# Statistics Paper: Normal Distrubution

Maths currently is that interesting content wise and we are just about to finish Core 2, the last core module in the course. However; we are doing a lot of revision and past papers so I’ll go through what we need to know as a class and talk you through the content as the year goes on bit by bit.

# The Paper

At the moments as a class, we have Statistics 1, one of the 3 modules we must do in year 12, as we have finished the course. In the paper, we had parts on every module in this course this included:

• Numerical measure: Mean, Interquartile Range and Mode
• Probability: The one I’m not that great at (I Think)
• Binomial Distribution: Mostly probability using a cool bell curve and tree diagrams
• Normal Distribution: Some more bell curves to talk about data
• Confidence Intervals: Bell curves again but talking about the mean more than anything
• Correlation and Regression: Lines of best fit and all that jazz

As you can see there is a lot so I won’t bore you and only go through 2 questions and as you see most of it is on bell curves.

If you don’t know what a bell curve it is relatively simple its and graph which looks like a bell and never really touches the x-axis or the bottom

## Q1

This time the start of the paper was nice it was about the Mean, Interquartile Range and Mode. This was made even easier by the fact that at a level you can put a table into your graphical calculator and it works it out for (like most stuff in stats).  I finished all 6 marks in 3 minutes YEY!!

## Normal Distribution

Now to first work out normal distribution you must first understand what it means so….

Normal Distribution is where the data when placed in a histogram (a type of bar chart). Makes the bell curve shape. This means the data is symmetrical of the mean so 50% of the data right of the mean and 50% left so overall there is 100% (1) underneath this bell curve. Just remember 100% = 1, 10% = 0.1, 1%=0.01 as we mainly use decimals instead of percentages.

With a normal distribution, you can work at how much of the data is below 50 or what is the value which contains 10% of the data. All you need is the mean and standard deviation (a fancy way to work out the distribution of the data.

Now to work information, we use something called a z-score this helps us work out what the probability less than X from a value (if this is confusing don’t worry I’m not the best teacher I have left a link at the bottom). to work out this z-score you can do your value = x take away the mean = μ divided by the standard deviation = σ or:

(x- μ)/ σ = z-score

With this z-score, we go into the table given to us in the exam and is in our book and then will give you your probability. Or if you are lazy like me put it in the calculator.

## Questions

Now to go through some questions like what were in the exam:

So, they tell you a made up story like Peter is a farmer and he measures the sizes the circumference of his watermelons. They had a mean of 50 cm and standard deviation of 1.5.

Then they will give you a few scenarios we are going to work such as:

1. The probability that the is less than 53 P(X<53)
2. The probability that it’s not exactly 48
3. The probability its greater than 54 P(X>51.5)

Now I always draw a curve to start to understand what I’m doing by putting the mean at the center point where 52 and shade in the are less than 52. This gives me an idea of the probability I should get. As the mean is in the shaded are it must be greater than 0.5 as the left side all together 0.5.

Now with a calculator you put in you lowest value which is this time is the most left we can go so -999999999999999 and greatest 52 and gives you your z-score and percentage. Or we can do it manually so we do:

(52-50)/1.5=

3/1.5=2

Then we look on our table which is available on the hyperlink below and find 2 and you get the value of 0.97725 and that question a

### Now 2

This one is very easy not calculator nothing. Because to get precise value 48 exactly is mathematically so so so small we write it as 0. So, to get everything else except is 0 is 1 as the graph contains an area/percentage of 1

### And lastly 3

We do the same calculation and draw the graph and we see that greater 51.5, the mean is not included so that means it’s going to be less than 0.5

(51.5-50)/1.5=

1.5/1.5=1

We look in our tables and 1= 0.53983. But that’s greater that 0.5 your telling that because a positive z-value works from left to right and to work out greater than we find percentage then minus it from 1.

So:

1-0.53983=0.46017

Or use your calculator ha-ha 😃

## Try at home

This is from a real question from an exam paper see if you can work them out then leave a comment down below with your answer. I will give a shout out to who ever gets them right and the mark scheme for guys to see where you went wrong. Good Luck (iii is hard to think about draw a bell curve working the probabilities for both values 😉 )

Wooden lawn edging is supplied in 1.8 m length rolls. The actual length, X meters, of a roll may be modeled by a normal distribution with mean 1.81 and standard deviation 0.08. Determine the probability that a randomly selected roll has a length:

(i) less than 1.90 m;

(ii) greater than 1.85 m;

(iii) between 1.81 m and 1.85 m.

[6 marks]

And that was pretty much all the questions in that part without going into much detail

### Link on maths is fun normal distribution

https://www.mathsisfun.com/data/standard-normal-distribution.html

### Table and how to use it on this hyperlink

https://www.mathsisfun.com/data/standard-normal-distribution-table.html