A Detailed Look Into Standard Deviation

A Guide To Standard Deviation In Descriptive Statistics

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Have you wondered why we humans study statistics? Weren’t mathematics & all its various branches frightening enough? Well,  the necessity of statistical interpretation stems from the need to observe and understand patterns, distributions and intrinsic behavior of datasets. Statistics enables us to analyze and study the nature of relationships between data and ascertain the accuracy of interpretive procedures.

In simple words, statistics help us study the nature and behavior of some data set. With powerful descriptive and inferential techniques, stats provide us with excellent, in-depth analytical tools to explore data sets’ properties.

One of the most common and potent statistical techniques for measuring the variability of data is the standard deviation procedure. This article intends to explore this simple but beneficial technique in details and look into its myriad applications.

Let’s start with some definition.

Understanding Standard Deviation

TO understand the idea behind standard deviation, we first need to know about the concept of variability and its necessity in data analysis.

For any data set or distribution, variability defines the spread of data in a particular distribution.  A simple concept tells us how scattered, dispersed or spread out the data is.

Above, we have a normal distribution, and the standard deviation defines its width or the spread of the data values concerning the midpoint of the dataset or the plot. Measures of central tendency such as mean, median and mode, allow statisticians to determine the center or midpoint of a distribution.

Standard deviation is a measure of variability or the variation of data concerning the distribution‘s central point. The standard deviation is used frequently to study, describe, and make inferences about distributions along with range and interquartile range.

From an application perspective, standard deviation allows us to discern vital underlying information or causes behind some data pattern.

Using Standard Deviation

As mentioned, the standard deviation allows statisticians to discern the variability of data from the central point. Let us find out how it does so.

Calculate the standard deviation of the following distribution:

6, 8, 10, 12, 14, 16, 18, 20

As per the definition of standard deviation, we need to determine the variability of the distribution elements with respect to some central value or midpoint. The Mean or Expected Value is the midpoint value that specifies the dispersion or SD primarily.

Mean is nothing but the average of the elements in a data set.

For the above distribution,

Let the dataset be denoted by X= 6, 8, 10, 12, 14, 16, 18, 20 and n= no. of elements in X

  • Mean or the Average ~= [∑ni=1Xi]/ n. The common notation or symbol of mean is µ.

In this case, µ is approximately equal to 13.

  • Next up, we need to calculate the variance of the data. It is nothing more than the spread of each data point from the average or the mean, µ.

 The variance of X is denoted by σ and calculated as follows.

 σ= Var (X) = (1/n-1) * ( ∑ni=1 (Xi – µ)2  )

 For this problem, the variance comes around to  

Variance:

((6-13)2 + (8-13)2 + (10-13)2 + (12-13)2 + (14-13)2 + (16-13)2 + (18-13)2 + (20-13)2    / (n-1) =

(49+25+9+1+1+9+25+49)/ 7=

(104)/7 = 24

  • Standard Deviation is the square root variance. For our sum, we have √ σ= √(24) = 4.582

And, that is how we calculate the standard deviation for any data distribution. Standard deviation calculators follow the above steps & formulas to determine sample or population standard deviation.

Larger standard deviations in data indicate greater variability in data, whereas smaller values indicate less variability.

Normal Distributions and Standard Deviations

Standard deviations are calculated from the mean, a measure of central tendency. If we plot the dispersion or variability, then we will obtain a bell-shaped or normal distribution curve.

Standard deviation defines the width of any normal distribution curve: the extent of dispersion of the data elements from the midpoint (mean).

Here’s the graph for the above data set done in Excel. 

As observable, the curve crests at the midpoint at 13, which is the mean or central tendency measure used to calculate the whole dataset’s standard deviation. Remember that not every data set follows a normal distribution pattern.

The above curve is also known as the Probability Density Function curve, as it denotes a variable’s density over a given range. (that is, the nature of the data values in our set)

Another curve called the cumulative distribution function curve shows the overall trend of change in the variable or a data set. For our problem, we obtain the following figure.

Standard deviation is a powerful statistical tool and finds widespread applications in data science & machine learning.

Applying Standard Deviation

The central idea behind the standard deviation is observing the data’s tendency to deviate or disperse from the mean or expected value. Calculating the SD of a sample or a whole population can help identify trends & patterns pertaining to that data.

  • In biostatistics, variance and standard deviation are used to observe, ascertain and investigate some biological trait or phenomenon in a population.
  • Population census employs standard deviation to determine age, gender and other demographic parameters.
  • The standard deviation has huge applications on the business front. From predicting share prices to risk management, the technique is used to measure and estimate probable events.
  • In data science and business intelligence, the standard deviation is a rudimentary tool to study, analyze and mine data. Coupled with probability, statistical techniques such as standard deviation help business analysts and data scientist make accurate predictions from BIG amounts of DATA.
  • Data is central to AI, machine learning, deep learning and the like. Like in data science, statistical methods, like standard deviation & hypothesis testing, study massive amounts of data. These methods enable the extraction of insightful information from large datasets, which are then used to train & teach machine learning models.

The standard deviation has been used to design loss functions in deep learning networks. Loss functions allow the AI system to fine-tune their predictions through careful analyses & comparison of training results. Standard deviation loss functions calculate the variability or deviation from the expected or desired value. The AI model then makes the necessary corrections to increase & improve the accuracy of results.

Well, that brings us to the end of this write-up.  Let’s hope this little article on standard deviation and descriptive statistics was informative & educative enough for you. Remember, hard work, practice and intelligence are crucial to becoming a stat pro. So give it your all and seek writing assistance from genuine statistics assignment writing services in case of any difficulty.

All the best!

Author-Bio: Ronald McLean is a statistics professor from a reputed university in Ohio, the USA. He is also an avid data science enthusiast, freelance writer, blogger and part-time tutor at MyAssignmenthelp.com, a leading digital academic writing service in the United States.

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