Simple Regression and Correlation


In this blog post, you’ll learn how to implement or perform a simple linear regression and correlation coefficient.


Simple Linear Regression is a statistical method that is used to estimate or predict the relationship between two variables:

  • Dependent variable (criterion variable) is the main factor you are trying to understand and predict.
  • Independent variables (explanatory variables, or predictors) are the factors that might influence the dependent variable.


1. Linearity: The relationship between X and the mean of Y is linear.

2. Homoscedasticity: The variance of residual is the same for any value of X.

3. Independence: Observations are independent of each other.

4. Normality: For any fixed value of X, Y is normally distributed.


Correlation Coefficient is a statistical method that is used to test and measures the statistical relationship, or association, between two continuous variables.  


1. Two variables should be measured on a continuous scale.

2. Two continuous variables should be paired.

3. There should be independence of cases.

4. There should be a linear relationship between your two continuous variables. 


A scatter plot is a special type of graph used to visually inspect the data to see whether X and Y are linearly related. It is designed to show the relationship between two variables in a simple regression analysis.

1. Linear – a relationship is linear when the points on a scatterplot follow a somewhat straight line pattern.

2. Non-linear – a relationship is non-linear when the points on a scatterplot follow a pattern but not a straight line.

3. No Correlation – a relationship has no correlation when the points on a scatterplot do not show any pattern.

A correlation exists between two variables when one of them is related to the other in some way. A scatterplot can identify several different types of relationships between two variables

1. Positive Correlation – as the x variable increases, so does the y variable.

2. Negative Correlation – as the x variable increase, the y variable decreases. 

3. No Correlation – there is no evident relationship between the two variables.  


To perform the Simple Linear Regression in Excel, you need a free Microsoft add-in called the Analysis ToolPak. (Load the Data Analysis ToolPak in Excel).

Example: Car Plant Electricity Usage

The manager of a car plant wishes to investigate how the plant’s electricity usage depends upon the plant’s production. Calculate the relationship between the variable.

The scatterplot for Car Plant Electronic Usage shows a positive linear relationship, with no extreme outliers or potentially influential observations.

Step 1: On the Data tab, in the Analysis group, click the Data Analysis button. Select Regression and click OK.

Step 2: In the Regression dialog box, configure the following settings:

  • Select your Input Y Range:  which is your dependent variable, You can do this in two ways: either select the data in the worksheet or type the location of your data into the “Input Y Range box.
  • Select your Input X Range:  which is your independent variable, select the data in the worksheet or type the location of your data into the “Input X Range box.”
  • Check the Labels box if there are headers at the top of your x and y ranges.
  • Select the location where you want your Output range

Step 3: Click OK and observe the regression analysis output created by Excel.

Regression analysis output: Regression Statistics

1. Multiple R –  is the absolute value of the correlation coefficient of the two variables (X and Y) being evaluated. The larger the absolute value, the stronger the relationship. The Multiple-R value represents the simple correlation and is 0.896, which indicates a high degree of correlation.

2. R-Square –  It is the Coefficient of Determination, it indicates how well the model or regression line “fits” the data. The higher the R-squared, the better the model fits your data. In this case, 80.21% is calculated, which is very large.

3. Adjusted R Square – Adjusted R-Squared is best for analyzing multiple regression output.

4. Standard Error – The standard error in the regression output is an important number to understand when interpreting regression data. It is the average distance that the observed values fall from the regression line. Approximately 95% of the observations should fall within +/- two standard error of the regression which is also a quick approximation of a 95% prediction interval.

5. Observations – It is simply the number of observations in your model.

Regression analysis output: ANOVA

1. df – stands for degrees of freedom.

2. SS – is the sum of the squares. The smaller the Residual SS compared with the Total SS, the better your model fits the data.

3. MS –  or the Mean Square.

4. F – is the F statistic or F-test for the null hypothesis. It is used to test whether any of the independent variables in a multiple linear regression model are significant.

5. Significance F – gives an idea of how reliable your results are. If Significance F is less than 0.05, your model is better. If it is greater than 0.05, you’d probably better choose another independent variable.

This table indicates that the regression model predicts the dependent variable significantly well. Since the value of the Significance F (0.0008) is less than 0.05, the model is better and indicates that, overall, the regression model statistically significantly predicts the outcome variable.

Regression analysis output: Coefficient

This section provides specific information about the components of your analysis

Testing the Hypothesis of No Linear Relationship

1. Null Hypothesis: H0 : β = 0 (There is no linear relationship)

2. Alternative Hypothesis:  Ha : β   0 (There is a linear relationship)

The p-value tests the null hypothesis that the coefficient is equal to zero (no effect). A p-value less than the significance level of 0.05 indicates that you can reject the null hypothesis; otherwise, the null hypothesis is not rejected. In our case, the calculated p-value is 0.000082, which indicates the null hypothesis is rejected and concludes that there is a significant linear relationship between x and y. In general,  this data provides very strong evidence to conclude that the distribution of electricity usage depends upon the level of production.

Regression analysis output: Residuals

If you want to add Residuals, click the Residual checkbox in the Regression dialog box. The residuals show you how far away the actual data points from the predicted data points.


To perform Correlation, you need a free Microsoft add-in called the Analysis ToolPak. (Load the Data Analysis ToolPak in Excel).


The table below shows the height (x) in inches, and the pulse rate (y) per minute, for 9 people. Find the correlation coefficient and interpret your result.

The scatterplot shows no correlation, means that there is no evident relationship between the two variables.  

Step 1: On the Data tab, in the Analysis group, click the Data Analysis button. Select Correlation and click OK.

Step 2: In the Correlation dialog box, specify your Input Range and Output Range.

Step 3: Click OK and observe the Correlation analysis output created by Excel.

Interpreting Correlation Analysis Results

A correlation of 1 shows a perfect positive correlation, a correlation of -1 shows a perfect negative correlation, and a correlation of 0 shows no relationship between the movement of the two variables. The correlation results (-0.15) appear to be extremely weak, which indicates that there is no relationship between the variables.

Related Link:

Excel 2016 Regression Analysis

Excel 2016 Correlation Analysis

89 thoughts on “Simple Regression and Correlation

  1. The next time I read a blog, Hopefully it does not fail me just as much as this particular one. I mean, Yes, it was my choice to read through, but I truly believed youd have something helpful to talk about. All I hear is a bunch of moaning about something you could possibly fix if you werent too busy seeking attention.

  2. I don’t need to tell you how important it is to optimize every step in your SEO pipeline. But unfortunately, it’s nearly impossible to cut out time or money when it comes to getting good content. At least that’s what I thought until I came across Article Forge… Built by a team of AI researchers from MIT, Carnegie Mellon, Harvard, Article Forge is an artificial intelligence (AI) powered content writer that uses deep learning models to write entire articles about any topic in less than 60 seconds. Their team trained AI models on millions of articles to teach Article Forge how to draw connections between topics so that each article it writes is relevant, interesting and useful. All their hard work means you just enter a few keywords and Article Forge will write a complete article from scratch making sure every thought flows naturally into the next, resulting in readable, high quality, and unique content. Put simply, this is a secret weapon for anyone who needs content. I get how impossible that sounds so you need to see how Article Forge writes a complete article in less than 60 seconds! order here.

  3. To kasyno ma kilka rzeczy które musi dopracować, jest to jednak świetny nowy dodatek do świata kasyn online, z wieloma rzeczami, które można pokochać! Jeśli szukasz szybkiej akcji i szybkich wypłat, kasyno Nitro powinno być Twoim następnym celem! Życzymy miłej zabawy oraz wysokich wygranych! Nie zapominajmy, że kasyna internetowe tak samo jak firmy bukmacherskie oferują nam w sieci bardzo atrakcyjne bonusy powitalne. Często z takich bonusów powitalnych gracz może zgarnąć nawet kilka tysięcy złotych. Jeżeli dojdzie do wykrycia, że posiadamy drugie konto w systemie i kasyno nam to udowodni, to musimy liczyć się z tym, iż środki zebrane na bonusowym koncie lub wszystkie pieniądze znajdujące się na naszym koncie zostaną skonfiskowane przez kasyno. Niestety takie są skutki łamania regulaminu. Dlatego na wszelki wypadek unikajmy dublowania kont, gdyż może się to dla nas skończyć zabraniem wygranych oraz banem w systemie na oba konta. Brak produktów 1. Poczta odbiór w punkcie – 3,9 zł2. Poczta Pocztex 48 – 8,9zł3. Kurier DHL,InPost – 12,9 zł4. Paczkomat InPost – 9,9 zł5. Powyżej 199zł – Dostawa GRATIS6. Płatność przy odbiorze +4,99 zł Okoliczności śmierci kwestionuje również rodzina Matthew. – Wiemy, że to nie była rosyjska ruletka – powiedziała mediom matka Minklera. – Mój syn został postrzelony w podbródek, jakby go chcieli uciszyć – dodała kobieta. Ruletka rosyjska lub husarska to ekstremalna gra, w której jeden z uczestników może zginąć od śmiertelnego strzału. Uważa się, że rosyjska ruletka była szeroko rozpowszechniona w Rosji, ale praktycznie nie ma żadnych udokumentowanych dowodów na to, jak ludzie grali w tę ekstremalną grę. Chciałbym stanąć w obronie tej kontrowersyjnej tematyki, jaką porusza ta gra. Dla mnie jest to temat neutralny, bo mi nie przeszkadza w zabawie to, że rzekomo promuje przemoc i samobójstwa grając w ten tytuł. To jest istne czepianie się dupereli. Równie dobrze, gry ze smokami i My Little Pony też powinny być zakazane, bo promują ideologię, którymi diabeł się interesuję, cytując pewną gwiazdę internetu. OOO, Gdyby była “Rosyjska ruletka: My Little Pony”, to by było dobre i z pewnością ładniejsze.

  4. Traditional subway tile has left the station, sconce lighting will be hot and earthy colors are in! As we ride the roller coaster of changes that continue in the wake of the COVID-19 pandemic, homeowners are embracing fresh home design ideas while holding on to a couple classic concepts. Read more … Welcome to Ivory Homes, Utah’ s Number One HomebuilderВ® for 34 years. Our architects and designers bring decades of experience to every home plan we offer. As Pro Builder Magazine’s National Builder of the Year, you’ll know you’re in good hands with a locally owned and operated company. We understand Utah’s rich landscape and climate—bringing you timeless homes that can be enjoyed for generations. We care about your home today and its value tomorrow! I appreciated the owner, Vernon. From the top down everyone genuinely cared about the process. They had a personal touch. They made the experience unique to me. Find Owner Financed Properties for sale at See more information about Owner financed oceanfront homes for sale in Ormond Beach Florida. Start a new search to find more Owner Financed Properties or other commercial listings on You may enter a US or Canadian location in the following formats: Rare find in growing Titusville! Currently zoned for multi family as well as residential with possible commercial frontage on Knox McRae. Small canal runs along side of p… Looking for Sachse Owner Financed Properties? If you are new to investing in commercial real estate, these guides can help provide some insight into investing in some of the other commercial opportunities available today. With over 7 million registered members, LoopNet is the leading marketplace for commercial real estate listings, news, loans and resources, attracting a large community of Tarrant County commercial real estate professionals many of whom specialize in Owner Financed Properties. The LoopNet user base is comprised of commercial real estate brokers, corporate executives, mortgage brokers, and over 3 million buyers, tenants and other principals from Tarrant County and other national and international markets.

  5. Das Online Casino beinhaltet ein umfangreiches Programm und bietet für jeden Spieler das passende Angebot an Casino Spielen. Die riesige Spielauswahl beinhaltet nebst unzähligen Slots, von renommierten Anbietern wie Red Tiger, Greentube by Novomatic und Gamanza, auch das klassische Grand Jeu. So finden Sie auf neben zahlreichen Slot Machines unter anderem Blackjack, Roulette, Baccarat und Poker. Ein weiteres Highlight in jedem Online Casino sind die beliebten Spiele des Live Casinos. Casino Jackpots Die verwendeten Daten stammen von Casino Listings. Den Major Millions Jackpot Slot von Microgaming gibt es in zwei verschiedenen Varianten, wobei die theoretische Chance auf den Gewinn des progressiven Jackpots in beiden Versionen identisch ist. Bei der ersten handelt es sich um einen klassischen einarmigen Banditen, bei der zweiten um einen herkömmlichen Video Slot auf fünf Walzen und 15 festgelegten Gewinnlinien. Um den progressiven Jackpot… So wie man Merkur Automatenspiele im Echtgeld zocken kann, lassen sich andererseits Spielautomaten Merkur Online kostenlos spielen und ohne Anmeldung im Merkur Casino ausprobieren. Wenn Sie Merkur Spiele kostenlos ohne Anmeldung spielen möchten, benötigen Sie keinen Merkur Casino Bonus ohne Einzahlung, denn es braucht nur der No Deposit Bonus eingesetzt werden. „Der Perseverance-Rover sammelt Proben aus dem Jezero-Krater, einem alten Seebecken mit einem Versorgungskanalsystem, das in die noachische (>3,7 Ga*) Stratigraphie eingraviert wurde. Unterschiedliche Arten von sedimentären, magmatischen, wasserverwitterten und durch Einschläge gebildeten Gesteinen, die in dieser Region zugänglich sind, werden eine geologische Aufzeichnung einer besonders wichtigen Zeitspanne für das Verständnis der Umweltentwicklung des Mars und möglicherweise seiner Biologie liefern.

  6. Uită de site-urile vechi! Este timpul să explorezi întregul univers al jocurilor ca la aparate, începând cu cele clasice și foarte populare, pe care le știe toată lumea și le vedem peste tot, inclusiv în sălile de păcănele, până la păcănele 3D sau Full HD din 2018. Nu conteaza dacă folosești PC, laptop, sau Windows, Android ori iOS, jocurile noastre merg perfect pe oricare dintre ele. Așa că ia-ți tableta sau telefonul mobil și ai acces nelimitat la toate jocurile noastre oriunde, oricând! După cum poți vedea și la păcănele online Wild Water gratis, jocul are câteva simboluri și funcții speciale despre care vom discuta acum: După doar câteva rotiri cu Queen of Hearts Deluxe gratis, vei vedea că există multe motive pentru care merită să-l încerci. Tematica pare desprinsă din basme, grafica este plăcută, cu animații, iar deasupra rolelor sunt câțiva îngeri, inclusiv Cupidon. Desigur, simbolurile jocului creat de compania Novomatic se încadrează perfect în poveste: castele, trandafiri, inimi, inele, prințese și simboluri clasice dar al căror font face parte din același tărâm magic. Telefon: 0731984272 Avantajul platilor online este mai presus de orice comoditate. In ritmul rapid de viata de azi, nu este necesar sa vizitati banca voastra. Prin intermediul serviciilor bancare digitale putem economisi timp pretios, banca este la dispozitia noastra fara o zi libera – 7 zile pe saptamana, 24 de ore pe zi; locatia noastra este irelevanta – putem obtine informatiile de care avem nevoie sau putem efectua plata necesara din cealalta parte a lumii. Eficienta nu este lipsita de importanta: ne putem gestiona toate conturile dintr-un singur loc, iar comisioanele pentru tranzactiile online sunt mai mici – adica economisim bani si timp. Trebuie să luați în considerare că nu toate cazinourile Paysafecard au adoptat aceleași limite pe care le are cardul preplătit. Operatorul poate stabili sume mai mici decât cele listate pe site-ul oficial al procesatorului de plăți. Depozitele sunt instantanee, așa cum am spus, așa că veți avea banii dvs. minute după confirmarea tranzacției. Mai jos puteți găsi informații detaliate despre fiecare dintre site-urile noastre de încredere în cazinourile Paysafecard online.

  7. Good post. I learn something totally new and challenging on blogs I stumbleupon every day. It will always be interesting to read articles from other writers and practice a little something from other sites.

  8. Otathan huomioon, että kaikki ajat ovat EST (Nykyinen aika) Las Vegasissa järjestettävä World Series of Poker pelataan vuosittain toukokuu-heinäkuu välisenä aikana 60 turnauksen voimin. Finaali on itseoikeutetusti 10 000 $ päätapahtuma, josta voittaja voi saada mukaansa jopa 10 miljoonaa dollaria. Voit lukea tästä viimeisimmät uutiset isosta turnaus-sarjasta, joka saa joka vuosi todella paljon huomiota osakseen maailmalla. Lue lisää meistä ja pokerista! Nm pokerisivustot saattavat olla enimmkseen pokeriin keskittyvi satoja tapahtumia ja palkitsee voittajat 18 WSOP Circuitin sormuksilla tmn Pokeriuutiset jossakin elvss elmss pelattavassa. Posted in nettipokeri, pokeri, pokeriuutiset Ota selvää, kuinka pelata pokeria! Pokerlistings Finland -sivusto on yksi suurimmista suomalaisista pokerioppaista netissä. Tarkoituksenamme on auttaa aloittelijoita ja kehittyneempiä pokerin pelaajia tarjoamalla heille yksilöllistä tietoa eri pokerivariaatioista sekä pokerihuoneista. Slot Hunter casino sisältää loistavia pelejä markkinoiden parhailta tekijöiltä. Vaikka kasinon nimen perusteella voisi kuvitella, että täältä ei löydy mitään muuta kuin kolikkopelejä, eli slotteja, ei hätää. Slot Hunter kasino sisältää toki paljon muutakin: voit pelata täällä mm. jackpotteja, korttipelejä, live kasinon pelejä sekä perinteisiä, automaattia vastaan pelattavia pöytäpelejä. Pelitarjontaan tutustuminen onnistuu parhaiten kasinobonusten siivellä, sillä tällöin pelejä pääsee testaamaan ilman liiallista oman rahan riskeeraamista. Ajoittain tarjolla on jopa no deposit bonus, jonka avulla kasinoa pääsee testaamaan täysin ilmaiseksi. Tämä talletusvapaa bonus voi olla joko ilmaista pelirahaa, jonka voi käyttää vapaasti kasinon peleissä, tai ilmaiskierroksia, jotka myönnetään netin suosituimpiin kolikkopeleihin. 

  9. よく見かけるコンクリートブロックですが、これには「〇キロ以上の重さには耐えられませんよ」という荷重制限があります。 これはコンクリートブロック1つあたりの耐荷重量ですので、並べて使用したりすれば耐荷重量は変わります。 これらをホームセンターで買ったらそこそこのお値段になるはずだが同社では無料で譲っている。 当店では、発泡面木・発泡目地棒(共に2Mもの)を在庫しています。 基本的なサイズとして、 面長15、20、25、30の面木と、 目地棒としては、 下が20で、高さが12で、上が14の、KC-15A(両テーパー両転び)や、 下が20で、高さが ホームセンターのエクステリアコーナー、インテリアコーナーとしても注目してみるのもよいかと思います。インダストリアルな雰囲気の部屋と相性良さそうだと思いますよ。 自分の手札2枚とコミュニティーカード(全てのプレイヤーが使用するカード)5枚、合計7枚の中から5枚を使用してポーカーの役を作り、一番強い役を作った人がチップを獲得します。 オンラインポーカーツールにつきましては、無料トライアル期間があるため、無料で使ってみて良さそうなら有料契約するのでも良いでしょう。 12月前半イベントスケジュール FAQ ポーカー クラブの定員は 501 人です。定員を引き上げたい場合はヘルプ センターからご連絡ください。開設できるクラブ数の上限は 2、加入できるクラブ数の上限は 10 です。ロビーの タブの で加入しているクラブを確認できます。

  10. Don’t believe the guys and gals who always bet too much then rant online. Bet within your means, play legit online roulette games and remember that the casino always has a slight advantage. With these practices, you’ll have an enjoyable and honest online gambling experience. Playing free online roulette is almost exactly the same as playing for real money, with the only difference being you don’t deposit any money. Check out our simple steps to start playing roulette for free today. Of course, when you play online roulette in Canada, you’ll have many other wagers of which to learn. However, these are some of the most important types you need to know about before you try roulette online. Although some online casinos focus mainly on slots and offer a limited selection of casino table games, others boast an extensive array of table games that include a good choice of online roulette variants. When we review an online roulette site, we look at the game library, the number of roulette game developers and the quality of games. Let’s just be honest. It may be easier to just list the games that Drake Casino does NOT have. Outside of slots, which of course is a great selection when playing at Drake Casino, they also provide Video Poker, Blackjack, Baccarat, Keno, Roulette, Craps, Scratch Cards and much more. It is one of the Drake Casino promotions or offers that give users the chance to begin from the bottom. With no Drake Casino coupon code required for activation, players get to climb up the ladder whereby they enjoy the perks such as 100% Drake Casino free bonus (reload promotion) of as much as $500 which translates into more playing chances and hence, more winning opportunities. At Drake Casino, the promotion terms are specified in a very simple way. Here are the most important things you need to keep in mind:

Leave a Reply

Your email address will not be published. Required fields are marked *

New Report