Normal Distribution

In a normal distribution, data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.

Data can be “distributed” (spread out) in different ways.

It can be spread out more on the left		Or more on the right

Or it can be all jumbled up

But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a “Normal Distribution” like this:

A Normal Distribution

The “Bell Curve” is a Normal Distribution.
And the yellow histogram shows some data that
follows it closely, but not perfectly (which is usual).

It is often called a “Bell Curve” because it looks like a bell.

Many things closely follow a Normal Distribution:

• heights of people
• size of things produced by machines
• errors in measurements
• blood pressure
• marks on a test

We say the data is “normally distributed”:

The Normal Distribution has:

Quincunx

You can see a normal distribution being created by random chance!It is called the Quincunx and it is an amazing machine.Have a play with it!

Standard Deviations

The Standard Deviation is a measure of how spread out numbers are (read that page for details on how to calculate it).

When we calculate the standard deviation we find that generally:

68% of values are within 1 standard deviation of the mean95% of values are within 2 standard deviations of the mean 99.7% of values are within 3 standard deviations of the mean

It is good to know the standard deviation, because we can say that any value is:

• likely to be within 1 standard deviation (68 out of 100 should be)
• very likely to be within 2 standard deviations (95 out of 100 should be)
• almost certainly within 3 standard deviations (997 out of 1000 should be)

Standard Scores

So to convert a value to a Standard Score (“z-score”):

• first subtract the mean,
• then divide by the Standard Deviation

And doing that is called “Standardizing”:

We can take any Normal Distribution and convert it to The Standard Normal Distribution.

The z-score formula that we have been using is:

• z is the “z-score” (Standard Score)
• x is the value to be standardized
• μ (‘mu”) is the mean
• σ (“sigma”) is the standard deviation

And this is how to use it:

Why Standardize … ?

It can help us make decisions about our data.

It also makes life easier because we only need one table (the Standard Normal Distribution Table), rather than doing calculations individually for each value of mean and standard deviation.

In More Detail

Here is the Standard Normal Distribution with percentages for every half of a standard deviation, and cumulative percentages:

A Practical Example: Your company packages sugar in 1 kg bags.

When you weigh a sample of bags you get these results:

• 1007g, 1032g, 1002g, 983g, 1004g, … (a hundred measurements)
• Mean = 1010g
• Standard Deviation = 20g

Some values are less than 1000g … can you fix that?

The normal distribution of your measurements looks like this:

31% of the bags are less than 1000g,
which is cheating the customer!

It is a random thing, so we can’t stop bags having less than 1000g, but we can try to reduce it a lot.

Let’s adjust the machine so that 1000g is:

So let us adjust the machine to have 1000g at −2.5 standard deviations from the mean.

Now, we can adjust it to:

• increase the amount of sugar in each bag (which changes the mean), or
• make it more accurate (which reduces the standard deviation)

Let us try both.

Adjust the mean amount in each bag

The standard deviation is 20g, and we need 2.5 of them:

2.5 × 20g = 50g

So the machine should average 1050g, like this:

Adjust the accuracy of the machine

Or we can keep the same mean (of 1010g), but then we need 2.5 standard deviations to be equal to 10g:

10g / 2.5 = 4g

So the standard deviation should be 4g, like this:

(We hope the machine is that accurate!)

Or perhaps we could have some combination of better accuracy and slightly larger average size, I will leave that up to you!

Video Tutorial