Typically, the software begins by collecting data points from a user input or another source of data. These data points collectively form the dataset for which you aim to calculate the standard deviation.
2. Calculation of the Mean
After collecting the data points, the software calculates the average of the dataset. This is achieved by summing all the data points and then dividing the total by the number of data points (n). In mathematical notation, it is expressed as:
Mean = (x1 + x2 + x3 + ... + xn) / n
3. Differences Squared
The software subsequently computes the squared variances between every data point and the average. This process quantifies how much each data point varies from the mean. The use of squares ensures that positive deviations are not nullified by negative ones.
SquaredDifference_i = (xi - Mean)^2, where i = 1 to n
4. Calculating variance
The variance, which calculates the mean of the squared variances, is calculated by the software. Variance measures the general diversity within the dataset and is expressed as the mean squared difference from the average.
Variance = (SquaredDifference_1 + SquaredDifference_2 + SquaredDifference_3 + ... + SquaredDifference_n) / n
5. Standard Deviation
Finally, the standard deviation is calculated by taking the square root of the variance. It provides insight into the dispersion of data points around the mean value.
Standard Deviation = sqrt(Variance)
Output
The software typically displays the computed standard deviation on the console or stores it for future utilization in data analysis or reporting purposes. In cases where relevant, the standard deviation is commonly presented with the appropriate units.
6. Error Handling
Reliable standard deviation algorithms include error management mechanisms to address scenarios such as inadequate data points (fewer than 2) and invalid inputs. This ensures the program operates reliably, maintaining consistency and preventing issues like division by zero or similar errors.
7. Functions of the library
The calculation of standard deviation in C can be achieved using one of the built-in functions from the standard library. The sqrt function from the math.h library is commonly employed for computing square roots. While this library is primarily used for square roots, other mathematical functions can also be leveraged to enhance both the speed and accuracy of the calculations.
Program:
Let's consider an example to illustrate a C program for calculating standard deviation:
#include <stdio.h>
#include <math.h>
int main()
{
int k, num;
float arr[25];
float sum = 0.0, mean, Stan_Deviation;
// Asking for input
printf("Enter the total number of elements: ");
scanf("%d", &num);
printf("Enter the value of elements: \n");
for (k = 1; k <= num; k++ )
scanf("%f", &arr[k]);
// Calculating Mean
for (k = 1; k <= num; k++)
sum += arr[k];
mean = sum/num;
// Calculating Standard Deviation
sum=0.0;
for (k = 1; k<= num; k++)
sum+=(arr[k] - mean) * (arr[k] - mean);
Stan_Deviation = sqrt(sum / num);
printf("Mean: %6.3f \n", mean);
printf("Standard Deviation: %.6f", Stan_Deviation);
return 0;
}
Output:
Enter the total number of elements: 8
Enter the value of elements:
20
15
29
35
43
56
67
78
Mean: 42.875
Standard Deviation: 21.044224
Explanation:
- Include header files
In this instance, the software starts by importing the necessary header files for handling input/output operations (stdio.h) and mathematical calculations (math.h).
Next, a range of variables is initialized.
k: variable for the loop counter.
- The dataset's overall element count is given by the suffix num .
- An array with up to 25 pieces called arr[25] is used to hold the dataset.
- Variables that hold the dataset's total, mean , and standard deviation are called sum, mean , and Stan_Deviation .
- Data input
After asking the user to enter the total number of elements in the dataset (num), the program enters a loop and inputs the values of each element into the array arr .
- Calculation of the Mean
After data collection, the program computes the mean (average) of the dataset by adding up all the elements in the array arr and dividing the total by the number of elements (num) . The mean variable holds the mean value.
- Calculation of Standard Deviation
- It uses the variance formula to determine the standard deviation. It first sets the sum's initial value to 0 before entering a loop that iterates through each element of the array.
- It computes the squared difference between each element and the mean during the loop, adds these squared differences to the sum, and then divides the sum by num.
- The standard deviation , which is kept in the Stan_Deviation variable, is obtained by taking the square root of the variance once the variance has been calculated.
- Output
Afterward, the software displays the approximate average and standard deviation on the console utilizing the printf function. The display format can be customized to specify the precision of decimal points by employing the format specifiers %6.3f and %.6f.