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Python Program to Calculate Standard Deviation
In this article, we will learn how to implement a python program to calculate standard deviation on a dataset.
Consider a set of values plotted on any coordinate axes. Standard deviation of these set of values, called population, is defined as the variation seen among them. If the standard deviation is low, the values are plotted closely to the mean. But if the standard deviation is high, the values are dispersed farther from the mean.
It is denoted by square root of the variance of a dataset. There are two types of standard deviations −
The population standard deviation is calculated from every data value of a population. Hence, it is a fixed value. The mathematical formula is defined as −
$$\mathrm{SD\:=\:\sqrt{\frac{\sum(X_i\:-\:X_m)^2}{n}}}$$
Where,
Xm is the mean of a dataset.
Xi is the elements of the dataset.
n is the number of elements in the dataset.
However, the sample standard deviation is a statistic calculated only on some datum values of a population, hence the value depends upon the sample chosen. The mathematical formula is defined as −
$$\mathrm{SD\:=\:\sqrt{\frac{\sum(X_i\:-\:X_m)^2}{n\:-\:1}}}$$
Where,
Xm is the mean of a dataset.
Xi is the elements of the dataset.
n is the number of elements in the dataset.
Input Output Scenarios
Let us now look at some input output scenarios for various sets of data −
Assume the dataset only contains positive integers −
Input: [2, 3, 4, 1, 2, 5] Result: Population Standard Deviation: 1.3437096247164249 Sample Standard Deviation: 0.8975274678557505
Assume the dataset only contains negative integers −
Input: [-2, -3, -4, -1, -2, -5] Result: Population Standard Deviation: 1.3437096247164249 Sample Standard Deviation: 0.8975274678557505
Assume the dataset only contains positive and negative integers −
Input: [-2, -3, -4, 1, 2, 5] Result: Population Standard Deviation: 3.131382371342656 Sample Standard Deviation: 2.967415635794143
Using Mathematical Formula
We have seen the formula of standard deviation above in the same article; now let us look at the python program to implement the mathematical formula on various datasets.
Example
In the following example, we are importing the math library and calculating the standard deviation of the dataset by applying sqrt() built-in method on its variance.
import math #declare the dataset list dataset = [2, 3, 4, 1, 2, 5] #find the mean of dataset sm=0 for i in range(len(dataset)): sm+=dataset[i] mean = sm/len(dataset) #calculating population standard deviation of the dataset deviation_sum = 0 for i in range(len(dataset)): deviation_sum+=(dataset[i]- mean)**2 psd = math.sqrt((deviation_sum)/len(dataset)) #calculating sample standard deviation of the dataset ssd = math.sqrt((deviation_sum)/len(dataset) - 1) #display output print("Population standard deviation of the dataset is", psd) print("Sample standard deviation of the dataset is", ssd)
Output
The output standard deviation obtained is as follows −
Population standard deviation of the dataset is 1.3437096247164249 Sample standard deviation of the dataset is 0.8975274678557505
Using std() function in numpy module
In this approach, we import the numpy module and only population standard deviation is calculated using the numpy.std() function on the elements of a numpy array.
Example
The following python program is implemented to calculate the standard deviation on the elements of a numpy array −
import numpy as np #declare the dataset list dataset = np.array([2, 3, 4, 1, 2, 5]) #calculating standard deviation of the dataset sd = np.std(dataset) #display output print("Population standard deviation of the dataset is", sd)
Output
The standard deviation is displayed as the following output −
Population standard deviation of the dataset is 1.3437096247164249
Using stdev() and pstdev() Functions in statistics module
The statistics module in python provides functions called stdev() and pstdev() to calculate the standard deviation of a sample dataset. The stdev() function in python only calculates the sample standard deviation whereas the pstdev() function calculates the population standard deviation.
The parameters and return type for both functions is the same.
Example 1: Using stdev() Function
The python program to demonstrate the usage of stdev() function to find the sample standard deviation of a dataset is as follows −
import statistics as st #declare the dataset list dataset = [2, 3, 4, 1, 2, 5] #calculating standard deviation of the dataset sd = st.stdev(dataset) #display output print("Standard Deviation of the dataset is", sd)
Output
The sample standard deviation of the dataset obtained as an output is as follows −
Standard Deviation of the dataset is 1.4719601443879744
Example 2: Using pstdev() Function
The python program to demonstrate the usage of pstdev() function to find the population standard deviation of a dataset is as follows −
import statistics as st #declare the dataset list dataset = [2, 3, 4, 1, 2, 5] #calculating standard deviation of the dataset sd = st.pstdev(dataset) #display output print("Standard Deviation of the dataset is", sd)
Output
The sample standard deviation of the dataset obtained as an output is as follows −
Standard Deviation of the dataset is 1.3437096247164249
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