🕷️ Crawler Inspector

 Sign In Courses Explore all Vedantu courses by class or target exam, starting at ₹9000  Find courses by class  Find courses by target [ Long Term Courses Guaranteed improvement in marks or get your fees back](https://www.vedantu.com/course/ai-live?show_bn=t1&couponCode=VNAV20&itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu) [ One-to-one LIVE classes Learn one-to-one with a teacher for a personalised experience](https://courses.vedantu.com/one-to-one-live/?itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu&utm_source=new_nav_menu&utm_medium=new_nav_menu&utm_campaign=new_nav_menu) [ Topic specific courses Master any topic at just ₹1](https://www.vedantu.com/microcourses-list)  Courses for Kids Courses for Kids Confidence-building & personalised learning courses for Class LKG-8 students [ English Superstar Age 4 - 8 Level based holistic English program](https://www.vedantu.com/english-superstar?&itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu) [ Summer Camp For Lkg - Grade 10 Limited-time summer learning experience](https://courses.vedantu.com/campsforkids?itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu) [ Spoken English Class 3 - 5 See your child speak fluently](https://www.vedantu.com/superspeakers?&itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu) [ Learn Maths Class 1 - 5 Turn your child into a Math wizard](https://www.vedantu.com/supermaths?&itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu) [ Coding Classes Class 1 - 8 Learn to build apps and games, be future ready](https://www.vedantu.com/supercoders?&itm_source=new_nav_menu&itm_medium=new_nav_menu&itm_campaign=new_nav_menu) Free study material Get class-wise, author-wise, & board-wise free study material for exam preparation [NCERT Solutions](https://www.vedantu.com/ncert-solutions) [CBSE](https://www.vedantu.com/cbse) [JEE Main](https://www.vedantu.com/jee-main) [JEE Advanced](https://www.vedantu.com/jee-advanced) [NEET](https://www.vedantu.com/neet) [Question and Answers](https://www.vedantu.com/question-answer) [Popular Book Solutions](https://www.vedantu.com/textbook-solutions) Subject wise Concepts [ICSE & State Boards](https://www.vedantu.com/state-boards) Kids Concept [Online Tuition](https://www.vedantu.com/online-tuition) [Competative Exams and Others](https://www.vedantu.com/exams) [Worksheets](https://www.vedantu.com/worksheets) [Offline Centres](https://www.vedantu.com/offline-centres?utm_source=web&utm_medium=navbar&utm_campaign=seopages) Online Tuition Get class-wise, subject-wise, & location-wise online tuition for exam preparation Online Tuition By Class Online Tuition By Subject Online Tuition By Location More Know about our results, initiatives, resources, events, and much more [ Our results A celebration of all our success stories](https://courses.vedantu.com/vedanturesults) [ Child safety Creating a safe learning environment for every child](https://www.vedantu.com/child-safety) [ Help India Learn Helps in learning for Children affected by the Pandemic](https://www.helpindialearn.in/?itm_source=Homepage&itm_medium=Homepage-Navigation-bar&itm_campaign=Navigation-bar) [ WAVE Highly-interactive classroom that makes learning fun](https://www.vedantu.com/wave) [ Vedantu Improvement Promise (VIP) We guarantee improvement in school and competitive exams](https://www.vedantu.com/vedantu-improvement-promise?itm_source=Web&itm_medium=Vip_Web_Homepage_Banner_Knowmore&itm_campaign=Vip_Launch) [ Master talks Heartfelt and insightful conversations with super achievers](https://www.vedantu.com/mastertalks)  Our initiatives  Resources [ About us Know more about our passion to revolutionise online education](https://www.vedantu.com/v/about-us) [ Careers Check out the roles we're currently hiring for](https://www.vedantu.com/careers) [ Our Culture Dive into Vedantu's Essence - Living by Values, Guided by Principles](https://www.vedantu.com/culture) [ Become a teacher Apply now to join the team of passionate teachers](https://www.vedantu.com/become-a-teacher) [ Vedantu Partner Program Vedantu powered coaching institute near you](https://courses.vedantu.com/vedantu-partner-program/) [ Contact us Got questions? Please get in touch with us](https://www.vedantu.com/v/contacts) [ Vedantu Store](https://www.vedantu.store/) - [Maths](https://www.vedantu.com/maths) - Exponential Smoothing: Concepts, Formulas & Applications # Exponential Smoothing: Concepts, Formulas & Applications Reviewed by: [Rama Sharma](https://vedantu.com/authors/rama-sharma)    Download PDF [NCERT Solutions](https://www.vedantu.com/ncert-solutions)  [NCERT Solutions for Class 12](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12) [NCERT Solutions for Class 11](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11) [NCERT Solutions for Class 10](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10) [NCERT Solutions for class 9](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9) [NCERT Solutions for class 8](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8) [NCERT Solutions for class 7](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-7) [NCERT Solutions for class 6](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-6) [NCERT Solutions for class 5](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-5) [NCERT Solutions for class 4](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-4) [NCERT Solutions for Class 3](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-3) [NCERT Solutions for Class 2](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-2) [NCERT Solutions for Class 1](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-1) [CBSE](https://www.vedantu.com/cbse)  [CBSE class 3](https://www.vedantu.com/cbse/class-3) [CBSE class 4](https://www.vedantu.com/cbse/class-4) [CBSE class 5](https://www.vedantu.com/cbse/class-5) [CBSE class 6](https://www.vedantu.com/cbse/class-6) [CBSE class 7](https://www.vedantu.com/cbse/class-7) [CBSE class 8](https://www.vedantu.com/cbse/class-8) [CBSE class 9](https://www.vedantu.com/cbse/class-9) [CBSE class 10](https://www.vedantu.com/cbse/class-10) [CBSE class 11](https://www.vedantu.com/cbse/class-11) [CBSE class 12](https://www.vedantu.com/cbse/class-12) [NCERT](https://www.vedantu.com/ncert) [CBSE Study Material](https://www.vedantu.com/study-material/cbse-study-material)  [CBSE Sample Papers](https://www.vedantu.com/sample-papers) [CBSE Syllabus](https://www.vedantu.com/syllabus/cbse-syllabus) [CBSE Previous Year Question Paper](https://www.vedantu.com/previous-year-question-paper/cbse-previous-year-question-papers) [CBSE Important Questions](https://www.vedantu.com/cbse/important-questions) [Marking Scheme](https://www.vedantu.com/cbse/cbse-marking-scheme) [Textbook Solutions](https://www.vedantu.com/textbook-solutions)  [RD Sharma Solutions](https://www.vedantu.com/rd-sharma-solutions) [Lakhmir Singh Solutions](https://www.vedantu.com/textbook-solutions/lakhmir-singh-solutions) [HC Verma Solutions](https://www.vedantu.com/hc-verma-solutions) [TS Grewal Solutions](https://www.vedantu.com/commerce/ts-grewal-solutions) [DK Goel Solutions](https://www.vedantu.com/commerce/dk-goel-solutions) [NCERT Exemplar Solutions](https://www.vedantu.com/textbook-solutions/ncert-exemplar-problems-solutions) [CBSE Notes](https://www.vedantu.com/revision-notes)  [CBSE Notes for class 12](https://www.vedantu.com/revision-notes/cbse-class-12-notes) [CBSE Notes for class 11](https://www.vedantu.com/revision-notes/cbse-class-11-notes) [CBSE Notes for class 10](https://www.vedantu.com/revision-notes/cbse-class-10-notes) [CBSE Notes for class 9](https://www.vedantu.com/revision-notes/cbse-class-9-notes) [CBSE Notes for class 8](https://www.vedantu.com/revision-notes/cbse-class-8-notes) [CBSE Notes for class 7](https://www.vedantu.com/revision-notes/cbse-class-7-notes) [CBSE Notes for class 6](https://www.vedantu.com/revision-notes/cbse-class-6-notes)      [](https://www.vedantu.com/bookademo?&utm_source=mau_crm&utm_medium=seoTopBanner&utm_campaign=pagelanding&utm_term=11neet_bod&couponCode=)  Latest Updates ***  Book Free Demo : [1 Teacher, 1 Student, 100% Attention (Classes start from ₹800 per hour)](https://www.vedantu.com/one-to-one-course?utm_source=seo_pages&utm_medium=top_widget_section&utm_campaign=maths&utm_content=/maths/exponential-smoothing&utm_term=mathematics&grade=) ## How Does Exponential Smoothing Improve Forecasting Accuracy? Exponential Smoothing Equations Exponential smoothing was initially introduced in the statistical literature without considering the past work done by Robert Goodell Brown in 1956 and then further expanded by Charles C. Holt in 1957. Exponential smoothing is a reliable principle for smoothing time series data through the exponential window function. The controlling input of the exponential smoothing calculation is stated as the smoothing factor or the smoothing constant. Forecast of the weighted averages of past observations are introduced using exponential smoothing methods, with the weights breaking down exponentially as the observations get formed. In other words, the more the latest the observation the higher the corresponding weight. As we are aware of the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It can be easily applied for making determinations on the basis of prior assumptions by the user, such as seasonality. Exponential smoothing is primarily used for time-series data analysis. Exponential Smoothing Formula The exponential smoothing formula is derived by: st = θxt+(1 – θ)st-1= st-1+ θ(xt – st-1) Here, st is a former smoothed statistic, it is the simple weighted average of present observation xt st-1 is former smoothed statistic θ is smoothing factor of data; 0 \< θ \< 1 t is time period If the value of the smoothing factor is greater, then the level of smoothing will be minimized. Value of α nearer to 1 minimum smoothing effect and offer higher weights to recent changes in the data, while the value of θ nearer to zero has higher smoothing effect and is less responsive to recent changes. There is no precise method of choosing θ. Accurate factors are selected on the basis of the statistician's judgments or else a statistical technique may be used to optimize the value of θ. For example, the method of least squares can be used to estimate the value of θ for which the sum of the quantities is diminished. Exponential Smoothing Methods The three exponential smoothing methods to determine exponential smoothing. They are: 1. Simple or single exponential smoothing 2. Double exponential smoothing 3. Triple exponential smoothing Single Exponential Smoothing If the data which is obtained has no trend and no seasonal pattern, then the single exponential smoothing method for forecasting the time series is primarily used. This method makes use of weighted moving averages with exponentially decreasing weights. The single exponential smoothing method formula is given by: st = θxt+(1 – θ)st-1 = st-1 + θ(xt – st-1) Double Exponential Smoothing The double exponential smoothing method is also known as Holt's trend corrected or second-order exponential smoothing. This method is primarily used to forecast the time series when the data has a linear trend and no seasonal pattern. The motive of double exponential smoothing is to introduce a term considering the possibility of a series indicating some form of trend. This slope component is itself reformed through exponential smoothing. The double exponential smoothing formula is derived by: S1 = y1 B1 = y1-y0 For t\>1, st = θyt + (1 – θ)(st-1 + bt-1) βt = β(st – st-1) + (1 – β)bt-1 Here, St is smoothed statistic, it is the simple weighted average of present observation yt st-1 = former smoothed statistic θ = smoothing factor of data; 0 \< θ \< 1 t = time period bt = accurate estimation of trend at time t β = trend smoothing factor; 0 \< β \<1 Triple Exponential Smoothing In the triple exponential smoothing method, exponential smoothing is used thrice. This method is primarily used to forecast the time series when the data has both linear trend and seasonal patterns.This method is also known as holt-Winters exponential smoothing. The triple exponential smoothing formula is derived by: s 0 \= x 0 s t \= α x t c t − L \+ (1 - α)(s t − 1 \+ b t − 1 ) b t \= β(s t \- s t − 1 \+ (1 - β)b t − 1 c t \= γ x t s t \+ (1 - γ)c t − L Here, st = smoothed statistic, it is the simple weighted average of present observation xt st-1 = previous smoothed statistic α = smoothing factor of data; 0 \< α \< 1 t = time period bt = accurate estimation of trend at time t β = trend smoothing factor; 0 \< β \<1 ct = sequence of seasonal error-free factors at time t γ = seasonal variation smoothing factor; 0 \< γ \< 1 Solved Examples 1\. The Sales of Books in a Bookstall for the Last 10 Months is Given Below in Tabulated Form. Calculate the Simple Exponential Smoothing Estimating α =0.3 for the Below Data. | | | |---|---| | Month | No. of Books Sold | | January | 30 | | February | 25 | | March | 35 | | April | 25 | | May | 20 | | June | 30 | | July | 35 | | August | 40 | | September | 30 | | October | 35 | ## Solution | | | | |---|---|---| | Month | No. of Books Sold | Exponential Smoothing(α =0.3) | | January | 30 | 30\.00 | | February | 25 | 30\.00 | | March | 35 | 28\.50 | | April | 25 | 30\.45 | | May | 20 | 14\.1 | | June | 30 | 15\.87 | | July | 35 | 20\.109 | | August | 40 | 24\.5763 | | September | 30 | 29\.20341 | | October | 45 | 29\.442387 | | November | \- | 34\.1096709 | Quiz Time 1\. The Use of Smoothing Technique is Accurate When 1. The primary source of variation is random behavior 2. Seasonality is includes 3. Data exhibit a strong trend 4. All the above are accurate 2\. Times Series Data is Classified in 1. Two components 2. Three components 3. Four components 4. Five components ## FAQs on Exponential Smoothing: Concepts, Formulas & Applications 1\. What is meant by exponential smoothing in time series analysis? Exponential smoothing is a **time series forecasting method** for univariate data that can be extended to support data with a systematic trend or seasonal component. It assigns exponentially decreasing weights to past observations, meaning more recent data points are given more importance in the forecast than older data points. This makes it particularly effective for short-term forecasting. 2\. What is the formula for simple exponential smoothing? The formula for simple exponential smoothing (SES) is used for data without a trend or seasonality. The forecast for the next period (St+1) is calculated as: **St+1 = αYt + (1 - α)St** Where: - **St+1** is the forecast for the next period. - **α (alpha)** is the smoothing constant (0 ≤ α ≤ 1). - **Yt** is the actual observed value in the current period t. - **St** is the forecast for the current period t. 3\. What is the role of the smoothing constant (alpha) in exponential smoothing? The smoothing constant, **alpha (α)**, determines how much weight is given to the most recent observation versus older observations. Its value ranges from 0 to 1. - A **high alpha** (e.g., 0.8) makes the forecast highly responsive to recent changes, as it gives more weight to the latest actual data. - A **low alpha** (e.g., 0.2) results in a smoother forecast that is less influenced by recent random fluctuations, as it gives more weight to past forecasts. The choice of alpha is a trade-off between smoothing out noise and quickly adapting to changes in the underlying pattern. 4\. When should you use exponential smoothing for forecasting? Exponential smoothing is best used for **short-term forecasting** when dealing with time series data. The specific type of exponential smoothing to use depends on the data's characteristics: - Use **Simple Exponential Smoothing** when the data is stationary (no trend or seasonality). - Use **Double Exponential Smoothing (Holt's method)** when the data has a clear trend but no seasonality. - Use **Triple Exponential Smoothing (Holt-Winters method)** when the data exhibits both a trend and seasonality. 5\. What are the key advantages of using the exponential smoothing method? The main advantages of exponential smoothing are: - **Simplicity:** The models are relatively easy to understand and computationally inexpensive, requiring only a few data points to start. - **Accuracy for Short-Term Forecasts:** It often provides highly accurate forecasts for the near future, especially when the underlying patterns are stable. - **Emphasis on Recent Data:** It intuitively gives more importance to recent events, which is often a realistic assumption for many business and economic time series. 6\. What are the limitations of exponential smoothing? Despite its benefits, exponential smoothing has some limitations: - **Lagging Forecasts:** The models tend to lag behind actual data, especially when there are sharp turns or shifts in the pattern. - **Parameter Selection:** Finding the optimal values for the smoothing constants (alpha, beta, gamma) can be subjective and may require trial and error. - **Difficulty with Long-Term Forecasts:** While excellent for the short term, its accuracy diminishes significantly for long-range forecasting, especially for simple models that cannot project trends far into the future. 7\. How does Double Exponential Smoothing improve upon the simple method? Double Exponential Smoothing, or **Holt's method**, improves upon simple smoothing by explicitly accounting for trends in the data. It uses two smoothing constants: alpha (α) for the level and beta (β) for the trend. By forecasting both the level and the slope of the data, it can generate more accurate predictions for time series that are consistently increasing or decreasing, a task where simple exponential smoothing would systematically lag behind. 8\. Which forecasting model is better: Exponential Smoothing or ARIMA? Neither model is universally better; the choice depends on the data. **Exponential Smoothing** models are simpler, more intuitive, and perform well for data with clear trend and seasonality patterns. They are essentially weighted averages of past observations. **ARIMA (AutoRegressive Integrated Moving Average)** models are more complex and powerful. They can model more intricate time series structures by using correlations between past values (autocorrelation). While ARIMA can be more accurate for complex data, it requires more statistical expertise to identify, fit, and validate the model. Recently Updated Pages [Master Maths Concepts & Formulas \| Fast Learning Guide](https://www.vedantu.com/maths) [Properties of Trapezium - Definition, Formulas & Examples](https://www.vedantu.com/maths/trapezium) [Relationship Between Zeroes & Coefficients of Polynomials](https://www.vedantu.com/maths/relationship-between-zeroes-and-coefficients-of-polynomials) [Properties of Rational Numbers: Closure, Commutative, Associative & More](https://www.vedantu.com/maths/properties-of-rational-numbers) [Precision in Math: Master Accuracy & Consistency](https://www.vedantu.com/maths/precision) [Volume of a Cuboid Formula: Definition, Examples & Practice](https://www.vedantu.com/maths/formula-of-volume-of-a-cuboid) [Master Maths Concepts & Formulas \| Fast Learning Guide](https://www.vedantu.com/maths) [Properties of Trapezium - Definition, Formulas & Examples](https://www.vedantu.com/maths/trapezium) [Relationship Between Zeroes & Coefficients of Polynomials](https://www.vedantu.com/maths/relationship-between-zeroes-and-coefficients-of-polynomials) [Properties of Rational Numbers: Closure, Commutative, Associative & More](https://www.vedantu.com/maths/properties-of-rational-numbers) [Precision in Math: Master Accuracy & Consistency](https://www.vedantu.com/maths/precision) [Volume of a Cuboid Formula: Definition, Examples & Practice](https://www.vedantu.com/maths/formula-of-volume-of-a-cuboid) - 1 - 2 Trending topics [Area of Square Using Diagonal: Formula & Step-by-Step Guide](https://www.vedantu.com/maths/area-of-square-using-diagonal) [1 Billion in Rupees: Explained with Examples & Conversion](https://www.vedantu.com/maths/1-billion-in-rupees) [How to Calculate Percentage Step by Step](https://www.vedantu.com/maths/how-to-calculate-percentage) [Mean, Median and Mode Explained with Examples](https://www.vedantu.com/maths/mean-median-mode) [Logarithm Values From 1 to 10: Complete Table](https://www.vedantu.com/maths/log-values-from-1-to-10) [Value of e in Maths: Explained with Examples](https://www.vedantu.com/maths/value-of-e) [Area of Square Using Diagonal: Formula & Step-by-Step Guide](https://www.vedantu.com/maths/area-of-square-using-diagonal) [1 Billion in Rupees: Explained with Examples & Conversion](https://www.vedantu.com/maths/1-billion-in-rupees) [How to Calculate Percentage Step by Step](https://www.vedantu.com/maths/how-to-calculate-percentage) [Mean, Median and Mode Explained with Examples](https://www.vedantu.com/maths/mean-median-mode) [Logarithm Values From 1 to 10: Complete Table](https://www.vedantu.com/maths/log-values-from-1-to-10) [Value of e in Maths: Explained with Examples](https://www.vedantu.com/maths/value-of-e) - 1 - 2 Other Pages [CBSE Date Sheet 2026 Revised for Class 10 and 12](https://www.vedantu.com/cbse/cbse-date-sheet) [Supply and Demand](https://www.vedantu.com/commerce/supply-and-demand) [JEE Main 2026: Exam Dates, Session 2 Updates, City Slip, Admit Card & Latest News](https://www.vedantu.com/jee-main) [JEE Main 2026 Jan 21 Shift 1 Question Papers with Solutions & Answer Keys – Detailed Day 1 Analysis](https://www.vedantu.com/jee-main/2026-question-paper-21-january-shift-1) [CUET 2026: Exam Dates, Syllabus, Application Form](https://www.vedantu.com/cuet) [CBSE Class 12 Date Sheet 2026 – Download the Updated 12th Board Exam Timetable PDF](https://www.vedantu.com/cbse/cbse-class-12-date-sheet-time-table) [CBSE Date Sheet 2026 Revised for Class 10 and 12](https://www.vedantu.com/cbse/cbse-date-sheet) [Supply and Demand](https://www.vedantu.com/commerce/supply-and-demand) [JEE Main 2026: Exam Dates, Session 2 Updates, City Slip, Admit Card & Latest News](https://www.vedantu.com/jee-main) [JEE Main 2026 Jan 21 Shift 1 Question Papers with Solutions & Answer Keys – Detailed Day 1 Analysis](https://www.vedantu.com/jee-main/2026-question-paper-21-january-shift-1) [CUET 2026: Exam Dates, Syllabus, Application Form](https://www.vedantu.com/cuet) [CBSE Class 12 Date Sheet 2026 – Download the Updated 12th Board Exam Timetable PDF](https://www.vedantu.com/cbse/cbse-class-12-date-sheet-time-table) - 1 - 2 [![]()](https://www.vedantu.com/bookademo?&utm_source=mau_crm&utm_medium=seoSideBanner&utm_campaign=pagelanding&utm_term=11neet_bod&couponCode=) Study material NCERT - [NCERT](https://www.vedantu.com/ncert?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions](https://www.vedantu.com/ncert-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Maths](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-maths?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Physics](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-physics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Chemistry](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-chemistry?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Biology](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-biology?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Business Studies](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-business-studies?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Economics](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-economics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Accountancy](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-accountancy?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 English](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-english?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 12 Hindi](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-hindi?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Maths](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-maths?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Physics](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-physics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Chemistry](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-chemistry?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Biology](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-biology?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Business Studies](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-business-studies?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Economics](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-economics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Accountancy](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-accountancy?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 English](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-english?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 11 Hindi](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-11-hindi?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 10](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 10 Maths](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10-maths?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 10 Science](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10-science?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 10 English](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10-english?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 10 Social Science](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8-social-science?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 10 Hindi](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-10-hindi?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 9](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 9 Maths](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9-maths?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 9 Science](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9-science?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 9 English](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9-english?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 9 Social Science](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9-social-science?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 9 Hindi](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-9-hindi?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 8](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 8 Maths](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8-maths?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 8 Science](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8-science?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 8 English](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8-english?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 8 Social Science](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8-social-science?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Solutions for Class 8 Hindi](https://www.vedantu.com/ncert-solutions/ncert-solutions-class-8-hindi?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Books](https://www.vedantu.com/ncert-books?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Books Class 12](https://www.vedantu.com/ncert-books/ncert-books-class-12?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Books Class 11](https://www.vedantu.com/ncert-books/ncert-books-class-11?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Books Class 10](https://www.vedantu.com/ncert-books/ncert-books-class-10?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Books Class 9](https://www.vedantu.com/ncert-books/ncert-books-class-9?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Books Class 8](https://www.vedantu.com/ncert-books/ncert-books-class-8?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) Reference book solutions - [Reference Book Solutions](https://www.vedantu.com/textbook-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [HC Verma Solutions](https://www.vedantu.com/hc-verma-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [RD Sharma Solutions](https://www.vedantu.com/rd-sharma-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [RS Aggarwal Solutions](https://www.vedantu.com/rs-aggarwal-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NCERT Exemplar Solutions](https://www.vedantu.com/textbook-solutions/ncert-exemplar-problems-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Lakhmir Singh Solutions](https://www.vedantu.com/textbook-solutions/lakhmir-singh-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [DK Goel Solutions](https://www.vedantu.com/commerce/dk-goel-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [TS Grewal Solutions](https://www.vedantu.com/commerce/ts-grewal-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Sandeep Garg](https://www.vedantu.com/commerce/sandeep-garg-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) Competitive Exams - [Competitive Exams](https://www.vedantu.com/exams?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main](https://www.vedantu.com/jee-main?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced](https://www.vedantu.com/jee-advanced?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET](https://www.vedantu.com/neet?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Olympiad Preparation](https://www.vedantu.com/olympiad?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NDA](https://www.vedantu.com/exams/nda-exam?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [KVPY](https://www.vedantu.com/exams/kvpy-entrance-exam-details?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NTSE](https://www.vedantu.com/exams/ntse-exam?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) CBSE - [CBSE](https://www.vedantu.com/cbse?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Syllabus](https://www.vedantu.com/syllabus/cbse-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Sample Paper](https://www.vedantu.com/sample-papers/cbse-sample-papers?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Worksheets](https://www.vedantu.com/cbse/cbse-ncert-worksheets?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Important Questions](https://www.vedantu.com/cbse/important-questions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Previous Year Question Papers Class 12](https://www.vedantu.com/previous-year-question-paper/cbse-previous-year-question-papers-class-12?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Previous Year Question Papers Class 10](https://www.vedantu.com/previous-year-question-paper/cbse-previous-year-question-papers-class-10?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Important Formulas](https://www.vedantu.com/formula?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) ICSE - [ICSE](https://www.vedantu.com/icse?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [ICSE Solutions](https://www.vedantu.com/icse/icse-solutions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [ICSE Class 10 Solutions](https://www.vedantu.com/icse/class-10-solutions-concise-selina-publishers?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [ICSE Class 9 Solutions](https://www.vedantu.com/icse/class-9-solutions-concise-selina-publishers?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [ICSE Class 8 Solutions](https://www.vedantu.com/icse/class-8-solutions-concise-selina-publishers?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) State boards - [State Boards](https://www.vedantu.com/state-boards?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [AP Board](https://www.vedantu.com/ap-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Bihar Board](https://www.vedantu.com/bihar-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Gujarat Board](https://www.vedantu.com/gujarat-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Karnataka Board](https://www.vedantu.com/karnataka-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Kerala Board](https://www.vedantu.com/kerala-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Maharashtra Board](https://www.vedantu.com/course/pro?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [MP Board](https://www.vedantu.com/mp-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Rajasthan Board](https://www.vedantu.com/rajasthan-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Telangana Board](https://www.vedantu.com/telangana-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [TN Board](https://www.vedantu.com/tn-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [UP Board](https://www.vedantu.com/up-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [WB Board](https://www.vedantu.com/west-bengal-board?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) Free Study Material - [Free Study Material](https://www.vedantu.com/study-material?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Previous Year Question Papers](https://www.vedantu.com/previous-year-question-paper?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Sample Papers](https://www.vedantu.com/sample-papers?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Study Materials](https://www.vedantu.com/study-material/jee-main-study-material?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Study Materials](https://www.vedantu.com/study-material/jee-advanced-study-material?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Study Materials](https://www.vedantu.com/study-material/neet-study-material?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Olympiad Study Materials](https://www.vedantu.com/study-material/olympiad-study-material?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Kids Learning](https://www.vedantu.com/kids-learning?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Ask Questions](https://www.vedantu.com/question-answer?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) Important Subjects - [Physics](https://www.vedantu.com/physics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Biology](https://www.vedantu.com/biology?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Chemistry](https://www.vedantu.com/chemistry?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Maths](https://www.vedantu.com/maths?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [English](https://www.vedantu.com/english?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Commerce](https://www.vedantu.com/commerce?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Geography](https://www.vedantu.com/geography?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Civics](https://www.vedantu.com/civics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) Revision Notes - [Revision Notes](https://www.vedantu.com/revision-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Class 12 Notes](https://www.vedantu.com/revision-notes/cbse-class-12-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Class 11 Notes](https://www.vedantu.com/revision-notes/cbse-class-11-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Class 10 Notes](https://www.vedantu.com/revision-notes/cbse-class-10-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Class 9 Notes](https://www.vedantu.com/revision-notes/cbse-class-9-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [CBSE Class 8 Notes](https://www.vedantu.com/revision-notes/cbse-class-8-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) JEE MAIN - [JEE Main](https://www.vedantu.com/jee-main?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Important Questions](https://www.vedantu.com/jee-main/important-questions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Important Chapters](https://www.vedantu.com/jee-main/topics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Notes](https://www.vedantu.com/jee-main/revision-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Formulas](https://www.vedantu.com/jee-main/formulas?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Difference between](https://www.vedantu.com/jee-main/difference-between?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Syllabus](https://www.vedantu.com/jee-main/syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Physics Syllabus](https://www.vedantu.com/jee-main/physics-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Mathematics Syllabus](https://www.vedantu.com/jee-main/maths-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Chemistry Syllabus](https://www.vedantu.com/jee-main/chemistry-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Main Previous Year Question Paper](https://www.vedantu.com/jee-main/previous-year-question-paper?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) JEE ADVANCED - [JEE Advanced](https://www.vedantu.com/jee-advanced?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Important Questions](https://www.vedantu.com/jee-advanced/important-questions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Important Chapters](https://www.vedantu.com/jee-advanced?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Notes](https://www.vedantu.com/jee-advanced/revision-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Formulas](https://www.vedantu.com/jee-advanced/formulas?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Difference between](https://www.vedantu.com/jee-advanced/difference-between?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Syllabus](https://www.vedantu.com/jee-advanced/syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Physics Syllabus](https://www.vedantu.com/jee-advanced/physics-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Mathematics Syllabus](https://www.vedantu.com/jee-advanced/maths-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Chemistry Syllabus](https://www.vedantu.com/jee-advanced/chemistry-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [JEE Advanced Previous Year Question Paper](https://www.vedantu.com/jee-advanced/previous-year-question-paper?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) NEET - [NEET](https://www.vedantu.com/neet?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Important Questions](https://www.vedantu.com/neet/neet-important-questions?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Important Chapters](https://www.vedantu.com/neet/neet-topics?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Notes](https://www.vedantu.com/neet/neet-revision-notes?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Diagrams](https://www.vedantu.com/neet/neet-diagrams?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Difference between](https://www.vedantu.com/neet/neet-difference-between?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Syllabus](https://www.vedantu.com/neet/neet-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Physics Syllabus](https://www.vedantu.com/neet/neet-physics-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Chemistry Syllabus](https://www.vedantu.com/neet/neet-chemistry-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Biology Syllabus](https://www.vedantu.com/neet/neet-biology-syllabus?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [NEET Previous Year Question Paper](https://www.vedantu.com/neet/neet-previous-year-question-paper?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) OFFLINE CENTRES - [Muzaffarpur](https://www.vedantu.com/offline-centres/muzaffarpur?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Chennai](https://www.vedantu.com/offline-centres/chennai?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Bangalore](https://www.vedantu.com/offline-centres/bangalore?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Patiala](https://www.vedantu.com/offline-centres/patiala?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Delhi](https://www.vedantu.com/offline-centres/delhi?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Hyderabad](https://www.vedantu.com/offline-centres/hyderabad?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Vijayawada](https://www.vedantu.com/offline-centres/vijayawada?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Visakhapatnam](https://www.vedantu.com/offline-centres/visakhapatnam?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Tiruchirapalli](https://www.vedantu.com/offline-centres/tiruchirapalli?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Madurai](https://www.vedantu.com/offline-centres/madurai?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Coimbatore](https://www.vedantu.com/offline-centres/coimbatore?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Pune](https://www.vedantu.com/offline-centres/pune?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Nagpur](https://www.vedantu.com/offline-centres/nagpur?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Guntur](https://www.vedantu.com/offline-centres/guntur?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Aurangabad](https://www.vedantu.com/offline-centres/aurangabad?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Raipur](https://www.vedantu.com/offline-centres/raipur?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Bilaspur](https://www.vedantu.com/offline-centres/bilaspur?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Srinagar](https://www.vedantu.com/offline-centres/srinagar?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Jammu](https://www.vedantu.com/offline-centres/jammu?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Bhubaneswar](https://www.vedantu.com/offline-centres/bhubaneswar?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) - [Jhunjhunu](https://www.vedantu.com/offline-centres/jhunjhunu?utm_source=SEO&utm_medium=Footer&utm_campaign=SEO&utm_content=Study_Materials) © 2026.Vedantu.com. All rights reserved [Privacy policy](https://www.vedantu.com/privacy-policy)[Terms and conditions](https://www.vedantu.com/tnc)

Exponential Smoothing Equations Exponential smoothing was initially introduced in the statistical literature without considering the past work done by Robert Goodell Brown in 1956 and then further expanded by Charles C. Holt in 1957. Exponential smoothing is a reliable principle for smoothing time series data through the exponential window function. The controlling input of the exponential smoothing calculation is stated as the smoothing factor or the smoothing constant. Forecast of the weighted averages of past observations are introduced using exponential smoothing methods, with the weights breaking down exponentially as the observations get formed. In other words, the more the latest the observation the higher the corresponding weight. As we are aware of the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It can be easily applied for making determinations on the basis of prior assumptions by the user, such as seasonality. Exponential smoothing is primarily used for time-series data analysis. Exponential Smoothing Formula The exponential smoothing formula is derived by: st = θxt+(1 – θ)st-1= st-1+ θ(xt – st-1) Here, st is a former smoothed statistic, it is the simple weighted average of present observation xt st-1 is former smoothed statistic θ is smoothing factor of data; 0 \< θ \< 1 t is time period If the value of the smoothing factor is greater, then the level of smoothing will be minimized. Value of α nearer to 1 minimum smoothing effect and offer higher weights to recent changes in the data, while the value of θ nearer to zero has higher smoothing effect and is less responsive to recent changes. There is no precise method of choosing θ. Accurate factors are selected on the basis of the statistician's judgments or else a statistical technique may be used to optimize the value of θ. For example, the method of least squares can be used to estimate the value of θ for which the sum of the quantities is diminished. Exponential Smoothing Methods The three exponential smoothing methods to determine exponential smoothing. They are: 1. Simple or single exponential smoothing 2. Double exponential smoothing 3. Triple exponential smoothing Single Exponential Smoothing If the data which is obtained has no trend and no seasonal pattern, then the single exponential smoothing method for forecasting the time series is primarily used. This method makes use of weighted moving averages with exponentially decreasing weights. The single exponential smoothing method formula is given by: st = θxt+(1 – θ)st-1 = st-1 + θ(xt – st-1) Double Exponential Smoothing The double exponential smoothing method is also known as Holt's trend corrected or second-order exponential smoothing. This method is primarily used to forecast the time series when the data has a linear trend and no seasonal pattern. The motive of double exponential smoothing is to introduce a term considering the possibility of a series indicating some form of trend. This slope component is itself reformed through exponential smoothing. The double exponential smoothing formula is derived by: S1 = y1 B1 = y1-y0 For t\>1, st = θyt + (1 – θ)(st-1 + bt-1) βt = β(st – st-1) + (1 – β)bt-1 Here, St is smoothed statistic, it is the simple weighted average of present observation yt st-1 = former smoothed statistic θ = smoothing factor of data; 0 \< θ \< 1 t = time period bt = accurate estimation of trend at time t β = trend smoothing factor; 0 \< β \<1 Triple Exponential Smoothing In the triple exponential smoothing method, exponential smoothing is used thrice. This method is primarily used to forecast the time series when the data has both linear trend and seasonal patterns.This method is also known as holt-Winters exponential smoothing. The triple exponential smoothing formula is derived by: s 0 \= x 0 s t \= α x t c t − L \+ (1 - α)(s t − 1 \+ b t − 1 ) b t \= β(s t \- s t − 1 \+ (1 - β)b t − 1 c t \= γ x t s t \+ (1 - γ)c t − L Here, st = smoothed statistic, it is the simple weighted average of present observation xt st-1 = previous smoothed statistic α = smoothing factor of data; 0 \< α \< 1 t = time period bt = accurate estimation of trend at time t β = trend smoothing factor; 0 \< β \<1 ct = sequence of seasonal error-free factors at time t γ = seasonal variation smoothing factor; 0 \< γ \< 1 Solved Examples 1\. The Sales of Books in a Bookstall for the Last 10 Months is Given Below in Tabulated Form. Calculate the Simple Exponential Smoothing Estimating α =0.3 for the Below Data. | | | |---|---| | Month | No. of Books Sold | | January | 30 | | February | 25 | | March | 35 | | April | 25 | | May | 20 | | June | 30 | | July | 35 | | August | 40 | | September | 30 | | October | 35 | ## Solution | | | | |---|---|---| | Month | No. of Books Sold | Exponential Smoothing(α =0.3) | | January | 30 | 30\.00 | | February | 25 | 30\.00 | | March | 35 | 28\.50 | | April | 25 | 30\.45 | | May | 20 | 14\.1 | | June | 30 | 15\.87 | | July | 35 | 20\.109 | | August | 40 | 24\.5763 | | September | 30 | 29\.20341 | | October | 45 | 29\.442387 | | November | \- | 34\.1096709 | Quiz Time 1\. The Use of Smoothing Technique is Accurate When 1. The primary source of variation is random behavior 2. Seasonality is includes 3. Data exhibit a strong trend 4. All the above are accurate 2\. Times Series Data is Classified in 1. Two components 2. Three components 3. Four components 4. Five components