Tag Archives: Statistics

How to Calculate Standard Deviation and Variance for Ungrouped Data: A Comprehensive Discussion

The variance and the standard deviation both are two useful and crucial statistical terms that are interlinked with one another. These terms are important when measuring the statistical values dispersion. A distribution’s dispersion is the amount by which its values vary from the distribution’s average. The amount of variance can be measured using a variety of metrics. The degree of dispersion is determined by measuring the variation of data points.

The terms variance and standard deviation are related because computing the square root of the variance provides the square root of the standard deviation for the specified data values. Meaningful measures of the variability of the data values are the variance, mean, and standard deviation. Measures of variance and standard deviation indicate how much and how far apart the data points are from the mean, respectively.

Two metrics are used to assess an investment’s risk: variance and standard deviation. The risk associated with investments rises as variance or standard deviation increases. Because the return on investment is determined using the mean, it is also significant. In this article, we will elaborate on the terms of the standard deviation and the variance in detail.

What is Variance?

The squared difference between the mean and each observation is the variance of these given data values. In 1918, R.A. Fisher delivered the idea of variance. Because of its significance, variance is widely used for measuring dispersion. Mathematically:

  • Sk 2 = ? (x – x?)2 / n where Sk 2 is variance and ? is the summation sign.


  • Sk 2 = (? x2 / n) – (?x / n)2

A set of data’s variance describes the degree of dispersion within it. The above-mentioned relationships indicate that as the values given approach one another and become equivalent, the variance decreases to zero.  Any values that are not zero have positive variances. 

When the data points are far separated from the mean and each other, it indicates a significant variation; when the data points are close to the mean and each other, it indicates minimal variance.

Defining the Standard Deviation (SD):

The positive square root of the variance signifies the important term, standard deviation. To compute the value of the standard deviation, three variables are required. In a data collection, the value of each point represents the initial variable, and a sum number represents each subsequent variable (x, x1, x2, x3, etc.) 

The mean is then applied to the values of the variables x and n, as well as the data values given to them. Symbolically

  • Sk = ? [? (x – x?)2 / n] where Sk is the standard deviation.


  • Sk = ? [(? x2 / n) – (?x / n)2]

The units used to represent the standard deviation match or correspond to the units used to represent the observations. One metric that shows how different something is from the mean is the standard deviation, often known as dispersion or spread. The standard deviation helps visualize a typical variation from the mean.

It is a preferred measure of variability since it goes back to the original units of data values of measurement. Just like with variance, there is a big variation if the data points are widely scattered from the mean and a small variation if the data points are near the mean.

The variance that the numbers depart from the average is determined by the standard deviation. Using the standard deviation, which is based on all data, is the simple method of evaluating dispersion. Consequently, a slight variation in one data point has an impact on the standard deviation.

How to calculate standard deviation and variance?

Using online tools is an easy way to calculate standard deviation and variance. Below are a few solved examples for calculating standard deviation and variance manually.

Example 1:

Calculate what will be the variance and the standard deviation for the following given scores of the students in the table.

Score (xi)786182593293442623


Step 1: Now we will perform the following necessary computations as given in the table:

xi786182593293442623?X = 498
xi 26084372167243481102486491936676529?X2 = 32824

Step 2: We will apply the relevant formula according to the computations that we perform in the above table.

The formula for variance

Sk 2 = (? x2 / n) – (?x / n)2

Putting the relevant values:

Sk 2 = (32824 / 9) – (498 / 9) ^ 2

S2 = 3647.11 – (248004 / 81)

Sk 2 = 3647.11 – 3061.78

Sk 2 = 585.33 scores ^ 2 Ans.

For standard deviation:

Sk = ? [(? x2 / n) – (?x / n) ^ 2]

Sk = ? (585.33)

Sk = 24.19 scores Ans.

Example 2:

Compute what will be the variance and the standard deviation for the values given in the following table.



Step 1: First of all, we are to compute the average of the given data values in the above table.

x? = (3 + 6 + 7 + 8 + 10 + 12 + 12 + 14) / 8

x? = (72) /8

x? = 9

Step 2: Now we are to compute the following necessary computations in the table to proceed to the next for determining the variance and the standard deviation.

x(xk x?)(xk x?)2
Total? (x – ?)2 = 94

Step 3: We will apply the relevant formula according to the computations that we perform in the above table.

The formula for variance

Sk 2 = ? (xk – x?) 2 / n

Putting the relevant values:

Sk 2 = (94) / 8

Sk 2 = 11.75

For standard deviation:

Sk = ? [? (x – x?)2 / n]

Sk = ? (11.75)

Sk = 3.4278 Ans.

Wrap Up:

In this article, we have covered the key ideas of variance and standard deviation in a lot of detail. We have discussed the meanings of these terms as well as the numerous mathematical relations that enable us to compute these crucial terms for observation and analysis of the given data. 

The New Technologies Used in The Statistics

The collecting, analysis, and interpretation of data are all covered by statistics, a significant area of mathematics. It provides essential tools to summarize and describe dataset through measures like mean, mode ,median and standard deviation. Inferential theory in statistics enables us to make big predictions about large populations based on smaller samples. They do it by hypothetical testing and confidence interval usage. 

One more important term probability is used in statistics which allow the assessment of likelihood in various situations. Correlation and regression helps in understanding relation between variables and facilitate in identifying patterns. 

Statistics has applications in science, business ,social studies which enable professionals to draw evidence based conclusions so that they can make informed decisions. They can gain valuable data insights.  You can use a test statistic calculator  to find the mean of one population or two. 

Here are few technologies to used in statistics that are in trends and gaining prominence in 2023

Big Data Analytics

With the rising growth of data , its important to have tools that can handle data in millions. The concept of big data is used in data handling and extracting valuable insight from massive datasets. Different tools like Apache Hadoop, Apache Spark and distributed databases enable statisticians to process and analyze vast amounts of data efficiently.

Machine Learning and Artificial Intelligence:

Statistical modeling and prediction have been transformed by machine learning techniques. Deep learning, random forests and support vector models are used in pattern recognition, classification. AI powered tools have automated data analysis processes making it easy for statisticians to find relevant patterns. 

Data Visualization 

New tools and libraries have appeared to create interactive data visualization and dashboards. These tools enable decision-makers to interactively study data and make a clear decision about their findings. They can easily express their reports about data through interactive dashboards.

Bayesian Statistics:

The statistical inference process can be guided by the Bayesian analysis approach, which bears the name of English mathematician Thomas Bayes. It enables the combination of prior knowledge about a population parameter with evidence from data in a sample. Markov Chain Monte Cario (MCMC)methods and other Bayesian inference techniques have made complex probabilistic models more accessible. 

Cloud computing:

Cloud computing has affected the statics field by providing scalable and cost-effective solutions for data storage. Since your data is stored you can use it anywhere for processing and analyzing. Now statisticians can take advantage of cloud based data without investment in local infrastructure. For more powerful and faster computation of dataset cloud servers also provide data analytics tools and machine learning algorithms.

 Additionally cloud computing provides multiple statisticians to work on single projects so as to provide remote working. It provides an environment for collaborative work. Cloud computing provides security to your data with backup options and data privacy. Embracing cloud technology empowers statisticians to conduct sophisticated analyses to derive meaningful insights of data. A test statistic calculator can be used to find the mean if deviation is given.


Statistics is a crucial tool in various fields including science, business, social sciences, economics and engineering. These technologies help researchers and decision makers to draw reliable conclusions. It helps in identifying patterns and making predictions according to the data set. The technology to be used in statistics is constantly evolving. For more information on new trends in statistics it’s best to refer to recent research articles , papers or conferences and industry publications.