Standard deviation is a widely used measure of dispersion or variability in a dataset. It quantifies the average distance or spread of data points from the mean of the dataset. The standard deviation provides valuable insights into the consistency, reliability, and distribution of data around the mean.

Formula for Standard Deviation:

For a sample:

s=n−1∑(xi​−xˉ)2​​

For a population:

σ=N∑(xi​−μ)2​​

Where:

• xi​ represents each individual data point

• 
  $\stackrel{}{<br>}$

  xˉ represents the sample mean
  
  $\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">represents\; the\; population\; mean</span>}$

• $n$ represents the sample size
• $N$ represents the population size
• 
  $\sum ({�}_{�}-\overline{�}{)}^2$

  ∑(xi​−xˉ)2 or 

  
  $$

  ∑(xi​−μ)2 represents the sum of squared differences between each data point and the mean

• 
  
  $\sqrt{}$

  ​ denotes the square root

Characteristics and Considerations:

1. Sensitivity to Variability: Standard deviation measures the spread of data points around the mean. A higher standard deviation indicates greater variability, while a lower standard deviation indicates less variability.
2. Units of Measurement: Standard deviation is expressed in the same units as the original data, making it interpretable and relevant in various contexts.
3. Square Root of Variance: Standard deviation is the square root of variance, providing a measure of dispersion that considers the original units of measurement.
4. Population vs. Sample: The formulas for standard deviation differ slightly depending on whether the data represents a population or a sample, with the sample formula (using
   
   $$

   n−1) providing an unbiased estimate of the population standard deviation.

5. Comparison with Mean Deviation: Compared to mean deviation, which considers the absolute differences, standard deviation considers the squared differences, giving more weight to larger deviations and providing a comprehensive measure of data variability.

Example:

Consider the following dataset representing the daily sales (in units) of a product for a week:

100,105,110,115,120

100,105,110,115,120

To calculate the standard deviation:

1. Calculate the sample mean:
   
   $<span\; style="text-transform:\; math-auto;">x</span>=\frac{100+105+110+115+120}{5}=\frac{550}{5}=110\text{ units<span class="base" style="font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif;"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="mord mathnormal"></span></span></span></span></span></span>}$

2. 
   $5100+105+110+115+120​=5550​=110 units$

3. Calculate the squared differences between each data point and the mean:
   
   $(100-110{)}^2=100$

    

   (100−110)2=100
   $(105-110{)}^2=25$

    

   (105−110)2=25
   $(110-110{)}^2=0$

    

   (110−110)2=0
   $(115-110{)}^2=25$

    

   (115−110)2=25
   $(120-110{)}^2=100$

    

   (120−110)2=100

4. Calculate the sum of squared differences:
   
   $\sum ({\mathrm{<span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal">x</span><span\; class="msupsub"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist"><span><span\; class="pstrut"></span><span\; class="sizing\; reset-size6\; size3\; mtight"><span\; class="mord\; mathnormal\; mtight">i</span></span></span></span><span\; class="vlist-s"></span></span><span\; class="vlist-r"><span\; class="vlist"><span></span></span></span></span></span></span><span\; class="mspace"></span><span\; class="mbin">-</span><span\; class="mspace"></span></span><span\; class="base"><span\; class="strut"></span><span\; class="mord\; accent"><span\; class="vlist-t"><span\; class="vlist-r"><span\; class="vlist"><span><span\; class="pstrut"></span><span\; class="mord\; mathnormal">x</span></span><span><span\; class="pstrut"></span><span\; class="accent-body"><span\; class="mord">ˉ</span></span></span></span></span></span></span></span></span>}}_{}\stackrel{}{ˉ}{)}^2=100+25+0+25+100=250$

    

   ∑(xi​−xˉ)2=100+25+0+25+100=250

5. Calculate the sample standard deviation:
   
   $s=\sqrt{\frac{250}{5-1}}=\sqrt{\frac{250}{4}}=\sqrt{62.5}\approx 7.91\text{ units}$

    

   s=5−1250​​=4250​​=62.5​≈7.91 units

The sample standard deviation of the daily sales for the week is approximately 7.91 units, indicating the average distance of daily sales from the mean of 110 units.

Standard deviation is a fundamental measure of dispersion that quantifies the average spread of data points around the mean. By considering the squared differences, standard deviation provides a comprehensive and interpretable measure of data variability, making it a valuable tool in descriptive statistics, data analysis, and statistical inference.