data set may have no mode if the frequency of all data points is the same. Also, we can have more than one mode if we encounter two or more data points having the same frequency.

The **mode() function returns the number with the maximum number of occurrences. If the passed argument is empty, **StatisticsError is raised.

**Example: Python code to calculate Mode

Python `

Python code to demonstrate the

working of mode() function

on a various range of data types

Importing the statistics module

from statistics import mode

Importing fractions module as fr

Enables to calculate harmonic_mean of a

set in Fraction

from fractions import Fraction as fr

tuple of positive integer numbers

data1 = (2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 7)

tuple of a set of floating point values

data2 = (2.4, 1.3, 1.3, 1.3, 2.4, 4.6)

tuple of a set of fractional numbers

data3 = (fr(1, 2), fr(1, 2), fr(10, 3), fr(2, 3))

tuple of a set of negative integers

data4 = (-1, -2, -2, -2, -7, -7, -9)

tuple of strings

data5 = ("red", "blue", "black", "blue", "black", "black", "brown")

Printing out the mode of the above data-sets

print("Mode of data set 1 is % s" % (mode(data1))) print("Mode of data set 2 is % s" % (mode(data2))) print("Mode of data set 3 is % s" % (mode(data3))) print("Mode of data set 4 is % s" % (mode(data4))) print("Mode of data set 5 is % s" % (mode(data5)))

`

**Output:

Mode of data set 1 is 5
Mode of data set 2 is 1.3
Mode of data set 3 is 1/2
Mode of data set 4 is -2
Mode of data set 5 is black

Refer to the below article to get detailed information about averages and Measures of central tendency.

Measure of Variability

Till now, we have studied the measure of central tendency but this alone is not sufficient to describe the data. To overcome this we need the **measure of variability. The measure of variability is known as the spread of data or how well our data is distributed. The most common variability measures are:

Measure of Variability

Range

The difference between the largest and smallest data point in our data set is known as the range. The range is directly proportional to the spread of data which means the bigger the range, the more the spread of data and vice versa.

Range = Largest data value – smallest data value

We can calculate the maximum and minimum values using the **max() and **min() methods respectively.

**Example: Python code to calculate Range

Python `

Sample Data

arr = [1, 2, 3, 4, 5]

#Finding Max Maximum = max(arr)

Finding Min

Minimum = min(arr)

Difference Of Max and Min

Range = Maximum-Minimum
print("Maximum = {}, Minimum = {} and Range = {}".format( Maximum, Minimum, Range))

`

**Output:

Maximum = 5, Minimum = 1 and Range = 4

Variance

It is defined as an average squared deviation from the mean. It is calculated by finding the difference between every data point and the average which is also known as the mean, squaring them, adding all of them, and then dividing by the number of data points present in our data set.

\sigma^2=\frac{\sum(x-\mu^2)}{N}

where N = number of terms

u = Mean

The statistics module provides the **variance() method that does all the maths behind the scene. If the passed argument is empty, **StatisticsError is raised.

**Example: Python code to calculate Variance

Python `

Python code to demonstrate variance()

function on varying range of data-types

importing statistics module

from statistics import variance

importing fractions as parameter values

from fractions import Fraction as fr

tuple of a set of positive integers

numbers are spread apart but not very much

sample1 = (1, 2, 5, 4, 8, 9, 12)

tuple of a set of negative integers

sample2 = (-2, -4, -3, -1, -5, -6)

tuple of a set of positive and negative numbers

data-points are spread apart considerably

sample3 = (-9, -1, -0, 2, 1, 3, 4, 19)

tuple of a set of fractional numbers

sample4 = (fr(1, 2), fr(2, 3), fr(3, 4), fr(5, 6), fr(7, 8))

tuple of a set of floating point values

sample5 = (1.23, 1.45, 2.1, 2.2, 1.9)

Print the variance of each samples

print("Variance of Sample1 is % s " % (variance(sample1))) print("Variance of Sample2 is % s " % (variance(sample2))) print("Variance of Sample3 is % s " % (variance(sample3))) print("Variance of Sample4 is % s " % (variance(sample4))) print("Variance of Sample5 is % s " % (variance(sample5)))

`

**Output:

Variance of Sample1 is 15.80952380952381
Variance of Sample2 is 3.5
Variance of Sample3 is 61.125
Variance of Sample4 is 1/45
Variance of Sample5 is 0.17613000000000006

Standard Deviation

It is defined as the square root of the variance. It is calculated by finding the Mean, then subtracting each number from the Mean which is also known as the average, and squaring the result. Adding all the values and then dividing by the no of terms followed by the square root.

\sigma=\sqrt\frac{\sum(x-\mu)^2}{N}

where N = number of terms

u = Mean

The **stdev()method of the statistics module returns the standard deviation of the data. If the passed argument is empty, **StatisticsError is raised.

**Example: Python code to calculate Standard Deviation

Python `

Python code to demonstrate stdev()

function on various range of datasets

importing the statistics module

from statistics import stdev

importing fractions as parameter values

from fractions import Fraction as fr

creating a varying range of sample sets

numbers are spread apart but not very much

sample1 = (1, 2, 5, 4, 8, 9, 12)

tuple of a set of negative integers

sample2 = (-2, -4, -3, -1, -5, -6)

tuple of a set of positive and negative numbers

data-points are spread apart considerably

sample3 = (-9, -1, -0, 2, 1, 3, 4, 19)

tuple of a set of floating point values

sample4 = (1.23, 1.45, 2.1, 2.2, 1.9)

Print the standard deviation of

following sample sets of observations

print("The Standard Deviation of Sample1 is % s" % (stdev(sample1)))

print("The Standard Deviation of Sample2 is % s" % (stdev(sample2)))

print("The Standard Deviation of Sample3 is % s" % (stdev(sample3)))

print("The Standard Deviation of Sample4 is % s" % (stdev(sample4)))

`

**Output:

The Standard Deviation of Sample1 is 3.9761191895520196
The Standard Deviation of Sample2 is 1.8708286933869707
The Standard Deviation of Sample3 is 7.8182478855559445
The Standard Deviation of Sample4 is 0.41967844833872525

Refer to the below article to get detailed information about the Measure of variability.[Statistical Functions in Python | Set 2 ( Measure of Spread)]