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When you have two or more numbers it is nice to find a value for the "center".

2 Numbers

With just 2 numbers the answer is easy: go half-way between.

Example: what is the central value for 3 and 7?

Answer: Half-way between, which is 5.

You can calculate it by adding 3 and 7 and then dividing the result by 2:

3 or More Numbers

We can use that idea of "adding then dividing" when we have 3 or more numbers:

Example: what is the central value of 3, 7 and 8?

Answer: You calculate it by adding 3, 7 and 8 and then dividing the results by 3 (because there are 3 numbers):

Notice that we divide by 3 because we have 3 numbers ... very important!

The Mean

So far we have been calculating the Mean (or the Average):

Mean: Add up the numbers and divide by how many numbers.

But sometimes the Mean can let you down:

Example: Birthday Activities

Uncle Bob wants to know the average age at the party, to choose an activity.

There will be 6 kids aged 13, and also 5 babies aged 1.

Add up all the ages, and divide by 11 (because there are 11 numbers):

(13+13+13+13+13+13+1+1+1+1+1) / 11 = 7.5...

The Mean was accurate, but in this case it was not useful.

The Median

But you could also use the Median: simply list all numbers in order and choose the middle one:

Example: Birthday Activities (continued)

List the ages in order:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

Choose the middle number:

The Median age is 13 ... so let's have a Disco!

Sometimes there are two middle numbers. Just average those two:

Example: What is the Median of 3, 4, 7, 9, 12, 15

There are two numbers in the middle:

So we average them:

The Median is 8

The Mode

The Mode is the value that occurs most often:

Example: Birthday Activities (continued)

Group the numbers so we can count them:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

"13" occurs 6 times, "1" occurs only 5 times, so the mode is 13.
How to remember? Think "mode is most"
But Mode can be tricky, there can sometimes be more than one Mode.

Example: What is the Mode of 3, 4, 4, 5, 6, 6, 7

Well ... 4 occurs twice but 6 also occurs twice.

So both 4 and 6 are modes.

When there are two modes it is called "bimodal", when there are three or more modes we call it "multimodal".

Outliers

Outliersare values that " lie out side" the other values.

They can change the mean a lot, so we can either not use them (and say so) or use the median or mode instead.

Example: 3, 4, 4, 5 and 104

Mean: Add them up, and divide by 5 (as there are 5 numbers):

24 does not represent those numbers well at all!

Without the 104 the mean is:

But please tell people you are not including the outlier.

Median: They are in order, so just choose the middle number, which is 4:

Mode: 4 occurs most often, so the Mode is 4

Harmonic Mean

The Harmonic Mean is good at handling large outliers:

Example: 2, 4, 6 and 100

But small outliers will make things worse!

Conclusion

There are other ways of measuring central values, but Mean, Median and Mode are the most common.

Use the one that best suits your data. Or better still, use all three!

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When you have two or more numbers it is nice to find a value for the "center". With just 2 numbers the answer is easy: go half-way between. Answer: Half-way between, which is 5.

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When two sets of data are strongly linked together we say they have a High Correlation.

The word Correlation is made of Co- (meaning "together"), and Relation

Correlation is Positive when the values increase together, and

Correlation is Negative when one value decreases as the other increases

A correlation is assumed to be linear (following a line).

Correlation can have a value:

1 is a perfect positive correlation

0 is no correlation (the values don't seem linked at all)

-1 is a perfect negative correlation

The value shows how good the correlation is (not how steep the line is), and if it is positive or negative.

Example: Ice Cream Sales

The local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for the last 12 days:

And here is the same data as a Scatter Plot:

We can easily see that warmer weather and higher sales go together. The relationship is good but not perfect.

In fact the correlation is 0.9575 ... see at the end how I calculated it.

Also try the Correlation Calculator.

Correlation Is Not Good at Curves

The correlation calculation only works properly for straight line relationships.

Our Ice Cream Example: there has been a heat wave!

It gets so hot that people aren't going near the shop, and sales start dropping.

Here is the latest graph:

The correlation value is now 0: "No Correlation" ... !

The calculated correlation value is 0 (I worked it out), which means "no correlation".

But we can see the data follows a nice curve that reaches a peak around 25° C.

But the correlation calculation is not "smart" enough to see this.

Moral of the story: make a Scatter Plot, and look at it!
You may see a relationship that the calculation does not.

"Correlation Is Not Causation"

A common saying is "Correlation Is Not Causation".

What it really means is that a correlation does not prove one thing causes the other:

One thing might cause the other

The other might cause the first to happen

They may be linked by a different thing

Or it could be random chance!

There can be many reasons the data has a good correlation.

Example: Sunglasses vs Ice Cream

Our Ice Cream shop finds how many sunglasses were sold by a big store for each day and compares them to their ice cream sales:

The correlation between Sunglasses and Ice Cream sales is high

Does this mean that sunglasses make people want ice cream?

Example: Poor suburbs are more likely to have high pollution.

Why?

Do poor people make pollution?

Are polluted suburbs the only place poor people can afford?

Is it a common link, such as factories with low paying jobs and lots of pollution?

Example: A Real Case!

A few years ago a survey of employees found a strong positive correlation between "Studying an external course" and Sick Days.

Does this mean:

Studying makes them sick?

Sick people study a lot?

Or did they lie about being sick so they can study more?

Without further research we can't be sure why.

How To Calculate

How did I calculate the value 0.9575 at the top?

I used "Pearson's Correlation". There is software that can calculate it, such as the CORREL() function in Excel or LibreOffice Calc ...

... but here is how to calculate it yourself:

Let us call the two sets of data "x" and "y" (in our case Temperature is x and Ice Cream Sales is y):

Step 1: Find the mean of x, and the mean of y

Step 2: Subtract the mean of x from every x value (call them "a"), and subtract the mean of y from every y value (call them "b")

Step 5: Divide the sum of ab by the square root of [(sum of a2) × (sum of b2)]

Here is how I calculated the first Ice Cream example (values rounded to 1 or 0 decimal places):

As a formula it is:

Where:

Σ is Sigma, the symbol for "sum up"

is each x-value minus the mean of x (called "a" above)

is each y-value minus the mean of y (called "b" above)

You probably won't have to calculate it like that, but at least you know it is not "magic", but simply a routine set of calculations.

Note for Programmers

You can calculate it in one pass through the data. Just sum up x, y, , and xy (no need for a or b calculations above) then use the formula:

Other Methods

There are other ways to calculate a correlation coefficient, such as "Spearman's rank correlation coefficient".

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When two sets of data are strongly linked together we say they have a High Correlation. The word Correlation is made of Co- (meaning "together"), and Relation Correlation is Positive when the values increase together, and Correlation is Negative when one value decreases as the other increases A correlation is assumed to be linear (following a line).

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Convert Quadratics from Standard form to Vertex Form

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4.03 Other Forms of a Quadratic Function

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4.03 Other Forms of a Quadratic Function

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Discriminant for types of solutions for a quadratic | Algebra II | Khan Academy

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Unit 3 Review- Choose Your Own Adventure

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Solving quadratic equations by factoring

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Algebra Basics: What Are Polynomials? - Math Antics

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Solving a System of Equations Using Elimination and Multipliers

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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