Algebra: Variance and Standard Deviation

Greetings math friends! In this post we’re going to go over variance and standard deviation. We will take this step by step to and explain the significance each have when it comes to a set of data. Get your calculators ready because this step by step although not hard, will take some serious number crunching! Check out the video below to see how to check your work using a calculator and happy calculating! 🙂

What is the Variance?

The variance represents the spread of data or distance each data point is from the mean.  When we have multiple observations in our data, we want to know how far each unit of data is from the mean.  Are all the data points close together or spread far apart?  This is what the variance tells us!

Don’t freak out but here’s the formula for variance, notated as sigma squared:

This translates to:

Let’s try an example:

What is Standard Deviation?

Standard deviation is a unit of measurement that is unique to each data set and is used to measure the spread of data. The formula for standard deviation happens to be very similar to the variance formula!

Below is the formula for standard deviation, notated as sigma:

Since this is the same exact formula as variance with a square root, all we need to do is take the square root of the variance to find standard deviation:

Now try calculating these statistics on your own with the following practice problems!

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Algebra: Combining Like Terms and the Distributive Property

Greetings math peeps! In today’s post we are going to review some of the basics: combining like terms and the distributive property. It’s so important to master the basics such as these, so you’re prepared and ready to handle the harder stuff that’s just around the corner, trust me they’re coming! And for those who already feel comfortable with these topics, great! Skip ahead and try the practice questions at the bottom of this post and happy calculating! 🙂

When do we combine “like terms?”

Combining like terms allows us to simplify and calculate our answer with terms that have the same variable and same exponent values only. For example, we can combine the following expression:

How do we combine like terms?

We add or subtract the whole number coefficients and keep the variable they have in common.

Why? We could not add these two terms together because their variables do not match! 2 is multiplied by x, while 3 is multiplied by the variable xy.

Why? We could not add these two terms together because their variables and exponents do not match! 2 is multiplied by x, while 3 is multiplied by the variable x^2 . Exponents for each variable must match to be considered like terms.

Distributive Property:

Combining like terms and the distributive property go hand in hand.  The distributive property rule states the following:

There are no like terms to combine in the example above, but let’s see what it would like to use the distributive property and combine like terms at the same time with the following examples:

Example #1:

Example #2:

In some cases, we also have to distribute is the -1 that can sometimes “hide” behind a parenthesis.

Try the following questions on your own on combining like terms and the distributive property and check out the video above for more!

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Algebra: Box Plots, Interquartile Range and Outliers, Explained!

Ahoy math friends! This post takes a look at one method of analyzing data; the box plot method. This method is great for visually identifying outliers and the overall spread of numbers in a data set.

Box plots look something like this:

Screen Shot 2020-09-02 at 11.19.22 AM.png

Why Box Plots?

Box Plots are a great way to visually see the distribution of a set of data.  For example, if we wanted to visualize the wide range of temperatures found in a day in NYC, we would get all of our data (temperatures for the day), and once a box plot was made, we could easily identify the highest and lowest temperatures in relation to its median (median: aka middle number).

From looking at a Box Plot we can also quickly find the Interquartile Range and upper and lower Outliers. Don’t worry,  we’ll go over each of these later, but first, let’s construct our Box Plot!

Screen Shot 2020-09-02 at 11.20.42 AM->  First, we want to put all of our temperatures in order from smallest to largest.
Screen Shot 2020-09-02 at 11.21.28 AM.png-> Now we can find Quartile 1 (Q1), Quartile 2 (Q2) (which is also the median), and Quartile 3 (Q3).  We do this by splitting the data into sections and finding the middle value of each section.Screen Shot 2020-09-05 at 11.19.22 PM

Q1=Median of first half of data

Q2=Median of entire data set

Q3=Median of second half of data

-> Now that we have all of our quartiles, we can make our Box Plot! Something we also have to take notice of, is the minimum and maximum values of our data, which are 65 and 92 respectively. Let’s lay out all of our data below and then build our box plot:

Screen Shot 2020-09-05 at 11.19.27 PM

Screen Shot 2020-09-05 at 11.20.45 PM

Now that we have our Box Plot, we can easily find the Interquartile Range and upper/lower Outliers.

Screen Shot 2020-09-05 at 11.21.54 PM

->The Interquartile Range is the difference between Q3 and Q1. Since we know both of these values, this should be easy!

Screen Shot 2020-09-05 at 11.22.02 PMNext, we calculate the upper/lower Outliers.

Screen Shot 2020-09-05 at 11.23.45 PM

-> The Upper/Lower Outliers are extreme data points that can skew the data affecting the distribution and our impression of the numbers. To see if there are any outliers in our data we use the following formulas for extreme data points below and above the central data points.

Screen Shot 2020-09-05 at 11.24.27 PM*These numbers tell us if there are any data points below 44.75 or above 114.75, these temperatures would be considered outliers, ultimately skewing our data. For example, if we had a temperature of  Screen Shot 2020-09-05 at 11.26.38 PMor Screen Shot 2020-09-05 at 11.29.25 PM these would both be considered outliers.

Screen Shot 2020-09-05 at 11.24.35 PM

Practice Questions:

Screen Shot 2020-09-05 at 11.34.21 PMSolutions:

Screen Shot 2020-09-05 at 11.37.06 PM

Screen Shot 2020-09-05 at 11.37.39 PM

Screen Shot 2020-09-05 at 11.38.10 PM

Screen Shot 2020-09-05 at 11.39.06 PM

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Algebra: How to use Recursive Formulas?

Welcome to Mathsux! This post is going to show you everything you need to know about Recursive Formulas by looking at three different examples. Check out the video below for more of an explanation and test your skills with the practice questions at the bottom of this page.  Happy calculating! 🙂

What is a Recursive Formula?

A Recursive Formula is a type of formula that forms a sequence based on the previous term value.  What does that mean?  Check out the example below for a clearer picture:

Example #1:

Screen Shot 2020-08-11 at 8.12.21 AM.pngScreen Shot 2020-08-11 at 8.12.33 AMScreen Shot 2020-08-11 at 9.18.18 AM

Screen Shot 2020-08-11 at 8.13.07 AMScreen Shot 2020-08-11 at 8.13.36 AM.pngScreen Shot 2020-08-11 at 9.21.01 AM.pngScreen Shot 2020-08-11 at 8.14.34 AM.pngScreen Shot 2020-08-11 at 8.14.49 AMExample #2:

Screen Shot 2020-08-11 at 8.15.19 AM.png

Screen Shot 2020-08-11 at 8.15.38 AMScreen Shot 2020-08-11 at 9.22.10 AMScreen Shot 2020-08-11 at 8.52.36 AMScreen Shot 2020-08-11 at 8.52.52 AM.pngScreen Shot 2020-08-11 at 9.23.54 AM.png

***Note this was written in a different notation but is solved in the exact same way!

Screen Shot 2020-08-11 at 8.53.24 AM.pngScreen Shot 2020-08-11 at 8.53.35 AM

Example #3:Screen Shot 2020-08-11 at 8.54.05 AM.pngScreen Shot 2020-08-11 at 8.54.19 AMScreen Shot 2020-08-11 at 9.24.42 AMScreen Shot 2020-08-11 at 8.54.49 AMScreen Shot 2020-08-11 at 9.25.41 AM.pngScreen Shot 2020-08-11 at 8.56.22 AMScreen Shot 2020-08-11 at 8.56.36 AM.pngPractice Questions:

Screen Shot 2020-08-11 at 10.04.21 AM

Solutions:

Screen Shot 2020-08-11 at 9.02.18 AM.png

Still got questions? No problem! Check out the video above for more or try the NYS Regents question below, and please don’t hesitate to comment with any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on Recursive Formulas?  Click here!

Algebra: Piecewise Function Review

Greetings, today’s post is for those in need of a piecewise function review!  This will cover how to graph each part of that oh so intimidating piecewise function.  There’s x’s, there are commas, there are inequalities, oh my! We’ll figure out what’s going on here and graph each part of the piecewise-function one step at a time.  Then check yourself with the practice questions at the end of this post. Happy calculating! 🙂

 

Wait, what are Piece-Wise Functions? Exactly what they sound like! A function that has multiple pieces or parts of a function.  Notice our function below has different pieces/parts to it.  There are different lines within, each with their own domain.Screen Shot 2020-07-21 at 10.01.59 AM

Now let’s look again at how to solve our example, solving step by step:

Screen Shot 2020-07-21 at 10.02.29 AM.pngScreen Shot 2020-07-21 at 10.02.41 AMScreen Shot 2020-07-21 at 10.03.06 AM.png

Translation: We are going to graph the line f(x)=x+1 for the domain where x > 0

To make sure all our x-values are greater than or equal to zero, we create a table plugging in x-values greater than or equal to zero into the first part of our function, x+1.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.04.33 AM

Screen Shot 2020-07-21 at 10.05.00 AM.png

 

Screen Shot 2020-07-21 at 10.06.46 AM

Translation: We are going to graph the line  f(x)=x-3 for the domain where x < 0

To make sure all our x-values are less than zero, let’s create a table plugging in negative x-values values leading up to zero into the second part of our function, x-3.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.07.33 AM.png

Screen Shot 2020-07-21 at 10.07.57 AM

Ready to try the practice problems below on your own!?

Practice Questions: Graph each piecewise function:

Screen Shot 2020-07-21 at 10.08.32 AM.png

 

 

 

 

 

 

 

 

Solutions:

Screen Shot 2020-07-21 at 10.09.20 AM

Screen Shot 2020-07-21 at 10.09.58 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on piecewise functions?  Click here.

 

 

Algebra: Absolute Value Equations

Happy Wednesday math friends! Today, we’re going to go over how to solve absolute value equations.  Solving for absolute value equations supplies us with the magic of two potential answers since absolute value is measured by the distance from zero.  And if this sounds confusing, fear not, because everything is explained below!

Also, if you have any questions about anything here, don’t hesitate to comment. Happy calculating! 🙂

Absolute Value measures the “absolute value” or absolute distance from zero.  For example, the absolute value of 4 is 4 and the absolute value of -4 is also 4.  Take a look at the number line below for a clearer picture:

Screen Shot 2020-07-08 at 2.02.40 PM.png

Now let’s see how we can apply our knowledge of absolute value equations when there is a missing variable!Screen Shot 2020-07-08 at 2.03.07 PMScreen Shot 2020-07-08 at 2.03.46 PM.pngScreen Shot 2020-07-08 at 2.04.00 PMScreen Shot 2020-07-08 at 2.04.26 PM.pngScreen Shot 2020-07-08 at 2.04.56 PM

Screen Shot 2020-07-08 at 2.05.17 PM.png

Screen Shot 2020-07-08 at 2.05.39 PMNow let’s look at a slightly different example:

Screen Shot 2020-07-11 at 4.49.57 PM.pngScreen Shot 2020-07-08 at 2.07.59 PM

Screen Shot 2020-07-08 at 2.07.41 PM.png

Screen Shot 2020-07-08 at 2.08.26 PM.png

Screen Shot 2020-07-08 at 2.08.46 PM

Screen Shot 2020-07-08 at 2.09.33 PMScreen Shot 2020-07-08 at 2.09.58 PM.png Screen Shot 2020-07-08 at 2.10.39 PM.pngScreen Shot 2020-07-08 at 2.10.50 PM

Practice Questions: Given the following right triangles, find the missing lengths and side angles rounding to the nearest whole number.

Screen Shot 2020-07-16 at 9.01.08 AM.png

 

 

 

 

 

Solutions:

Screen Shot 2020-07-08 at 2.12.04 PM

Still got questions?  No problem! Check out the video the same examples outlined above. Happy calculating! 🙂

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Algebra: How to Graph y=mx+b

Hi everyone, welcome back to Mathsux! This week we’ll be reviewing how to graph an equation of a line in y=mx+b form. And if you have not checked out the video below, please do! Happy calculating! 🙂

 

Graphing an Equation of Line: An equation of a line can be represented by the formula:Screen Shot 2020-06-17 at 9.07.16 PM

Y-Intercept: This is represented by b, the stand-alone number in y=mx+b. This represents where the line hits the y-axis.  This is always the first point you want to start with when graphing at coordinate point (0,b).

Slope: This is represented by m, the number next to x in y=mx+b. Slope tells us how much we go up or down the y-axis and left or right on the x- axis in fraction form:

Screen Shot 2020-06-17 at 9.09.42 PM

Now let’s check out an Example!

Graph the equation of a line Screen Shot 2020-06-17 at 9.10.42 PM.

Screen Shot 2020-06-17 at 9.12.01 PM

Screen Shot 2020-06-17 at 9.12.35 PM

Screen Shot 2020-06-17 at 9.13.34 PM

Screen Shot 2020-06-17 at 9.14.20 PMTry the following practice questions on your own!

Practice Questions:

Screen Shot 2020-06-17 at 9.15.22 PM

Screen Shot 2020-06-17 at 9.16.21 PM

Want more Mathsux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

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Algebra: 4 Ways to Factor Quadratic Equations

*If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-06-02 at 3.03.55 PMScreen Shot 2020-06-02 at 3.04.24 PM____________________________________________________________________Screen Shot 2020-06-02 at 3.20.04 PM____________________________________________________________________

Screen Shot 2020-06-02 at 3.07.02 PM____________________________________________________________________
Screen Shot 2020-06-02 at 3.07.42 PM
Choose the factoring method that works best for you and try the practice problems on your own below!

Practice Questions:

Screen Shot 2020-06-02 at 3.09.58 PMSolutions:

Screen Shot 2020-06-02 at 3.10.30 PM

Want a review of all the different factoring methods out there?  Check out the ones left out here (DOTS and GCF) and happy calculating! 🙂

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Algebra: Completing the Square

Learn how to Complete the Square by clicking on the Youtube video and trying the practice problems below. Happy Calculating! 🙂

Click the picture below to view the Youtube video.

Complete the Square copy

Screen Shot 2020-05-23 at 5.28.18 PMPractice Questions:

Screen Shot 2020-05-23 at 5.28.54 PM

Solutions:

Screen Shot 2020-05-23 at 5.29.19 PM

Need more of an explanation?  Check out why we complete the square in the first place here and please don’t forget to subscribe! 🙂

Algebra: Rate of Change

Screen Shot 2018-12-28 at 8.21.26 AMScreen Shot 2018-12-28 at 8.22.23 AMScreen Shot 2018-12-28 at 8.23.21 AMScreen Shot 2018-12-28 at 8.24.00 AMScreen Shot 2018-12-28 at 8.24.30 AM.pngScreen Shot 2018-12-28 at 8.25.03 AM.png

Extra Tip! Notice that we added the labels feet/second to our answer.  Why does this make sense?? The question tells us that P(t) represents feet and that t is equal to seconds.  Another way to look at this question when applying it to the slope formula is to realize that we are finding the change of feet divided by the change of seconds.                                                          ____________________________________________________________________________________

Still got questions?  Let me know in the comments and as always happy calculating!:)

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