Measures of dispersion like variance and standard deviation of ungrouped data are used to illustrate the collection and visualization of data in statistical research. A deeper understanding of this concept is improved by this variation in the research data obtained.
Variance and standard deviation are concepts that are used to examine and comprehend the interpretation and representation of the data under examination. The only possible values for variance are zero or positive and it can never be negative. The variance of the spread of the data can be zero if all of the data values that are being considered are identical.
In this comprehensive guide, we will discuss the calculations of the standard deviation for ungrouped data precisely. We will elaborate on here how to compute the population standard deviation and the sample standard deviation with some solved examples to apprehend this useful concept thoroughly.
Defining Standard Deviation (SD):
The standard deviation signifies how much a set of collective data values is varied or dispersed. The values are fairly close to the chosen mean when the standard deviation value is small. A large value of standard deviation elaborates on that the given data values are distributed over a wide range.
Additionally, standard deviation comes into two categories: sample standard deviation and population standard deviation. The formula we use to compute the standard deviation depends on whether the data is interpreted as a sample that represents a larger population or as a population regarding itself.
Below, we’ll explore these terms in detail.
What is Population Standard Deviation (PSD)?
The concept of population standard deviation is used to find the standard deviation for the total population. We need to collect information from each member of the population in order to calculate the standard deviation with accuracy.
If we consider the data to be a population in and of itself, we divide the data by the total number of data points (N).
σk = √ (∑ (xk – μ)2 / N)
What is Sample Standard Deviation (SSD)?
Using the concept of the sample standard deviation is one method of determining the variation for a particular sample. The sample standard deviation can also be used to approximate or draw conclusions about the standard deviation of the entire population.
Divide the sample size by one less than the total number of data points (n – 1) if the data reflects a sample drawn from a bigger population.
Sk = √ [∑ (xk – x̅)2 / (n -1)]
How to Calculate Standard Deviation for Ungrouped Data?
Below are a few steps to find the standard deviation for ungrouped data.
Step 1: Calculate the Mean (x̄ or µ)
Formula: Mean = Sum of all data points/Number of data points
Step 2: Find the Deviations from the Mean (xi – x̄ or xi – µ)
Formula: Deviation = Data point− Mean
Step 3: Square Each Deviation [∑ (xi – x̄)2 or∑ (xi – µ)2]
Formula: Squared Deviation = Deviation2
Step 4: Calculate the Mean of the Squared Deviations [∑(xi – x̄)2/n-1 or∑(xi – µ)2/N]
Formula: Mean of Squared Deviations = Sum of Squared Deviations / Number of data points
Step 5: Take the Square Root of the Mean of the Squared Deviations
Now have a look at the below examples to understand precisely.
Example 1:
Find the population and the sample standard deviation for the given scores in the following table.
| Sub | Eng | Phy | Chem | Math | ES | IR | GS |
| xk | 4 | 1 | 3 | 5 | 3 | 2 | 3 |
Solution:
Step 1: First of all, we will compute the average for the given scores in the table.
μ = x̅ = (4 + 1 + 3 + 5 + 3 + 2 + 3) / 7
μ = x̅ = 21 / 7
μ = x̅ = 3
Step 2: Now we will make the following calculations in the following table:
| xk | (xk – μ) = (xk – x̅) | (xk – μ)2 = (xk – x̅)2 |
| 4 | 1 | 1 |
| 1 | -2 | 4 |
| 3 | 0 | 0 |
| 5 | 2 | 4 |
| 3 | 0 | 0 |
| 2 | -1 | 1 |
| 3 | 0 | 0 |
| Total | 10 |
Step 3: The mathematical relation for the population standard deviation is:
σk = √ (∑ (xk – μ)2 / N)
Put the relevant values in the above mathematical formula for PSD.
σk = √ [ (10) / 7]
σk = √ (1.43)
σk = 1.1958 Ans.
Now the mathematical relation for the population standard deviation is:
Sk = √ [∑ (xk – x̅)2 / (n -1)]
Put the relevant values in the above mathematical formula for SSD.
Sk = √ [ (10) / (7 – 1)]
Sk = √ [ (10) / (6)]
Sk = √ (1.6667)
Sk = 1.2910 Ans.
Example 2:
Find the population and the sample standard deviation for the given values in the following table.
| 10 | 12 | 12 | 13 | 14 | 18 | 19 | 20 |
Solution:
Step 1: First of all, we will compute the average for the given values in the table.
μ = x̅ = (10 + 12 + 12 + 13 + 14 + 18 + 19 + 20) / 8
μ = x̅ = 118 / 8
μ = x̅ = 14.75
Step 2: Now we will make the following calculations in the following table:
| xk | (xk – μ) = (xk – x̅) | (xk – μ)2 = (xk – x̅)2 |
| 10 | -4.75 | 22.5625 |
| 12 | –2.75 | 7.5625 |
| 12 | -2.75 | 7.5625 |
| 13 | -1.75 | 3.0625 |
| 14 | -0.75 | 0.5625 |
| 18 | 3.25 | 10.5625 |
| 19 | 4.25 | 18.0625 |
| 20 | 5.25 | 27.5625 |
| Total | 97.5027 |
Step 3: The mathematical relation for the population standard deviation is:
σk = √ (∑ (xk – μ)2 / N)
Put the relevant values in the above mathematical formula for PSD.
σk = √ [ (97.5027) / 8]
σk = √ (12.1878)
σk = 3.4911 Ans.
Now the mathematical relation for the population standard deviation is:
Sk = √ [∑ (xk – x̅)2 / (n -1)]
Put the relevant values in the above mathematical formula for SSD.
Sk = √ [ (97.5027) / (8 – 1)]
Sk = √ [ (97.5027) / (7)]
Sk = √ (13.9290)
Sk = 3.7322 Ans.
Wrap Up:
In this comprehensive discussion, we have addressed the topic of the standard deviation in detail. We have elaborated on its important categories precisely i.e. population standard deviation and the sample standard deviation. We have also given some solved examples of both of its categories.
We hope you had fun reading this article and that it will be helpful for you the next time you need to solve the problems about the standard deviation and its important categories.