Mastering the Calculation of Standard Deviation for Ungrouped Data

In statistics and data analysis, the standard deviation stands as a fundamental concept that illuminates the spread or dispersion of data. It helps us understand how much the data points differ from each other and how they’re spread out in a group of observations.

This is derived from the square root of variance and is mostly denoted as SD and sigma (𝜎). Standard deviation is a reliable indicator of the degree of diversity or uniformity within data.

This introductory article exploration into standard deviation aims to unravel the following points:

  • Formulas of SD to calculate.
  • Determining Standard Deviation: Calculation Steps
  • Standard Deviation Calculation for Ungrouped Data
  • Solved Problems related to Standard Deviation.
  • Wrap up

Definition: Standard Deviation:

Standard deviation is a statistical measure that tells us how much individual data points in a dataset differ from the average (mean) of the dataset. It indicates the extent of variability or dispersion within the data.

  • A higher standard deviation means that data points are more dispersed out of the mean
  • A lower standard deviation suggests that data points are nearer to the mean.

It is commonly used in statistics to understand the distribution and consistency of data values.

Formulas to calculate SD:

There are distinct formulas to calculate the Standard deviation for grouped and ungrouped data. Additionally, distinct formulas exist to compute the SD of the Sample and population. Let’s explore each of these comprehensively.

Standard Deviation: Population

σ population=v​(Xi​−μ)2​​ / N

  • σ signifies the population SD.
  • Xi​denotes every data point in the population.
  • μ is the population mean.
  • N is the total number of data values in the population.
  • signifies the summation of the squared differences between each value and the population mean.
  • Vindicates the square root to find the final standard deviation

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Standard Deviation: Sample

σ Sample = v [∑​(xi​−xˉ)2​​ / n−1]

The denominator (n−1) is used to correct for bias in sample standard deviation calculations and is known as Bessel’s correction.

Determining Standard Deviation: Calculation Steps

Here are the steps, which help us how to calculate the standard deviation:

Step 1:Compute the Mean

  • Estimatethe average of all the values in the dataset.

Step 2:Find the Differences

  • Subtract the average from each data point in the dataset.

 Step 3:Determine the squares of the divergences.

  • Square every single one of the differences obtained in the previous step.

Step 4: Find out the Variance

  • Locate the average of the squared differences.

Step 5:Figure the Standard Deviation

  • Take the square rootof the variance(σ2) to get the standard deviation.

Standard Deviation Calculation for Ungrouped Data

The computation of Standard deviation is distinct for distinct data. The variance from the mean or average position of data is gauged by distribution. There are two ways available to calculate the SD of ungrouped data.

  1. Method of Actual Means
  2. Method of Assumed Means

Method of Actual Mean:

This method involves computing the SD by the actual mean method of the dataset.  Here is the formula to compute this:

σ = √(∑(x−¯x)/n)

Method of Assumed Mean:

A shortcut method for calculating the mean (average) of a set of data that is particularly useful when dealing with large or grouped data. D = x -A calculates the deviation from this assumed mean.

Formula:σ = Sqrt [(∑(d)2 /n) – (∑d/n)2]

Solved Examples related to Standard deviation:

Example 1:

Assume data of exam scores 75, 80, 85, 90, 95.  Compute the standard deviation of these scores.

Solution:

Step 1:Compute the Mean

Average (Mean)= (75 + 80 + 85 + 90 + 95) / 5

= 425 / 5 = 85

Step 2:Find the Differences

 

Xi Xi – X
75 -10
80 -5
85 0
90 5
95 10

Step 3:Square the Differences

(Xi – X)2
100
25
0
25
100
∑ (Xi – X)2 = 250

Step 4:Find out the Variance

σ = (100 + 25 + 0 + 25 + 100) / 5 = 250 / 5 = 50

Step 5:Determine the Standard Deviation

Standard deviation = √50 ≈ 7.07

You can take assistance from online tools to compute std form sample and population data values such as:

https://www.standarddeviationcalculator.io/

Example 2:

Consider a dataset of weights 10, 12, 15, 18, 20 pounds.

Calculate the standard deviation using the Assumed Mean Method with an assumed mean of 15 pounds.

Solution:

Step 1:

Assumed mean (A) = 15

Step 2:

D = X -A

 

-5
3
0
3
5
∑ d= 6

 

Step 3: Square the differences:

D = (X –A)2

 

25
9
0
9
25
∑ = 68

 

Step 4:Evaluate the variance using the formula:

σ2 = (68/5) – (6/5)2 = 12.16

Step 5:Take the square root of the previous step value for SD

√ σ2 = √ 12.16 =

σ = 3.48

Wrap up:

In this article, we explored the world of standard deviation, a vital statistical measure for understanding data’s spread. We delved into its definition, formulas, and calculation steps, tackling both grouped and ungrouped data. Solved examples illuminated its practical application.

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