Understanding Standard Deviation: Explanation & Calculation
Table of Contents
Standard Deviation is a fundamental statistical tool used to measure the extent of variability or dispersion within a dataset. It serves as an important indicator of how much individual data points depart or differ from the meanof the dataset. The standard deviation helps to clarify the distribution and patterns in the data by measuring this dispersion and offering insightful information about the range of values.
A low standard deviation indicates that the values are generally near the meanwhile a large standard deviation indicates that the values are significantly off from the mean.
This article will explain the following points:
- Definition
- Representation
- Formula
- How to calculate the SD
- Solved Problems of SD
Defining STD:
Standard Deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of data points around their mean (average). It indicates how spread out the data is from the central value.
Representation:
Standard deviationis often abbreviated as SD. It is commonly represented in mathematical texts and equations. It’s typically denoted by the lowercase Greek letter sigma (σ)for the population standard deviation.It is s represented by the Latin letter swhen indicating the sample standard deviation.
Standard Deviation Formulas:
The formulasfor SD calculation depend on whether we are dealing with a population or a sample:
| Population Standard Deviation:
|
Sample Standard Deviation:
|
| Formula:σ = √ (Σ (x – μ) ² / N)
Where: · σ is the population SD · Σ represents the sum of all values · x is an individual value · μ is the population mean (average) · N is the total number of values in the population
|
Formula:s = √ (Σ (x – x̄) ² / (n – 1))
Where: · s is the sample standard deviation · Σ represents the sum of all values · x is an individual value · x̄ is the representative value of the sample. · n is the number of values in the sample |
Key Differences:
- Sample SD uses the sample mean x̄ and sample size n.
- Population SD uses the population mean μ and total population size N.
- The sample SD includes an extra term in the denominator to correct for underestimation due to sampling.
Steps to calculate the STD:
Here is the procedure to calculate the STD:
Step 1:Collect the set of numerical data for which we want to calculate the standard deviation.
Step 2:Calculate the Mean by adding up all the numbers in the data set and then divide the sum by the total count of numbers.
Step 3:Find the Difference by Subtracting the mean from each data point. These variances can be either favorable (positive)or unfavorable (negative).
Step 4:Square each of the differences obtained in the previous step.
Step 5:Find the average of the squared differences. This is done by adding up all the squared differences and dividing the sum by the total count of numbers. The result is the variance.
Step 6:Calculate the Standard Deviation by taking the square root of the variance. The square root of the variance gives the standard deviation.
Solved Examples of STD
These examples demonstrate how we calculate the standard deviation
Problem 1:
Dataset: Daily temperatures in Celsius for a week in a particular city:
| 25 | 24 | 26 | 23 | 27 | 25 | 28 |
Solution:
Calculate the Mean:
Mean (x̄) = (25 + 24 + 26 + 23 + 27 + 25 + 28) / 7 = 178 / 7 = 25.43 (approx.)
Calculate the Difference
Difference from the mean:
| X | X – x̄ |
| 25 | -0.42859 |
| 24 | -1.4285 |
| 26 | 0.5714 |
| 23 | -2.4285 |
| 27 | 1.5714 |
| 25 | -0.4285 |
| 28 | 2.5714 |
Square of each Difference
| (X – x̄)2 |
| 0.1837 |
| 2.0409 |
| 0.3265 |
| 5.8981 |
| 2.4693 |
| 0.1837 |
| 6.6121 |
| Σ (x – x̄) ² = 17.7143 |
Calculate S.D
Putting the values in the formula:
s = √ (Σ (x – x̄) ² / (n – 1))
s = √ (1 / 7 -1) (17.7143)
s = √ 2.9524
s = 1.72
An sd calculator could be a handy tool for finding sample and population standard deviation in order to get rid of lengthy manual calculations.
Problem 2:
Consider a dataset representing the ages of individuals in a town:
| Ages | 30 | 35 | 28 | 40 | 25 |
Solution:
To find the population standard deviation for this dataset:
- Calculate the Mean (μ):
Mean (μ) = (30 + 35 + 28 + 40 + 25) / 5 = 158 / 5 = 31.6
- Calculate the Differenceand square each Difference
| X | X – μ | (X – μ)2 |
| 30 | -1.6 | 2.56 |
| 35 | 3.4 | 11.56 |
| 28 | -3.6 | 12.96 |
| 40 | 8.4 | 70.56 |
| 25 | -6.6 | 43.56 |
| – | – | ∑ (Xi – μ)2 = 141.2 |
- Calculate the Variance:
Variance = (2.56 + 11.56 + 12.96 + 70.56 + 43.56) / 5 = 141.2 / 5 = 28.24
- Calculate the Population Standard Deviation (σ):
σ = √ (Σ (x – μ) ² / N)
σ = √ (1 / 5) * (141.2)
σ = √ (0.2) (142.2)
σ = √ 28.24
σ = 5.31
Wrap up:
In this article, we explored the concept of Standard Deviation (SD. Represented by σ for populations and s for samples, SD quantifies data spread. Calculating it involves steps like finding differences, squares, variance, and taking square roots. We solved problems to illustrate its application, offering clarity on both population and sample SD.