Passy's World of Mathematics

Distance Between Two Points

Posted on September 18, 2013 by Passy

Image Source: http://northrofneutral.wordpress.com

GPS Navigation Units are continually calculating the Distance Between Two Points.

Eg. The Distance from the Car to the next Turn.

This is worked out by using the current Coordinates of the Car, and the known GPS grid map coordinates of the next Intersection where we need to turn.

In this lesson, we show how to work out the Distance between Two Points on the Cartesian Grid, using Pythagoras Theorem.

If you are interested in knowing about How Sat Nav Locates Positions, then watch the following five minute video.

Distance Between Points Using Pythagoras

Let’s work through an example to show how this is done.

We can measure the “Across” distance and the “Up” distance by counting squares.

This creates a right triangle on our grid, as shown below.

We can apply “Pythagoras Theorem” to the right angled triangle to work out what the distance AB is between the two points.

If you do not know about Pythagoras Theorem for Right Triangles, then see our previous lesson about this at the following link:

http://passyworldofmathematics.com/pythagoras-and-right-triangles/

Distance Between Points Formula

Rather than have to draw Pythagoras Trianges onto our graph of the two points, we can develop an Algebra Formula to save us a lot of time and work.

We can subtract coordinate values to work out the “Across” and the “Up”, instead of drawing a graph and counting squares.

We now substitute our subtracted coordinates expressions: XB – XA and YB – YA into the Pythagoras Theorem, and thereby create an Algebra Formula for the Distance between any two points labelled “A” and “B”.

We will do some examples on how to find Distance using the subtracting coordinates Formula later in this lesson, but first we recommend watching the following videos to consolidate the material we have covered so far in this lesson.

Distance Using Pythagoras Video

The following video shows how to find the distance between points using the Pythagoras Theorem.

Pythagoras Theorem is a great way of understanding how the Distance Between Points works; however we believe that people should move on from this to using the Formula Method.

For the Pythagoras method, we need to spend a lot of time manually drawing our points onto a Cartesian grid and forming Right Triangles around them.

However, if we identify the x1,y1 and x2,y2 coordinate values of our points and use these in the Distance Formula, we do not have to draw any graphs at all.

Once mastered, the Distance Formula saves a lot of time, and can even be programmed into computerised apps.

Distance Formula Video

The following video from Spiro at Vivid Maths shows how to do a typical Distance Formula question.

Distance Formula Steps

When asked to find the Distance between two points, label these two points “A” and “B”.

It does not matter which way around they are labelled, as the formula will give the same answer for both scenarios.

It is not necessary to graph the two points on a Cartesian Grid, as the formula only requires the number values of their (x,y) coordinates.

After labelling the two points, follow these working out steps:

Distance Formula – Example One

Here is an example with the working out steps marked in.

See if you can fill in the blanks and get to the correct answer.

(The fully worked answer is further down the page).

Here is the fully worked solution for Example 1:

Note that we have labelled the Coordinates (x1,y1) rather than (xA,yA), as “A” is usually called “Point 1”, and “B” is known as “Point 2”.

Distance Formula – Example Two

Here is an example with the working out steps marked in.

See if you can fill in the blanks and get to the correct answer.

(The fully worked answer is further down the page).

Here is the fully worked solution for Example 2:

Note that we have labelled the Coordinates (x2,y2) rather than (xB,yB), as “B” is usually called “Point 2”, and “A” is known as “Point 1”.

Remember that it does not matter which way around you label the two points “A” and “B”, as the answer should be the same in both cases.

Note that it is not necessary to draw the points on an X-Y Grid, if the Point Coordinates are given in the question:

eg. If you are given (3,2) and (6,8), then you can simply just work off the x,y coordinate values of these two points.

Eg. It is not necessary to plot these two points onto an X-Y Cartesian Grid.

Distance Formula Calculators

In the Real World, people do not calculate Distance manually like we have done, they use a Calculator App to do it.

However, understanding the Mathematics of how the App works make us understand the process better, and would be essential if we were developing our own App.

The following page has a great Distance Between Points Calculator, which shows full working out steps and answers:

http://ncalculators.com/geometry/length-between-two-points-calculator.htm

Here is another easy to use Distance Calculator that is on the Web.
You can use this on screen calculator below, right here on this lesson page:

If the on screen calculator does not work then use:

The following web page for this easy to use Distance Calculator:

http://easycalculation.com/analytical/distance.php

Blank X-Y Grid

Here is a blank X-Y Grid you can print out and use for plotting question Points, and working out distances using Pythagoras, or the Distance Formula.

Distance Between Points Worksheets

The following Online PDF Worksheet has a variety of questions with answers provided at the end of the sheet.

Click here for Worksheet 1

Here is another worksheet on Distance Between Points which also has answers at the end of it.

This worksheet is totally over the top and has 900 questions on it!

Just pick about 10 questions to do from it that cover a variety of positive and negative coordinates.

Click here for Worksheet 2

Distance Formula Online Test / Worksheet

Finally if you need any extra practice then try out the online test on the following web page:

http://www.onlinemathlearning.com/distance-formula-worksheet.html

Calculating GPS Distance

Image Source: http://www.ga.gov.au

Because the Earth is not flat, and is a curved surface, calculating the distance between two GPS Coordinates uses more complex mathematical formulas involving spherical trigonometry.

Spherical Trigonometry is an important part of mathematics which people who sail ships at sea, and aircraft pilots need to know. These calculations are done by computer apps in the real world.

However Pilots and Mariners still need to know the maths involved with these calculations, in case the computer or GPS ever breaks down!

The following web page shows some of these formulas, and also supplies the javascript code for programming a calculator for them into a computer.

http://www.movable-type.co.uk/scripts/latlong.html

If you enjoyed this lesson, why not get a free subscription to our website.
You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

If you are a subscriber to Passy’s World of Mathematics, and would like to receive a free PowerPoint version of this lesson, that is 100% free to you as a Subscriber, then email us at the following address:

Please state in your email that you wish to obtain the free subscriber copy of the “Distance Between Points” Powerpoint.

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

Like Us on Facebook

Our Facebook page has many additional items which are not posted to this website.

These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.

Check it out at the following link:

https://www.facebook.com/PassysWorldOfMathematics

While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

PayPal does accept Credit Cards, but you will have to supply an email address and password so that PayPal can create a PayPal account for you to process the transaction through. There will be no processing fee charged to you by this action, as PayPal deducts a fee from your donation before it reaches Passy’s World.

Enjoy,
Passy

Posted in Straight Line Graphs | Tagged cartesian distance, cartesian grid, Cartesian Plane, coordinate geometry, crow flies distance, distance between 2 points, distance between points calculation, distance between points calculator, distance between points formula, distance between two points, distance between two points formula, distance calculator, distance formula, how to caculate distance, points distance formula, xy points distance | 6 Comments

Linear Relationship Rules

Posted on September 17, 2013 by Passy

Image Modified from www.chiropractic.com

Linear Rules or Functions are mathematical algebra equations which tell us how to get the output Y-values for a given set of input X-values.

The Rule tells us the “Relationship” between all of the x and y values.

If we graph any of these input and output (x,y) values, a straight line will be created.

We need to be able to find the Rule for Sets of (x,y) pairs, and for any straight line graphs we have drawn, so that:

1) We can find the “Y” values for very large x-values, without having to draw a giant graph and manually read them off

2) We can find the “Y” values for very small x-values, like x = 0.001, which would be way too small to try and read off a normal sized graph

3) We can type the rule into a Computer Graphing Application, and have the computer draw the graph for us

In relation to point 3) above, nobody draws graphs in the real world any more, they are all drawn by computer apps.

If you would like to try out an Online Graph Maker, then try the ones out at these two links:

http://webgraphing.com/plotting_basic.jsp#

http://rechneronline.de/function-graphs/

Real World Straight Line Graphs

If you would like to see where straight lines are used in the real World, then have a look at our lessons about this at the links below:

http://passyworldofmathematics.com/real-world-line-graphs/

and

http://passyworldofmathematics.com/straight-line-graphs/

Types of Linear Relationship Rules

There are basically 3 types of Linear Rules:

1) Simple Addition or Subtraction

Eg. y = x +2, y = x -5, y = x + 7, y = x -3 etc

2) Simple Multiplication or Division

Eg. y = 2x, y = -5x, y = x/2, y = -x/5 etc

3) Combination Rule using y = mx + c

Eg. y = 2x + 3, y = -3x + 1, y = 4x -2, y = -x + 7 etc

When we are given a Table of Input and Output x and y values, we check for:

1) Simple Addition or Subtraction

and if this does not work out then check for

2) Simple Multiplication or Division

and if this does not work out then check for

3) Combination Rule using y = mx + c

Image Source: http://media2.newsobserver.com

Think of this as being like a Doctor in a Hospital Emergency Ward checking an incoming patient’s injuries.

They might check for :

1) Do they just have some cuts and bleeding

2) Do they just have broken bones

3) Do they have a combination of injuries

Just like the Doctor case, if we have situation 3), involving a combination of issues, then this situation will require the most work.

Simple Addition Rule

In this situation the difference between the y and x values will be the same number every time as we work down our set of (x,y) values.

If we cannot identify this relationship by simply looking at the numbers, then we can mathematically check for it.

We check by adding an extra “y-x” column to our given table of input “x” and output “y” values, and then doing the math.

In every case the y-x difference is always the same number value
of positive “1”.

The Linear Rule is y = x + 1

Check:

y = x + 1 (sub in our x,y is 0,1 value)

1 = 0 + 1

1 = 1

Our y = x + 1 rule works, and so our y = x + 1 answer is correct!

Zero Addition Rule

If the X-Y difference check produces zeroes, then we have the y=x rule.

This relationship should be obvious by just looking at the x and y values and seeing that they are the same.

We do not really need to do the Y-X column maths.

Simple Subtraction Rule

This is the exact same approach as the Simple Addition Rule, but our y-x values all come out the same NEGATIVE NUMBER each time.

In every case the y-x difference is always the same number value of NEGATIVE 2 OR “-2”.

The Linear Rule is y = x + -2

which we write as y = x – 2

Check:

y = x – 2 (sub in our x,y is 0,-2 value)

-2 = 0 – 2

-2 = -2

Our y = x – 2 rule works, and so our y = x – 2 answer is correct!

If we do not find a simple Addition or Subtraction Rule that works for our (x,y) points, then we need to move on to checking for a Simple Multiplication Relationship.

Simple Multiplication Rule

If there is a specific number that we can multiply each x value by, and this produces the y-value, then we have a Multiplication Relationship.

In this situation y divided by x for all values will be the same number every time, EXCEPT FOR X = 0

(It is impossible to divide any number by zero, and so we have written “–” in our table).

If we cannot identify this relationship by simply looking at the numbers, then we can mathematically check for it.

We can check for a “Multiplication Relationship” by adding an extra “y/x” column to our given table of input “x” and output “y” values, and then doing the math.

In every case y/x is always the same number value of positive “3”.

The Linear Rule is y = 3x

Check:

y = 3x (sub in our x,y is 1,3 value)

3 = (3)(1)

3 = 3

Our y = 3x rule works, and so our y = 3x answer is correct!

Simple Multiplication – Negative Values

Note that y/x can come out a negative number, such as -2.

If this happens the Linear Rule is y = -2x

Simple Multiplication – y/x value is 1

The y/x value can sometimes turn out as “1”.

If this happens the Linear Rule is y = 1x

which we write as:

y = x

Simple Division Rule

If our y/x values from a Simple Multiplication checking table turn out to be all the same fraction value, then we have a Simple Division Linear Rule.

The same y/x value we get each time is 1/4 .

Our Linear Rule is: y = 1/4 x

We can write this as: y = x/4

Combination Rule: y = mx + c

If we cannot identify a simple Addition, Subtraction, Multiplication, or Division relationship, then there is probably a “Combination Rule” involved.

Eg. If our y-x values, and our y / x values, do NOT give us a common number answer pattern, then we probably have a more complicated “Combination Rule” involved.

Combination Rules are more complicated, and involve both Multiplying and Adding to make a rule in the form:

y = mx + c or y = mx – c

where m = gradient, and c = y-intercept

Note that Americans, and some other countries, call the combination rule

y = mx + b and y = mx – b,

Combination Rule – Working Out Steps

There are four steps, as follows:

Step 1 involves working out the Change in x and y values.

Step 2 involves Calculating the Gradient or Slope as:

m = Change in Y / Change in X

If we have a Graph given to us, instead of an x,y Values Table, then we need to find the gradient “m” directly from the graph.

Now let’s look at Steps 3 and 4, where we find the “C” value for y = mx + c

People have a lot of trouble with this step, because it looks like there is too much working out to do.

Another way to approach Step 3 is to realize that we have half the rule already, because we know m = 2.

This means the Y values are all 2x + ?

If we now go back and visually inspect the X and Y-values Table, knowing that we need to double each “x” value and then also do something else,

we might be able to figure out what that something else is.

Once we figure out this something else is “adding 1” then we know that C = + 1, and the full rule is y = 2x + 1.

Once we have completed Steps 1 to 3, we know the number values for “m” and “c”, so all we need to do is put these values into y = mx + c

As we discussed above, once we know the gradient “m” we can look at the table values and should be able to figure out “c”.

We combine Steps 3 and 4 together into a single step:

Video About Finding y = mx + c Rules

Finding y = mx + c rules is not easy for beginners, and it might take a while until you fully figure out what is going on.

Watching the following video will help you better undertand what needs to be done for these types of questions.

Finding Rules for Line Graphs

When we have a Line Graph, the procedure is the same, except that our (x,y) points are on the line, rather than being in a values table.

The Procedure is:

Find the Gradient “m” between any two points on the line using Rise / Run

(The points are best located where the line crosses the corners of a set of four grid squares, so that we have whole number values for Rise and Run).

We then put our number value for “m” into y = mx + c, and use a set of x,y point values from the line to work out the value of “c”.

Finding Rule for Graphs – Example One

In the example below, we find the Gradient slope, and then from there on we have half the rule done.

All we need to do is work out what else is done to “x” to get the “Y” value, and this will tell us what “c” is.

Gradient “m” is always the multiply part of the rule, and “c” is always the add on or subtract part of the rule.

Finding Rule for Graphs – Example Two

In this next example, we do the same procedure:

First use “m” = Rise / Run to get the first part of the rule.

Then pick any point from the line, multiply the “x” coordinate by the “m” value, and then figure out what we need to add on, (or subtract away), to get the Y-Value.

The above example shows that we can use the Combination Rule y = mx + c for any type of straight line.

The Example 2 line was actually a simple “Addition Rule” relationship.

Even though we went through the more complicated “Combination Rule” steps, we still got to the correct answer.

Blank X-Y Grid

Here is a blank X-Y Grid you can print out and use for drawing question lines onto and working out rules.

Linear Rule Generator

The Maths Warehouse website has a great free java app on their page where we can move around the given straight line, and the appp works out the y = mx + c (as y = mx + b) rule for us.

It can be found on the following web page:

http://www.mathwarehouse.com/algebra/linear_equation/slope-intercept-form.php

If you enjoyed this lesson, why not get a free subscription to our website.
You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

Please state in your email that you wish to obtain the free subscriber copy of the “Linear Relationships Rules” Powerpoint.

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

Like Us on Facebook

Our Facebook page has many additional items which are not posted to this website.

These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.

Check it out at the following link:

https://www.facebook.com/PassysWorldOfMathematics

While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

Enjoy,
Passy

Posted in Pythagoras, Straight Line Graphs | Tagged calculating y-intercept, coordinate geometry, equation of a line, find rule for coordinates, find rule for x, finding gradient, finding graph rule, finding line equation, finding line rule, finding linear rule, finding y-intercept, gradient, gradient intercept form, how to find equation of a line, how to find graph rule, how to find line rule, how to find linear equation, how to find linear rule, how to find rule for a straight line, how to find y=mx+b, how to find y=mx+c, linear relationship, linear relationship rule, linear rule, rule for a straight line, slope, slope intercept form, straight line graph, straight line rule, values tables, xy points relationship, xy points rule, y points, y-intercept | 7 Comments

Back to Back Stem and Leaf Plots

Posted on September 7, 2013 by Passy

Image Source: Modified from http://ex-interim.blogspot.com.au

In this lesson we look at Back to Back Stem and Leaf Plots.

These types of graphs are used for comparing two sets of statistics where the same thing has been measured.

Using a Back to Back Plot we can directly compare the statistics of the two items: eg. Males versus Females, Brand A versus Brand B, Natural versus Synthetic, Team X Performance versus Team Y performance, X-Box versus Playstation, Apple versus Samsung, and so on.

A Back to Back Plot is simply two Stem and Leaf Plots combined together, like the two person seat designed by Eleanor Hutchinson in the opening image for this lesson.

Basic Stem and Leaf Plots

A Stem and Leaf Plot is basically the same as a Horizontal Bar Chart.

Instead of Grouping the Data and drawing Bars, we Group the Data, but we then write the actual number values of the data to create the bars.

It is necessary to understand Basic Stem and Leaf Plots before attempting Back to Back Stem and Leaf Plots.

If you have never made a Stem and leaf Plot before, then do our previous lesson on this at the link below:

http://passyworldofmathematics.com/stem-and-leaf-plots/

Example Back to Back S & L Plot

Two Mobile Phones have been tested for Battery Life in between charges.

We need to create a Back to Back Stem and Leaf Plot to show the
comparison of the battery life results.

The data for the Battery Life of the two phones is as follows:

The first step to creating any Stem and Leaf Plot is to write our values out from lowest to highest, and group them into Tens; eg. ones, tens, twenties, thirties, etc

When we do this “Tens” grouping for our Mobile Phone Data, we obtain the following:

The following diagram shows how we draw the Stem and Leaf Plot for the “Brand A” Phone.

Notice that in the Stem and Leaf Plot we do not show the full values of battery life on the rows.

Eg. In a Stem and Leaf Plot, the data values are put in order from the Lowest to Highest.

We then group them into :

Ones = 0 to 9
Tens = 10 to 19
and so on.

These groups are called the “Stems”.

The “Leaf” is the last digit on each of the original data values.

So we only show the last digit of each battery life measurement.

Here is how we make the Stem and Leaf Plot for the “Brand B” phone, following the exact same method we used for our “Brand A” plot.

Let’s now take a look at our two Stem and Leaf Plots and the final steps we need to take to combine them into a single plot.

We keep the right hand side blue plot as it is, but we flip around the red left hand side plot so it is reversed like in a mirror.

The situation is like taking two people and turning one of them around, so that they are now “back to back” facing in opposite directions.

Here is how we reverse the left hand side “Brand B” Stem and Leaf Plot.

We now have our two Stem and Leaf Plots and we are ready to join them together “Back to Back”.

We move the two plots towards each other until their stems overlap, and thereby create the combined single back to back plot.

The resulting single diagram looks like this:

Usually our Stem and Leaf Plots we do in Math Class are not as brightly colored as this, and a typical maths workbook version of the above diagram would look like this:

Videos About Back to Back Stem and Leaf Plots

This first video covers Back to Back Stem and Leaf Plots in reasonable detail.

This short video shows how to draw a back to back stem and leaf plot from two related sets of data.

Here is a quick one minute video showing the structure of back to back plots.

Stem and Leaf Worksheets

This worksheet has Questions 4, 5, and 6 on Stem and Leaf Plots, and contains answers to these questions.

Click here for Stem and Leaf Worksheet 1

This second worksheet covers back to Back Stem and Leaf Plots, but does not have any answers supplied.

Click here for Stem and Leaf Worksheet 2

If you enjoyed this lesson, why not get a free subscription to our website.
You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

Please state in your email that you wish to obtain the free subscriber copy of the “Back to Back Stem and Leaf Plots” Powerpoint.

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

Like Us on Facebook

Our Facebook page has many additional items which are not posted to this website.

These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.

Check it out at the following link:

https://www.facebook.com/PassysWorldOfMathematics

While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

Enjoy,
Passy

Posted in Graphs, Statistics | Tagged back to back plots, back to back stem and leaf plot video, back to back stem and leaf plot worksheet, back to back stem plots, comparison stem and leaf diagram, comparison stem and leaf graph, comparison stem and leaf plot, comparison stem plots, how to create a back to back stem and leaf plot, how to do a back to back stem and leaf plot, how to make a back to back stem and leaf plot, S&L diagram, S&L plot, statistics graphs, stem and leaf, stem and leaf diagram, stem and leaf graph, stem and leaf plot, stem plots | 2 Comments

Symmetry and Skew

Posted on September 3, 2013 by Passy

This statistics lesson is all about Symmetry Skew and the Kangaroo !

The shape of a Histogram, Bar Chart, or Stem and Leaf plot tells us the type of data distribution we have.

If the tallest area (Mode) is in the middle of the Graph, with even reducing on each side of this, the Graph is called Symmetrical.

If the shape of the graph is not like this, and is bunched up on either the left or right side, we say the data is “Skewed”.

Skewed Data results for situations where there are either a lot of measured items that are very small, or a lot of items that are very large.

Skewed Graphs with their uneven shape affect the Mean Average Value.

This often means that we should use the Median as our average in these situations, and not use the distorted Mean value.

Positive Skew

The following example demonstrates and shows “Positive Skew”, where a Histogram stretches out to the right.

This particular class of students has not performed well on the Algebra Test, and so we have many low scores with tall bars on the left.

We can think of the shape of the Graph as resembling an Australian Kangaroo, and the direction of the Roo’s tail tells us the direction of the skew.

We call this the “Skew and Kangaroo Rule”.

Having an uneven skew shape in the Graph also affects the Mean,Median, and Mode.

The Mode is to the left of the middle Median, because the tall bars are all on the left hand side of the graph.

Due to the long tail of high values stretching to the right, the Mean will be dragged to the right of the Median.

Negative Skew

The following example demonstrates and shows “Negative Skew”, where a Histogram stretches out to the left.

This particular class of students has performed very well on the Algebra Test, and we have many high scores with tall bars on the right.

Placing a Kangaroo onto the graph shows that the tail goes to the left.

This means that the skew is to the Left, which is the negative direction.

Having an uneven skew shape in the Graph also affects the Mean,Median, and Mode.

The Mode is to the right of the middle Median, because the tall bars are all on the right hand side of the graph.

Due to the long tail of low scores stretching to the left, the Mean will be dragged to the left of the Median.

Symmetrical Histogram

If our Histogram is a pyramid type balanced shape, with both sides of the Median pretty much the same shape, then we say it is “Symmetrical”.

The “Kangaroo Rule” fits onto a symmetrical graph as shown below.

This particular maths class is fairly normal, and the sudents have produced both low and high scores.

The Mean, Median, and Mode will all be close together in the middle.

Skewed Histograms Video

The following video uses Microsoft Excel to show examples of Skewed Histograms, and how their shape affects the Mean Average Value.

Stem and Leaf Plot Symmetry and Skew

Measurement of Skew can also be applied to Stem and Leaf Plots.

However, we need to view the S&L Plot sideways to see whether the Skew is to the left or the right.

Summary

The shape of a Histogram, Bar Chart, or Stem and Leaf plot tells us the type of data distribution we have.

There is a great one page summary about Skew on the following web page:

http://www.mathsisfun.com/data/skewness.html

If the tallest area (Mode) is in the middle of the Graph, with even reducing on each side of this, the Graph is called Symmetrical.

If the shape of the graph is not like this, and is bunched up on wither the left or right side, we say the data is “Skewed”.

Skewed Data results for situations where there are either a lot of measured items that are very small, or a lot of items that are very large.

Skewed Graphs with their uneven shape affect the Mean Average Value.

This often means that we should use the Median as our average in these situations, and not use the distorted Mean value.

Kangaroo Statistics

Image Source: http://www.wildlifepark.com.au

The Australian Red Kangaroo is the world’s largest marsupial.

Females have one baby at a time, which at birth is smaller than a cherry.

Males are up to 1.8m (6 ft) tall and weigh up to 84 kg (187lbs).

Kangaroos continue to grow during all their life, which can be up to 20 years.

Giant short faced kangaroos (Procoptodon), extinct 40,000 years ago, were 3 m (10 feet tall) and weighed 200 kg.

Kangaroos live in groups made of 8 to 25 individuals, called mobs. A mob is made up of a male (the leader), several females and the young.

The stomach represents 15 % of the kangaroo’s weight.

Kangaroos can exist without water for up to 2 to 3 months.

Kangaroo’s back feet are made of two joined toes. The toe is extremely long bearing a large nail that makes a kangaroo’s kick extremely dangerous for dogs and even humans.

The tail is up to 1.1 m or 3.6 ft long in large kangaroos, and is used for leaning and for balancing during jumps.

Normal jumps are 1.2-1.9 m (4-6 ft) long, but at high speed they can be 13.5 m (45 ft) long and 3.3 m (11 ft) high.

The kangaroo can speed up to 50 km (30 mi) per hour, but it cannot sustain this speed for long periods.

Kangaroo Meat is one of the highest red meat protein sources and is very lean and healthy to eat.

There are set numbers for how many Kangaroos can be “harvested” yearly so that the natural population of Roos stays in environmental balance and is preserved.

“Harvested” means killed commercially for the meat and fur markets.

There are lots of statistics on Kangaroo Populations at the following Government Website:

Click here for Kangaroo Statistics

For those worried about harvesting making Kangaroos in danger of extinction, statistics show that in 2010 there were around 25 million Kangaroos in Australia. Eg. More Kangaroos than People!

If you enjoyed this lesson, why not get a free subscription to our website.
You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

Please state in your email that you wish to obtain the free subscriber copy of the “Symmetry and Skew” Powerpoint.

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

Like Us on Facebook

Our Facebook page has many additional items which are not posted to this website.

These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.

Check it out at the following link:

https://www.facebook.com/PassysWorldOfMathematics

While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

Enjoy,
Passy

Posted in Graphs, Statistics | Tagged histogram graphs, histogram skew, kangaroo facts, kangaroo mathematics, kangaroo rule, kangaroo rule for skew, kangaroo statistics, Mean, median, mode, skew, skew distribution, skew in graphs, skew kangaroo, skewed graph, statistics graphs, stem and leaf plot, stem and leaf skew, symmetrical distribution, symmetrical Graphs, symmetry, symmetry and skew, what is skew | Leave a comment

Mean Median Mode for Grouped Data

Posted on September 1, 2013 by Passy

Image Source: WordPress.com

A survey was conducted at a Cafe which sells food and coffees.

The reason for the survey was that they were having trouble keeping up with the demand for Cappuccino coffees during peak periods.

The Barista suggested that they get a bigger machine to cope with the high demand.

A bigger machine is very expensive to buy, and so the owner had a two day survey done to find out how many Cappuccinos were being made per hour in the Cafe.

From the survey results, they would be able to do some Graphs and Statistics, and better understand the current problem situation.

Gathering and Analysing Statistical Data is a key part of Business and Marketing, and provides a mathematical picture of current situations and future initiatives.

In this lesson we look at finding the Mean, Median, and Mode Averages for Grouped Data containing Class Intervals.

If you do not have any previous knowledge of Mean, Median, and Mode, then we suggest you do our previous lesson on this at the following link:

http://passyworldofmathematics.com/averages-mean-median-mode/

If you do not have any previous knowledge of Grouped Data, then we suggest you do our previous lesson on this at the following link:

http://passyworldofmathematics.com/grouped-data-histograms/

One final thing to note before starting this lesson is that in Australia, we refer to subgroups of data as “Intervals” or “Class Intervals” or “Classes”; whereas some other countries call these groups “Bins”.

So whenever we say “Interval”, we mean the same thing as a “Bin”.

Grouped Data Histogram

The Histogram shows an even spread of data, indicating that sometimes the Coffee Shop is very busy, while other times they are making less than eight cappuccinos per hour.

We now want to find the Average Number of Cappuccinos made every hour.

There are three types of Averages: the Mean, the Median, and the Mode.

In this lesson we calculate all three of these averages for the coffee shop example.

Finding the Range

The “Range” is the easiest Statistic to determine for Grouped Data.

We simply take the end of the Highest Interval, and subtract the Beginning of the first Interval.

Range = Maximium – Minimum

For our Coffee Statistics, the Highest Group is 16-19, so our High Value “Maximum” is 19.

The Lowest Group is 0-3, so the Low Value “Minimum” is zero.

Range = Maximium – Minimum = 19 – 0 = 19

The Range can also be stated as “0 to 19”

The Modal Class

The “Mode” is what happens most of the time, or on most occassions.

The “Mode” is the simplest Grouped Average to find.

It can be read straight from the Frequency Table, or straight from the Graph.

Sometimes we have more than one Group which is the most popular.

In these situations, we can have a two modes or a “Bimodal” situation, or three modes which is called “Trimodal”.

The Median Class

Finding the Median Class involves some working out steps to be applied to our original Frequency Table.

There are three Main Steps:

1) Finding the half-way midpoint in the Frequency values

2) Adding a third column to our Frequency Table where we calculate “Cumulative Frequency” values

3) Locating the half-way point in the Cumulative Frequency Column, and then seeing which Class Interval lines up with this half-way point.

How we do each of these steps is as follows.

There are two ways to find the half-way midpoint in the Frequency values.

We can either write out the numbers from 1 to the Total frequency value and manually find the middle; or we can use a simple math formula to find this value.

Rather than writing out a long list of numbers, It is much easier to use the formula: Middle = Total Frequency + 1 and then divide by 2.

Once we have the half-way point, the next step is to do “Cumulative Addition” of the Frequencies.

The following real world example shows how we do “Cumulative Addition”.

A Quicker way to get the “Cumulative” values is to use the “Zig Zag Adding Method”.

If you would like to check out some of Glenn’s music, then visit his website at the link below:

http://www.glennbrace.com/

Here is how we apply “Cumulative Addition” to Finding the Median Class.

Here is the complete working out for the Median Class, which turns out to be the Interval group “8-11”.

The MEAN Average for Grouped Data

Finding the MEAN involves more working out than finding the Modal Class and Median Class.

To find the MEAN (or “Mathematically Estimated Average Number”), involves adding two extra columns to our original Frequency Table.

There are three main steps:

1) Find all of our Interval Midpoints, and write these in the third column

2a) Multiply each Frequency x Midpoint value and put the answer each time into the fourth column

2b) Find the Total of the fourth column, eg. Total of Frequency x Midpoints

3) Apply the Formula: Grouped MEAN = ( Total of Frequency x Midpoints ) divided by Total Frequency

The first step is to find all of the Interval Midpoints.

There are two ways of doing step one: we can write each interval out as a list of numbers, or we can use a simple math formula to find the Midpoint.

Use whichever method you find easier to work with, they both give the same result.

Here is how we do the list of numbers method

To use the Midpoint Formula, add together the low and high values and divide each answer by two:

The second step on the way to finding the Grouped MEAN estimated average is to complete the final column of our table.

This involves multiplying each Midpoint by its Interval’s Frequency.

The third and final step is to apply the Grouped Mean Formula:

MEAN Average = Total of (Freq x Midpt) / Total Frequency

Mathematical Notation for MEAN

Our example of the MEAN calculation has working out steps written into the table.

These steps are usually not shown, once people are good at finding the MEAN.

There are also some special mathematical symbols used for the MEAN formula, which are as follows:

Our previous example worked out using correct mathematical notation looks like this:

Coffee Shop Conclusion

The Histogram Graph shows that around one quarter of the time the Cafe is very busy making Cappuccinos, but the other one quarter of the time they are not very busy at all.

The calculated Averages, (Mean Median and Mode), indicate that on average they are making 8 to 11 Cappuccinos per hour, which along with their other coffee offerings should be manageable.

There does not seem to be enough demand overall for Cappuccinos to justify buying a bigger coffee machine at this stage.

Videos about Grouped Mean Median and Mode

This first video is Part 1 of 2, and it shows how to find the median and mode for grouped data.

This next video is Part 2, and shows how to find the MEAN for grouped data.

The first 7 minutes of the following video shows how to calculate the Mean Average for Grouped Data.

This final video shows how to find Mean Median and Mode.

Worked Examples for Grouped Mean Median Mode

The following link is to the BBC Bitesize lesson on Grouped Data Averages.

Click here for BBC Bitesize Lesson and Examples

This next link is to a PDF document containing worked examples.

Click here for PDF of Worked mean median mode Examples

TI-83 and TI-84 Calculator Examples

Image Source: http://education.ti.com

The following link gives step by step instructions on how to use a Texas Intruments Calculator to find the Mean, Median, and Mode, of Grouped Data.

http://mathbits.com/MathBits/TISection/Statistics1/MMMgrouped.htm

If you enjoyed this lesson, why not get a free subscription to our website.
You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

Please state in your email that you wish to obtain the free subscriber copy of the “Grouped Data Mean Median Mode” Powerpoint.

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

Like Us on Facebook

Our Facebook page has many additional items which are not posted to this website.

These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.

Check it out at the following link:

https://www.facebook.com/PassysWorldOfMathematics

While you are there, LIKE the page so you can receieve our FB updates to your Facebook News Feed.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

Enjoy,
Passy

Posted in Graphs, Statistics | Tagged class interval, data midpoints, frequency table, grouped data, grouped data statistics, grouped data videos, Histogram, how to calculate averages, how to calculate medican class, how to calculate modal class, how to do statisitics, interpreting statistics, Mean, mean average from bins, mean formula, Mean Median, Mean Median and Mode for Grouped Data, mean median mode bins, median, median class, modal class, mode, Mode for Grouped Data, statistics bins, statistics formulas | 4 Comments

Passy's World of Mathematics

Distance Between Two Points

Linear Relationship Rules

Back to Back Stem and Leaf Plots

Symmetry and Skew

Translate Website

Popular Lessons

Latest Lessons

Our Facebook Page

Categories

Who has visited Map

Linking to Us

Visitor Statistics

User Feedback

Meta

Donate to Passy’s World

Like us on Facebook