# Normal Distributions

Powell Family

Created on September 4, 2024

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NormalDistributions

A normal distribution is a distribution of data that is symmetric around its mean value.

If you think about it, the shape of a normal distribution is of a bell curve.

WOW

How do you remember such a boring shape?

Boring

Lets look at some examples

111, 80, 95, 105, 70, 76, 82, 99, 80, 88, 100, 112

Amounts of Hot dogs purchased each day at a Carnival:

Put the data set in numerical order.

I am going to seperate the values into quartiles.(sry I switched the 3rd and 1st quartile around)

The box-and-whisker plot shows that the data is normally distributed. But how do we find Z-scores?

To find a Z-score you will first need to find the standard deviation of the set.

Take all of the raw scores and subtract by the mean.

Then we will square all of the raw scores and add them together.

Next, we divide 2193 by the size of the dataset. This would be 2193/ 12= 182.75

We square root it! √182.75 = 13.513.5 is the standard deviation

After finding the variance

Now that we have the standard deviation I can make a bell curve with it!

Now that we know the standard deviation is 13.5 we can find the z-score!The Z-score formula is this: z = (x-μ)/σ

In this equation z = (x-μ)/σZ is for z-scorex is for a raw scoreμ is the mean and σ is the standard deviation

When calculatingz-scores with real data

Let's plug 80 into the equation as our raw data!z = 80 - 91.5/13.5 = First we would subtract 80-91 and then divide by 13.5=-0.85

We previously saw that one day the hot dog stand only sold 80 hot dogs

In this equation z = (x-μ)/σZ is for z-scorex is for a raw scoreμ is the mean and σ is the standard deviation

(A bit of behind the scenes work) I pulled up the z-score table and found where the row -0.8 and the column 0.05 intersected

The z-score for 80 is 0.1977! This means that 80 is 19.77% away from the mean.

What I got as an answer:

But what if we wanted to see what percent of hot dogs the stand makes that is more than 112 hot dogs?

Let's plug 112 into the equation as our raw data!z = 112 - 91.5/13.5 = =1.51(more behind the scenes work of me finding the answer on the z-score table)

We would do something similar to what we did earlier when we used 80 as a raw score.

In this equation z = (x-μ)/σZ is for z-scorex is for a raw scoreμ is the mean and σ is the standard deviation

The z-score for 112 is .9345! or 93.45%.To find what the precent of days the stand makes more than 112 hot dogs we would subtract 1-.9345

What I got as an answer:

of the time, the hot dog stand makes more than 112 hot dogs!

6.55%

But who would even use Normal distribution and z-scores at work?

Market researchers, teachers, scientists, and engineers use these methods!

What I got as an answer:

Scientists need to understand normal distribution and probability because it will help them to interpret data and make calculations and hypothesis off of it.

Teachers need to be able to understand distributions and probability to see how their students rank for tests and to see if their class is actually learning!

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A great closing