# Program to implement standard error of mean

Last Updated : 11 Mar, 2024

Standard error of mean (SEM) is used to estimate the sample mean dispersion from the population mean. The standard error along with sample mean is used to estimate the approximate confidence intervals for the mean. It is also known as standard error of mean or measurement often denoted by SE, SEM or SE.

Examples:

`Input : arr[] = {78.53, 79.62, 80.25, 81.05, 83.21, 83.46}Output : 0.8063Input : arr[] = {5, 5.5, 4.9, 4.85, 5.25, 5.05, 6.0}Output : 0.1546`

Sample mean

Â

Sample Standard Deviation

Â

Estimate standard error of mean

Explanation:

given an array arr[] = {78.53, 79.62, 80.25, 81.05, 83.21, 83.46} and the task is to find standard error of mean.Â
mean = (78.53 + 79.62 + 80.25 + 81.05 + 83.21 + 83.46) / 6Â
= 486.12 / 6Â
= 81.02Â

Sample Standard deviation = sqrt((78.53 – 81.02)2 + (79.62- 81.02)2 + . . . + (83.46 – 81.02)2 / (6 – 1))Â
= sqrt(19.5036 / 5)Â
= 1.97502Â

Standard error of mean = 1.97502 / sqrt(6)Â
= 0.8063

## C++

 `// C++ Program to implement ` `// standard error of mean.` `#include ` `using` `namespace` `std;`   `// Function to find sample mean.` `float` `mean(``float` `arr[], ``int` `n)` `{   ` `    ``// loop to calculate ` `    ``// sum of array elements.` `    ``float` `sum = 0;` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``sum = sum + arr[i];` `    `  `    ``return` `sum / n;` `}`   `// Function to calculate sample` `// standard deviation.` `float` `SSD(``float` `arr[], ``int` `n)` `{` `    ``float` `sum = 0;    ` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``sum = sum + (arr[i] - mean(arr, n))` `                    ``* (arr[i] - mean(arr, n));`   `    ``return` `sqrt``(sum / (n - 1));` `}`     `// Function to calculate sample error.` `float` `sampleError(``float` `arr[], ``int` `n)` `{    ` `    ``// Formula to find sample error.` `    ``return` `SSD(arr, n) / ``sqrt``(n);` `}`   `// Driver function` `int` `main()` `{` `    ``float` `arr[] = { 78.53, 79.62, 80.25,` `                    ``81.05, 83.21, 83.46 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``cout << sampleError(arr, n);    ` `    ``return` `0;` `}`

## Java

 `// Java Program to implement` `// standard error of mean.`   `class` `GFG {`   `    ``// Function to find sample mean.` `    ``static` `float` `mean(``float` `arr[], ``int` `n)` `    ``{` `        ``// loop to calculate` `        ``// sum of array elements.` `        ``float` `sum = ``0``;` `        ``for` `(``int` `i = ``0``; i < n; i++)` `            ``sum = sum + arr[i];`   `        ``return` `sum / n;` `    ``}`   `    ``// Function to calculate sample` `    ``// standard deviation.` `    ``static` `float` `SSD(``float` `arr[], ``int` `n)` `    ``{` `        ``float` `sum = ``0``;` `        ``for` `(``int` `i = ``0``; i < n; i++)` `            ``sum = sum + (arr[i] - mean(arr, n)) ` `                  ``* (arr[i] - mean(arr, n));`   `        ``return` `(``float``)Math.sqrt(sum / (n - ``1``));` `    ``}`   `    ``// Function to calculate sample error.` `    ``static` `float` `sampleError(``float` `arr[], ``int` `n)` `    ``{` `        ``// Formula to find sample error.` `        ``return` `SSD(arr, n) / (``float``)Math.sqrt(n);` `    ``}`   `    ``// Driver function` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``float` `arr[] = { ``78``.53f, ``79``.62f, ``80``.25f, ` `                       ``81``.05f, ``83``.21f, ``83``.46f };` `        ``int` `n = arr.length;` `        ``System.out.println(sampleError(arr, n));` `    ``}` `}`   `// This code is contributed ` `// by  prerna saini`

## C#

 `// C# Program to implement` `// standard error of mean.` `using` `System;`   `class` `GFG {`   `    ``// Function to find sample mean.` `    ``static` `float` `mean(``float` `[]arr, ``int` `n)` `    ``{` `        `  `        ``// loop to calculate` `        ``// sum of array elements.` `        ``float` `sum = 0;` `        ``for` `(``int` `i = 0; i < n; i++)` `            ``sum = sum + arr[i];`   `        ``return` `sum / n;` `    ``}`   `    ``// Function to calculate sample` `    ``// standard deviation.` `    ``static` `float` `SSD(``float` `[]arr, ``int` `n)` `    ``{` `        ``float` `sum = 0;` `        ``for` `(``int` `i = 0; i < n; i++)` `            ``sum = sum + (arr[i] - mean(arr, n)) ` `                      ``* (arr[i] - mean(arr, n));`   `        ``return` `(``float``)Math.Sqrt(sum / (n - 1));` `    ``}`   `    ``// Function to calculate sample error.` `    ``static` `float` `sampleError(``float` `[]arr, ``int` `n)` `    ``{` `        `  `        ``// Formula to find sample error.` `        ``return` `SSD(arr, n) / (``float``)Math.Sqrt(n);` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `Main()` `    ``{` `        ``float` `[]arr = {78.53f, 79.62f, 80.25f, ` `                       ``81.05f, 83.21f, 83.46f};` `        ``int` `n = arr.Length;` `        ``Console.Write(sampleError(arr, n));` `    ``}` `}`   `// This code is contributed by Nitin Mittal.`

## JavaScript

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## PHP

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## Python3

 `# Python 3 Program to implement ` `# standard error of mean.` `import` `math`     `# Function to find sample mean.` `def` `mean(arr, n) :`   `    ``# loop to calculate ` `    ``# sum of array elements.` `    ``sm ``=` `0` `    ``for` `i ``in` `range``(``0``,n) :` `        ``sm ``=` `sm ``+` `arr[i]` `     `  `    ``return` `sm ``/` `n`     `# Function to calculate sample` `# standard deviation.` `def` `SSD(arr, n) :` `    ``sm ``=` `0` `    ``for` `i ``in` `range``(``0``,n) :` `        ``sm ``=` `sm ``+` `(arr[i] ``-` `mean(arr, n)) ``*` `(arr[i] ``-` `mean(arr, n))` ` `  `    ``return` `(math.sqrt(sm ``/` `(n ``-` `1``)))` ` `  ` `  `# Function to calculate sample error.` `def` `sampleError(arr, n) :`   `    ``# Formula to find sample error.` `    ``return` `SSD(arr, n) ``/` `(math.sqrt(n))`   ` `  `# Driver function` `arr ``=` `[ ``78.53``, ``79.62``, ``80.25``, ``81.05``, ``83.21``, ``83.46``]` `n ``=` `len``(arr)` `print``(sampleError(arr, n))` `    `  `  `  `# This code is contributed` `# by Nikita Tiwari.`

Output

```0.8063

```

Time Complexity: O(N2), for calculation of mean N times while calculating Sample Standard Deviation.
Auxiliary Space: O(1), as constant extra space is required.

### Python Solution(Using Statistics):

1. Importing statistics: The code begins by importing the statistics module, which provides functions for mathematical statistics of numeric data.
2. sample_error Function: This function takes an array (arr) containing numeric data as input and calculates the sample error.
3. Calculating Sample Standard Deviation: Inside the sample_error function, the sample standard deviation is calculated using the statistics.stdev() function. This function computes the sample standard deviation for the given data.
4. Calculating Sample Error: Once the sample standard deviation is obtained, the sample error is computed by dividing the standard deviation by the square root of the sample size (len(arr)). This follows the formula for sample error calculation, where sample error equals standard deviation divided by the square root of the sample size.

## Python3

 `import` `statistics`   `# Function to calculate the sample error` `def` `sample_error(arr):` `    ``# Calculate the sample standard deviation using statistics.stdev() for an unbiased estimator` `    ``std_dev ``=` `statistics.stdev(arr)` `    ``# Divide the standard deviation by the square root of the sample size to get the sample error` `    ``sample_err ``=` `std_dev ``/` `(``len``(arr) ``*``*` `0.5``)` `    ``return` `sample_err`   `# Sample data` `arr ``=` `[``78.53``, ``79.62``, ``80.25``, ``81.05``, ``83.21``, ``83.46``]`   `# Calculate and print the sample error` `print``(sample_error(arr))`

Time Complexity: O(N)

Auxiliary Space: O(1)