The standard deviation is a statistical metric that measures the degree of diversity or disperse within a dataset. It gauges how spread out the values are from the mean of the data. It helps understand the extent to which individual data points deviate from the overall average.

A small SD suggests that the data points are clustered about the mean while a high standard deviation suggests that the data points are more dispersed from the mean.

This measurement is widely used in various fields, including finance, science, and social sciences, to analyze and interpret data patterns.

This article will explore the following concepts of Standard deviation:

•       Definition.
•       Computation of Standard Deviation.
•       Calculating Standard Deviation with Grouped Data
•       Calculating Standard Deviation with grouped Data.
•       Examples of Standard Deviation.

The Concept of Standard Deviation:

The STD shows the extent of the spread or dispersion of data points around their average. The degree to which data points deviate from the mean displays how the data is spread throughout the dataset. It’s important to realize that the standard deviation increases with the degree of data point divergence from the mean.

This variation is calculated as the square root of the variance, which is the average of the squared differences from the Mean. Therefore, standard deviation provides a clear, computable suggestion of the spread of a dataset, whether it refers to a statistical population, a random variable, a specific dataset, or a probability distribution.

Significance of Standard Deviation

The importance of Standard deviation arises from various factors. Here are a few of them:

•       It simplifies the comprehension of outcomes when data shows greater dispersion.
•       The standard deviation of a distribution or dataset tends to be higher when the data is more uniformly spread out.
•       Business executives leverage standard deviation in Excel for financial analysis to understand risk management and make more informed investment decisions.
•       It aids in calculating the margins of error commonly found in survey outcomes.

How to Compute the Standard Deviation?

Calculating the standard deviation is a critical process in statistical analysis that allow us to understand the dispersion of a given sample or population dataset around its mean. Whether you’re working with grouped or ungrouped data, the steps to calculate the standard deviation slightly differ.

Here are some steps to calculate SD.

1.     Calculate the Mean.
2.     Find the Differences by subtracting the mean from each value in the data set.
3.     Square the Differences obtained in the previous step.
4.     Calculate the Variance by calculating the mean of the squared difference.
5.     Compute the Square Root of the variance.

SD of Grouped Data:

In cases of grouped data or a grouped frequency distribution, the occurrence or frequency of data values serves as a basis for computing the standard deviation. Let’s solve an example of group data to shed more light on this method:

Example:

Calculate the SD of the following grouped Data:

Suppose we have data representing the ages of individuals in a survey, grouped into age intervals with their respective frequencies:

Age Interval	Frequency
21 – 30	5
31 – 40	8
41 – 50	12
51 – 60	10
61 – 70	6

Solution:

Class Interval	Frequency (f)	Mid Value (xi)	fxi	fxi2
21 – 30	5	25.5	127.5	3251.5
31 – 40	8	35.5	284	10082
41 – 50	12	45.5	546	24843
51 – 60	10	55. 5	555	30802.5
61 – 70	6	65. 5	393	25741.5
	∑f = 41		∑fxi = 1905.5	∑fxi2= 94720.5

 

x̄ (Mean) = ∑fxi / ∑f = 1905.5 / 41 = 46.4756

Variance = 1/ (N – 1) [∑fxi2 – 1/N(∑fxi)2] 

 = 1 / 40 [94720.5 – (1 /41) (1905.5)2] = 154.0253

SD = √variance = √ 154.0253 = 12.4107

SD of Ungrouped Data:

Different techniques for standard deviation calculation are employed for various datasets. These methods gauge how data diverges from the mean or central point. There are two primary approaches to determining standard deviation:

Actual Mean Method:

The actual mean method can be calculated by the following formula:

•       Formula: σ = √(∑x−x̄)2 /n) 

Let’s imagine an example for more understanding:

Example:

Calculate the SD by actual mean method 4, and 5,7,8.

Solution:

x̄ = 6

X	(X – x̄)2
78	4
82	1
85	1
72	4
–	∑ (Xi – X)2 = 10

Formula: σ = √(∑x−x̄)2 /n)

σ = √10 / 4 = 1.58

Assumed Mean Method:

When dealing with large x values, the mean is established by choosing a specific value (A), alternatively referred to as the Assumed mean. The discrepancy from this selected mean is calculated as d = x – A.

σ = √[(∑(d)2 /n) – (∑d/n)2]

Example of Standard Deviation:

Let’s solve some examples to gain more demonstration of Standard Deviation.

Example 1:

Heights of Students In a school, the heights of ten students (in inches) are recorded:

hgt

65

68

71

63

72

66

70

67

69

64.

Calculate the SD for the heights of these students.

Solution:

Step 1:

Calculate the mean height:

Mean height = (65 + 68 + 71 + 63 + 72 + 66 + 70 + 67 + 69 + 64) / 10

Mean height ≈ 67.5 inches

Step 2:

Find the differences between each height and the mean:

Differences:

 

X – x̄
-2.5
0.5
3.5
-4.5
4.5
-1.5
2.5
-0.5
1.5
-3.5

 

 

Step 3:

Square these differences:

(X – x̄)2
6.25
0.25
12.25
20.25
20.25
2.25
6.25
0.25
2.25
12.25

 

 

Step 4:

Calculate the variance:

Variance = (6.25 + 0.25 + 12.25 + 20.25 + 20.25 + 2.25 + 6.25 + 0.25 + 2.25 + 12.25) / 10

Variance = 82.5 / 10 Variance = 8.25

Step 5:

Find the square root of the variance to get the standard deviation:

Standard Deviation ≈ √8.25 ≈ 2.87 inches

This example shows the calculation of the standard deviation for a set of heights.

Final Words

This article explored Standard Deviation. Calculating it involves steps like finding variances and utilizing different methods for grouped and ungrouped data. Examples clarified its application, from student heights to commute times.